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Generalized uniform spaces, uniformly locally contractive set-valued dynamic systems and fıxed points

Kazimierz Włodarczyk* and Robert Plebaniak

Author Affiliations

Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Ł ódź, Banacha 22, 90-238 Łódź, Poland

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Fixed Point Theory and Applications 2012, 2012:104  doi:10.1186/1687-1812-2012-104

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/104


Received:25 October 2011
Accepted:21 June 2012
Published:21 June 2012

© 2012 Włodarczyk and Plebaniak; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Motivated by classical Banach contraction principle, Nadler investigated set-valued contractions with respect to Hausdorff distances h in complete metric spaces, Covitz and Nadler (Jr.) investigated set-valued maps which are uniformly locally contractive or contractive with respect to generalized Hausdorff distances H in complete generalized metric spaces and Suzuki investigated set-valued maps which are contractive with respect to distances Qp in complete metric spaces with τ-distances p. Here, we provide more general results which, in particular, include the mentioned ones above. The concepts of generalized uniform spaces, generalized pseudodistances in these spaces and new distances induced by these generalized pseudodistances are introduced and a new type of sequential completeness which extended the usual sequential completeness is defined. Also, the new two kinds of set-valued dynamic systems which are uniformly locally contractive or contractive with respect to these new distances are studied and conditions guaranteeing the convergence of dynamic processes and the existence of fixed points of these uniformly locally contractive or contractive set-valued dynamic systems are established. In addition, the concept of the generalized locally convex space as a special case of the generalized uniform space is introduced. Examples illustrating ideas, methods, definitions, and results are constructed, and fundamental differences between our results and the well-known ones are given. The results are new in generalized uniform spaces, uniform spaces, generalized locally convex and locally convex spaces and they are new even in generalized metric spaces and in metric spaces.

MSC: 54C60; 47H10; 54E15; 46A03.

Keywords:
generalized uniform space; generalized pseudodistance; dynamic system; uniformly locally contractivity; contractivity; dynamic process; fixed point; generalized locally convex space; generalized metric space

Introduction

Let 2X denotes the family of all nonempty subsets of a space X. Recall that a set-valued dynamic system is defined as a pair (X, T), where X is a certain space and T is a set-valued map T : X → 2X; in particular, a set-valued dynamic system includes the usual dynamic system where T is a single-valued map.

Let (X, T) be a set-valued dynamic system. By Fix(T) and End(T) we denote the sets of all fixed points and endpoints (or stationary points) of T, respectively i.e., Fix(T) = {w X : w T(w)} and End(T) = {w X : {w} = T (w)}.

A dynamic process or a trajectory starting at w0 X or a motion of the system (X, T) at w0 is a sequence (wm : m ∈ {0} ∪ ℕ) defined by wm T(wm-1) for m ∈ ℕ (see, [1,2]).

If (X, T) is a dynamic system and w0 X then, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M1">View MathML</a>, we denote the set of all dynamic processes of the system (X, T) starting at w0.

A beautiful Banach's contraction principle [3] has inspired a large body of work over the last 50 years and there are several ways in which one might hope to improve this principle.

Theorem 1 [3]Let (X, d) be a complete metric space. Let T : X X be a single-valued map satisfying the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M2">View MathML</a>

(1)

Then: (i) T has a unique fixed point w in X, i.e. Fix(T) = {w}; and (ii) the sequence {T[m](u)} converges to w for each u X.

Let (X, d) be a metric space and let CB(X) denote the class of all nonempty closed and bounded subsets of X. If h : CB(X) × CB(X) → [0, ∞) represents a Hausdorff metric induced by d, it has the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M3">View MathML</a>

where d(x, C) = infcC d(x, c), x X, C CB(X).

A natural question to ask is whether the single-valued dynamic system in this principle can be replaced by the set-valued dynamic system. One of the first results in this direction was established in [4].

Theorem 2 [[4], Th. 5] Let (X, d) be a complete metric space. Assume that the set-valued dynamic system (X, T) satisfying T : X CB(X) is (h, λ)-contractive, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M4">View MathML</a>

(2)

Then T has a fixed point w in X, i.e. w T(w).

There are other important ways of extending the Banach theorem. In particular, many interesting theorems in this setting, proposed by Covitz and Nadler, Jr. [[5], Theorem 1], concern the set-valued dynamic systems in generalized metric spaces.

The concepts of generalized metric spaces and the canonical decompositions of these spaces appeared first in Luxemburg [6] and Jung [7]. Recall that a generalized metric space is a pair (X, d) where X is a nonempty set and d : X2 → [0, ∞] satisfies: (a) ∀x,yX {d(x, y) = 0 iff x = y}; (b) ∀x,yX {d(x, y) = d(y, x); (c) ∀x,y,zX {[d(x, z) < +∞ ∧ d(y, z) < +∞] ⇒ [d(x, y) < + ∞ ∧ d(x, y) ≤ d(x, z) + d(z, y)]}. Some characterizations of these spaces were presented by Jung [7] who proved the essential theorems about decomposition of a generalized metric spaces and discovered the way to obtain generalized (complete) metric spaces. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M5">View MathML</a>, -index set, be a family of disjoint metric spaces. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M7">View MathML</a> and, for any x, y X,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M8">View MathML</a>

then (X, d) is a generalized metric space. Moreover, if for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M9">View MathML</a>, (Xβ, dβ) is complete then (X, d) is a generalized complete metric space. Also, in generalized metric spaces (X, d) he introduced the following equivalence relation on X:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M10">View MathML</a>

Therefore, X is decomposed uniquely into (disjoint) equivalence classes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M11">View MathML</a>, which is called a canonical decomposition. We may read these results as follows.

Theorem 3 [7]Let (X, d) be a generalized metric space, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M7">View MathML</a>be the canonical decomposition and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M12">View MathML</a>. Then: (I) For each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M13">View MathML</a>, (Xβ, dβ) is a metric space; (II) For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M14">View MathML</a>, with β1 β2, d(x, y) = +∞ for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M15">View MathML</a>and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M16">View MathML</a>; and (III) (X, d) is a generalized complete metric space iff, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M13">View MathML</a>, (Xβ, dβ) is a complete metric space.

Before presenting the results of Covitz and Nadler, Jr. [5] we recall some notations.

Definition 1 Let (X, d) be a generalized metric space.

(a) We say that a nonempty subset Y of X is closed in X if Y = Cl(Y) where Cl(Y), the closure of Y in X, denote the set of all x X for which there exists a sequence (xm : m ∈ ℕ) in Y which is d-convergent to x.

(b) The class of all nonempty closed subsets of X is denoted by C(X), i.e. C(X) = {Y : Y ∈ 2X Y = Cl(Y)}.

(c) A generalized Hausdorff distance H : C(X) × C(X) → [0, ∞] induced by d is defined by: for each A, B C(X),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M17">View MathML</a>

where, for each E C(X) and ε > 0, N(ε, E) = {x X : ∃eE {d(x, e) < ε}}.

Theorem 4 [[5], Theorem 1] Let (X, d) be a generalized complete metric space and let w0 X. Assume that a set-valued dynamic system (X, T) satisfying T : X C(X) is (H, ε, λ)-uniformly locally contractive, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M18">View MathML</a>

Then the following alternative holds: either

(A) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M19">View MathML</a>; or

(B) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20">View MathML</a>.

It is not hard to see that each (H, λ)-contractive set-valued dynamic system defined below is, for each ε ∈ (0, + ∞), (H, ε, λ)-uniformly locally contractive.

Theorem 5 [[5], Corollary 1] Let (X, d) be a generalized complete metric space and let w0 X. Assume that the set-valued dynamic system (X, T) satisfying T : X C(X) is (H, λ)-contractive, i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M21">View MathML</a>

(3)

Then the following alternative holds: either

(A) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M22">View MathML</a>; or

(B) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20">View MathML</a>.

The following follows from Theorem 5 and generalize Nadler's Theorem 2.

Theorem 6 [[5], Corollary 3] Let (X, d) be a complete metric space and let w0 X. Assume that a set-valued dynamic system (X, T) satisfying T : X C(X) is (h, λ)-contractive, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M23">View MathML</a>

(4)

Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M20">View MathML</a>.

Recall that the investigations of fixed points of maps in complete generalized metric spaces appeared for the first time in Diaz and Margolis [8] and Margolis [9].

Another natural problem is to extend the Nadler's [[4], Th. 5] theorem to set-valued dynamic systems which are contractive with respect to more general distances. In complete metric spaces, this line of research was pioneered by Suzuki [10], who developed many crucial technical tools.

Definition 2 [11] Let (X, d) be a metric space. A map p : X × X → [0, ∞) is called a τ-distance on X if there exists a map η : X × [0, ∞) → [0, ∞) and the following conditions hold: (S1) ∀x,y,zX {p(x, z) ≤ p(x, y) + p(y, z)}; (S2) ∀xX t>0{η(x, 0) = 0 ⋀ η(x, t) ≥ t} and η is concave and continuous in its second variable; (S3) limn→∞ xn = x and limn→∞ supmn η(zn, p(zn, xm)) = 0 imply that ∀wX {p(w, x) ≤ lim infn→∞ p(w, xn)}; (S4) limn→∞ supmn p(xn, ym)) = 0 and limn→∞ η(xn, tn) = 0 imply that limn→∞ η(yn, tn) = 0; and (S5) limn→∞ η(zn, p(zn, xn)) = 0 and limn→∞ η(zn, p(zn, yn)) = 0 imply that limn→∞ d(xn, yn) = 0.

Theorem 7 [[10], Theorem 3.7] Let (X, d) be a complete metric space and let p be a τ-distance on X. Let a set-valued dynamic system (X, T) satisfying T : X C(X) be (Qp, λ)-contractive, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M24">View MathML</a>

(5)

where Qp(A, B) = supaA infbB p(a, b). Then there exists w X such that w T(w) and p(w, w) = 0.

Remark 1 Let us observe that this beautiful Suzuki's theorem include Covitz-Nadler's Theorem 6. Indeed, first we see that each metric d is τ-distance (cf. [11]) and next we see that each (h, λ)-contractive set-valued dynamic system (X, T) satisfying T : X C(X) is (Qd, λ)-contractive; in fact, Qd h on C(X) (cf. [12]). Moreover, there exist (Qd, λ)-contractive set valued dynamic systems (X, T) satisfying T : X C(X) which are not (h, λ)-contractive.

It is worth noticing that a number of authors introduce the new various concepts of set-valued contractions of Nadler type in complete metric spaces, study the problem concerning the existence of fixed points for such contractions and obtain the various generalizations of Nadler's result which are different from the mentioned above; see, e.g., Takahashi [13], Jachymski [[14], Theorem 5], Feng and Liu [12], Zhong et al. [15], Mizoguchi and Takahashi [16], Eldred et al. [17], Suzuki [18], Kaneko [19], Reich [20,21], Quantina and Kamran [22], Suzuki and Takahashi [23], Al-Homidan et al. [24], Latif and Al-Mezel [25], Frigon [26], Klim and Wardowski [27], Ćirić [28] and Pathak and Shahzad [29].

The above are some of the reasons why in nonlinear analysis the study of uniformly locally contractive and contractive set-valued dynamic systems play a particularly important part in the fixed point theory and its applications.

Let us notice that in the proofs of the results of [3-29], among other things, the following assumptions and observations are essential: (O1) The completeness of metric and generalized metric spaces is necessary; (O2) In Theorems 1, 2 and 4-7, the maps T : (X, d) → (X, d), T : (X, d) → (CB(X), h), T : (X, d) → (C(X), H) and T : (X, p) → (C(X), Qp) are investigated and the conditions (1)-(5) imply that these maps between spaces (X, d), (X, p), (CB(X), h), (C(X), H) and (C(X), Qp), respectively, are continuous; (O3) By Theorems 1, 2 and 4-7, for each w Fix(T) the following equalities d(w, w) = 0, h(T(w), T(w)) = 0, H(T(w), T(w)) = 0, Qp(T(w), T(w)) = 0 and p(w, w) = 0 hold, respectively; (O4) The distances h, H, and Qp are defined only on the spaces CB(X) or C(X), respectively.

Also, let us observe that in [30-36] we studied some families of generalized pseudodistances in uniform spaces and generalized quasipseudodistances in quasigauge spaces which generalize: metrics, distances of Tataru [37], w-distances of Kada et al. [38], τ- distances of Suzuki [11] and τ-functions of Lin and Du [39] in metric spaces and distances of Vályi [40] in uniform spaces.

Motivated by the comments and observations stated above our main interest of this article is the following:

Question 1 Are there spaces X, new distances on X which are more general than d, h, H, p and Qp, and set-valued dynamic systems (X, T) which are uniformly locally contractive or contractive with respect to new distances, such that the analogous assertions as in Theorems 1, 2 and 4-7 hold but, unfortunately: (M1) Spaces X (metric, generalized metric and more general) are not necessarily complete; (M2) If new distances we replaced by d, h, H, p or Qp then maps T are not necessarily continuous in the sense defined by inequalities (1)-(5), respectively; (M3) For T, w Fix(T) and for new distances the properties in (O3) do not necessarily hold in such generality; (M4) The new distances are defined on 2X, and thus not only on CB(X) or C(X) as in (O4)?

Our purpose in this article is to answer our question in the affirmative and providing the illustrating examples. More precisely, inspired by ideas of Diaz and Margolis [8], Margolis [9], Luxemburg [6], Jung [7], Nadler [[4], Th. 5], Covitz and Nadler [5] and Suzuki [10] and the above comments and observations, the concepts of the families <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M25">View MathML</a> (-index set) of generalized pseudometrics on a nonempty set X and the generalized uniform spaces (X, ) are introduced, the classes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a> of -families of generalized pseudodistances in (X, ) are defined and, in (X, ), a new type of -sequentially completeness with respect to -families (which extend the usual sequentially completeness in uniform and locally convex spaces and completeness in metric and generalized metric spaces) are studied (see the following section). Moreover, some partial quasiordered space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a> is defined (see Section "Partial quasiordered space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>") and, using <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distances on 2X (i ∈ {1, 2}) with respect to -families are introduced (see Section "<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distances on 2X, i ∈ {1,2}"). Also, we introduce the definitions of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a> -uniformly locally contractive and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive set-valued dynamic systems (X, T) (i ∈ {1, 2}) satisfying T : X → 2X (see Section "<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive set-valued dynamic systems (X, T), i ∈ {1, 2}") and, for w0 X, we establish the conditions guaranteeing the convergence of dynamic processes <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M1">View MathML</a> and the existence of fixed points for such contractions and, additionally, a special case when T : X C(X) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34">View MathML</a> is studied (see Sections 6-8). Also the concept of the generalized locally convex space as a special case of the generalized uniform space is introduced (see Section "Generalized locally convex spaces <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M35">View MathML</a>"). By generality of spaces and -families, our results, in particular, include and essentially generalize Theorems 1, 2 and 4-7. The examples illustrating ideas, methods and results are constructed and comparisons of our results with the results of Nadler [[4], Th. 5], Covitz and Nadler [5] and Suzuki [10] are given (see Sections 10-13). Finally, a natural question is formulated (see Section "Concluding remarks"). The results are new in generalized uniform spaces, uniform spaces, generalized locally convex and locally convex spaces and are new even in generalized metric spaces and in metric spaces.

Generalized uniform spaces (X, ) and the class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a> of -families of generalized pseudodistances on (X, )

The following terminologies will be much used.

Definition 3 Let X be a nonempty set. (a) The family

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M36">View MathML</a>

is said to be a -family of generalized pseudometrics on X (-family on X, for short) if the following three conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M38">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M39">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M40">View MathML</a>; and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a> If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42">View MathML</a> and x, y, z X and if dα(x, z) and dα(y, z) are finite, then dα(x, y) is finite and dα(x, y) ≤ dα(x, z) + dα(z, y).

(b) If is -family, then the pair (X, ) is called a generalized uniform space.

(c) Let (X, ) be a generalized uniform space. A -family is said to be separating if

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M43">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M44">View MathML</a>.

(d) If a -family is separating, then the pair (X, ) is called a Hausdorff generalized uniform space.

(e) Let (X, ) be a generalized uniform space and let (xm : m ∈ ℕ) be a sequence in X. We say that (xm : m ∈ ℕ) is -Cauchy sequence in X if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M45">View MathML</a>. We say that (xm : m ∈ ℕ) is -convergent in X if there is an x X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M46">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M47">View MathML</a>, for short).

(f) If every -Cauchy sequence in X is -convergent sequence in X, then a pair (X, ) is called a -sequentially complete generalized uniform space.

Definition 4 Let X be a nonempty set. The family

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M48">View MathML</a>

is said to be a -family of generalized quasi pseudometrics on X (-family on X, for short) if the following two conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M50">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M51">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M52">View MathML</a>) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42">View MathML</a> and x, y, z X and if qα(x, z) and qα(z, y) are finite, then qα(x, y) is finite and qα(x, y) ≤ qα(x, z) + qα(z, y).

Definition 5 Let (X, ) be a generalized uniform space.

(a) The family

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M53">View MathML</a>

is said to be a -family of generalized pseudodistances on X (-family on X, for short) if the following two conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a>) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42">View MathML</a> and x, y, z X and if Lα(x, z) and Lα(z, y) are finite, then Lα(x, y) is finite and Lα(x, y) ≤ Lα(x, z) + Lα(z, y); and

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>) For any sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M56">View MathML</a>

(6)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M57">View MathML</a>

(7)

the following holds

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M58">View MathML</a>

(8)

(b) Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a> be a class defined as follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M59">View MathML</a>

Remark 2 Let (X, ) be a generalized uniform space. (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M60">View MathML</a> since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M61">View MathML</a>. (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M62">View MathML</a>; see Sections 10-13.

Definition 6 Let (X, ) be a generalized uniform space, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a> and let (xm : m ∈ ℕ) be a sequence in X.

(a) We say that (xm : m ∈ ℕ) is -Cauchy in X if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M64">View MathML</a>.

(b) We say that (xm : m ∈ ℕ) is -convergent in X if there exists x X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M65">View MathML</a>.

(c) We say that (X, ) is -sequentially complete if each -Cauchy sequence in X is -convergent in X.

In the following remark, we list some basic properties of -families.

Remark 3 Let (X, ) be a generalized uniform space and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a>. (i) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M66">View MathML</a>, then is a -family on X; examples of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a> which are not -families on X are given in Section "Examples of the decompositions of the generalized uniform spaces". (ii) There exist -sequentially complete spaces which are not -sequentially complete; see Example 15. (iii) If (xm : m ∈ ℕ) in X is -convergent in X, then its limit point is not necessary unique; see Example 1.

Example 1 Let (ℝ, |·|) be a metric space. Define the family of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M67">View MathML</a> to be

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M68">View MathML</a>

It is obvious that is -family on ℝ and the sequence (1/m : m ∈ ℕ) is -convergent to each point w ∈ (0, +∞).

One can prove the following proposition:

Proposition 1 Let (X, ) be a Hausdorff generalized uniform space and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a>.

(I) If x y, x, y X, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M69">View MathML</a>.

(II) If (X, ) is -sequentially complete and if (xm : m ∈ ℕ) is -Cauchy sequence in X, then (xm : m ∈ ℕ) is -convergent in X.

Proof. (I)) Assume that there are x y, x, y X, such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M70">View MathML</a>. Then, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M71">View MathML</a>, since, by using (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a>), it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M72">View MathML</a>. Defining the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X by xm = x and ym = y for m ∈ ℕ, and observing that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M73">View MathML</a>, this implies that (6) and (7) for these sequences hold. Then, by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>), (8) holds, so it is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M74">View MathML</a>. On the other hand, is separating, so, since x y, it is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M75">View MathML</a>. This leads to a contradiction.

(II) Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M76">View MathML</a>, by Definition 6(c), this proves the existence of x X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M65">View MathML</a>. We can apply (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>) to sequences (xm : m ∈ ℕ) and (ym = x/ : m ∈ ℕ) and then we find that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M77">View MathML</a>. The uniqueness of the point of x follows from the fact that is separating. □

Partial quasiordered space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>

Proposition 2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>be a set of elements <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M78">View MathML</a>defined by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M79">View MathML</a>

and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M80">View MathML</a>. The relation <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M81">View MathML</a>on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M82">View MathML</a>

is a partial quasiordered on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M30">View MathML</a>and the pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M83">View MathML</a>is a partial quasiordered space.

Proof. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M84">View MathML</a> the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M85">View MathML</a> holds. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M86">View MathML</a>, the conditions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M87">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M88">View MathML</a> imply <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M89">View MathML</a>. For all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M90">View MathML</a>, the conditions <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M91">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M92">View MathML</a> imply Θ = Ω. □

Notation. The following notation is fixed throughout the article:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M93">View MathML</a>;

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M94">View MathML</a>;

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M95">View MathML</a>;

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M96">View MathML</a>.

In the sequel, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M90">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M97">View MathML</a> will stand for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M91">View MathML</a> and Θ ≠ Ω.

Definition 7 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M98">View MathML</a> be a nonempty subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M99">View MathML</a>. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M100">View MathML</a> is a infimum of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M98">View MathML</a> if the following two conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M101">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M102">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M104">View MathML</a>.

Example 2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M105">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M106">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M107">View MathML</a> then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M108">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M109">View MathML</a> does not exist since (3, 5, 7) and (4, 1, 8) are not comparable. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M110">View MathML</a> then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M111">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M112">View MathML</a>.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distances on 2X, i ∈ {1, 2}

Definition 8 Let (X, ) be a Hausdorff generalized uniform space and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a>.

(a) For C ∈ 2X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M113">View MathML</a>, let us denote

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M114">View MathML</a>

(9)

(b) For A, B ∈ 2X let us denote:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M115">View MathML</a>

(10)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M116">View MathML</a>

(11)

(c) Let i ∈ {1, 2}. The map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M117">View MathML</a> of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M118">View MathML</a>

A, B ∈ 2X, is called a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distance on 2X generated by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distance on 2X, for short).

Remark 4 For each A, B ∈ 2X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M119">View MathML</a>.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive set-valued dynamic systems (X, T), i ∈ {1, 2}

Definition 9 Let (X, ) be a Hausdorff generalized uniform space, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a> and let i ∈ {1,2}.

(a) Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a> be a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distance on 2X and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122">View MathML</a>. We say that a set-valued dynamic system (X, T), T : X → 2X, is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive on X if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M123">View MathML</a>

(12)

(b) Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a> be a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distance on 2X and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124">View MathML</a>. We say that a set-valued dynamic system (X, T), T : X → 2X, is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>- contractive on X if

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M125">View MathML</a>

(13)

Remark 5 Let (X, ) be a Hausdorff generalized uniform space, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a> be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122">View MathML</a>.

(i) If (X, T), T : X → 2X, is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M126">View MathML</a>-uniformly locally contractive on X then it is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M127">View MathML</a>-uniformly locally contractive on X.

(ii) If (X, T), T : X → 2X, is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M128">View MathML</a>-contractive on X then it is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M129">View MathML</a>-contractive on X.

(iii) Let i ∈ {1, 2}. If (X, T), T : X → 2X, is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive on X then it is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive on X.

Statement of results

Definition 10 Let (X, ) be a Hausdorff generalized uniform space and let x X/We say that a set-valued dynamic system (X, T), T : X → 2X, is closed at x if whenever (xm : m ∈ ℕ) is a sequence -converging to x in X and (ym : m ∈ ℕ) is a sequence -converging to y in X such that ym T(xm) for all m ∈ ℕ, then y T(x).

The main existence and convergence result of this article we can now state as follows.

Theorem 8 Assume that (X, ) is a Hausdorff generalized uniform space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a>and one of the following properties holds:

(P1) (X, ) is -sequentially complete; or

(P2) (X, ) is -sequentially complete.

Let i ∈ {1, 2}, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M117">View MathML</a>be a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M31">View MathML</a>-distance on 2X and assume that a set-valued dynamic system (X, T), T : X → 2X, has the property

(C) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M130">View MathML</a> {limm→∞ wm = w T is closed at w}.

(I) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a>and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a>satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122">View MathML</a>and (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(A1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M131">View MathML</a>; or

(A2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M132">View MathML</a> {w Fix(T) ⋀ limm→∞ wm = w ⋀ (wm : m ∈ {0} ∪ ℕ) is -Cauchy}.

(II) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a>satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124">View MathML</a>and (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive on X then, for each w0 X, the following alternative holds: either

(B1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M133">View MathML</a>; or

(B2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M132">View MathML</a> {w Fix(T) ⋀ limm→∞ wm = w ⋀ (wm : m ∈ {0} ∪ ℕ) is -Cauchy}.

Definition 11 Let (X, ) be a Hausdorff generalized uniform space.

(a) We say that a nonempty subset Y of X is closed in X if Y = Cl(Y) where Cl(Y), the closure of Y in X, denotes the set of all x X for which there exists a sequence (xm : m ∈ ℕ) in Y which is -convergent to x.

(b) The class of all nonempty closed subsets of X is denoted by C(X), i.e. C(X) = {Y : Y ∈ 2X Y = Cl(Y)}.

Theorem 8 has the following corresponding when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34">View MathML</a> and when T : X C(X).

Theorem 9 Let (X, ) be a Hausdorff -sequentially complete generalized uniform space, let i ∈ {1, 2} and assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M134">View MathML</a>is a <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M135">View MathML</a>-distance on C(X).

(I) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a>and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a>satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M122">View MathML</a>and if a set-valued dynamic system (X, T) satisfying T : X C(X) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M136">View MathML</a>-uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(F1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M131">View MathML</a>; or

(F2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M137">View MathML</a>.

(II) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M121">View MathML</a>satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M124">View MathML</a>and a set-valued dynamic system (X, T) satisfying T : X C(X) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M138">View MathML</a>-contractive on X then, for each w0 X, the following alternative holds: either

(G1) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M139">View MathML</a>; or

(G2) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M137">View MathML</a>.

Proof of Theorem 8

(I) Let i ∈ {1, 2}. The proof is divided into three steps.

Step 1. Assume that w0 X and suppose that the assertion (A1) does not hold; that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M140">View MathML</a>

(14)

Then there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141">View MathML</a>which is -Cauchy sequence on X; that is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M142">View MathML</a>

(15)

Indeed, since (14) holds, thus, by (12), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M143">View MathML</a>

(16)

It follows from (16) and Definition 8(c), that there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M144">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M145">View MathML</a>

(17)

From this, denoting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M146">View MathML</a>, we deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M147">View MathML</a>. Consequently, by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103">View MathML</a>), there exists

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M148">View MathML</a>

(18)

such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M97">View MathML</a> which implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M149">View MathML</a>

(19)

If i = 1, then we note that, by (18), (9), and (10), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M150">View MathML</a>. Clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M151">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M152">View MathML</a> and the conclusion

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M153">View MathML</a>

follows directly from (9), (10), (18), and (19).

If i = 2, then we also note that, by (18), (9) and (11), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M150">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M154">View MathML</a>. Clearly, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M151">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M152">View MathML</a> and the conclusion

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M155">View MathML</a>

follows directly from (9), (11), (18), and (19).

This proves

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M156">View MathML</a>

(20)

Since, by (20), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M157">View MathML</a>, it follows, using (12) and (20), that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M158">View MathML</a>

That is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M159">View MathML</a>

(21)

Denoting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M160">View MathML</a>, we see that condition (21) implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M161">View MathML</a>. Hence, by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M103">View MathML</a>), there exists

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M162">View MathML</a>

(22)

such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M163">View MathML</a>. This means

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M164">View MathML</a>

(23)

Let i = 1. Clearly, by (9), (10), and (22), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M165">View MathML</a>. Moreover, by (20), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M166">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M167">View MathML</a>. This, by (9), (10) and (21)-(23), implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M168">View MathML</a>

Let i = 2. Clearly, by (9)-(11) and (22), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M165">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M169">View MathML</a>. Moreover, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M166">View MathML</a>. Therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M167">View MathML</a>. This, by (9)-(11) and (21)-(23), implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M170">View MathML</a>

That is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M171">View MathML</a>

(24)

By (24), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M172">View MathML</a> and, using (12) and (24), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M173">View MathML</a>

This means

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M174">View MathML</a>

(25)

By induction, a similar argument as in the proofs of (17)-(25) shows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M175">View MathML</a>

(26)

It is clear that (26) implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141">View MathML</a> where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M176">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M177">View MathML</a>. Additionally, this sequence (wm : m ∈ {0} ⋂ ℕ) is a -Cauchy sequence on X, i.e., (15) holds.

Step 2. Assume that the condition (C) and the property (P1) hold. If w0 X and the assertion (A1) does not hold, then (A2) holds.

By Step 1, Definition 8(c) and (P1) (note that then (X, ) is -sequentially complete), we have that there exists w X satisfying

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M178">View MathML</a>

(27)

Applying (15), (27), and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>) (where (xm = wm : m ∈ ℕ) and (ym = w : m ∈ ℕ)), we find that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M179">View MathML</a>

(28)

Clearly, since (X, ) is Hausdorff, condition (28) implies that such a point w is unique.

We observe that w Fix(T). Indeed, we have that a dynamic process (wm : m ∈ {0} ∪ ℕ) satisfies (28). Hence, by (C), T is closed at w and, since ∀m∈ℕ{wm T(wm-1)}, we get w T(w). This proves that the assertion (A2) holds.

This yields the result when (C) and (P1) hold.

Step 3. Assume that the condition (C) and the property (P2) hold. If w0 X and the assertion (A1) does not hold, then (A2) holds.

If (A1) does not hold, then, by Step 1, there exists a sequence (wm : m ∈ {0} ⋂ ℕ) which satisfies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M141">View MathML</a> and, additionally, this sequence is a -Cauchy sequence on X, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M180">View MathML</a>

(29)

We prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M181">View MathML</a> is a -Cauchy sequence on X, i.e. that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M182">View MathML</a>

(30)

Indeed, by (29), we claim that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M183">View MathML</a>

Hence, in particular,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M184">View MathML</a>

(31)

Let now r0, j0 ∈ ℕ, r0 > j0, be arbitrary and fixed. If we define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M185">View MathML</a>

(32)

then (31) implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M186">View MathML</a>

(33)

Therefore, by (29), (33), and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M187">View MathML</a>

(34)

From (32)-(34), we then claim that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M188">View MathML</a>

(35)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M189">View MathML</a>

(36)

Let now <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M190">View MathML</a> and ε0 > 0 be arbitrary and fixed, let n0 = max{n2(α0, ε0), n3(α0, ε0)} + 1 and let s, l ∈ ℕ be arbitrary and fixed such that s > l > n0. Then s = r0 + n0 and l = j0 + n0 for some r0, j0 ∈ ℕ such that r0 > j0 and, using (35) and (36), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M191">View MathML</a>

Hence, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M192">View MathML</a>

The proof of (30) is complete.

Now we see that there exists a unique w X such that limm→∞ wm = w. Indeed, since (X, ) is a Hausdorff -sequentially complete generalized uniform space and the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M181">View MathML</a> is a -Cauchy sequence on X, thus there exists a unique w X such that limm→∞ wm = w.

Moreover, we observe that w Fix(T). Indeed, we have that a dynamic process (wm : m ∈ {0} ∪ ℕ) satisfies limm→∞ wm = w. Hence, by (C), T is closed at w and, since ∀m∈ℕ{wm T(wm-1)}, we get w T(w). We proved that the assertion (A2) holds.

This yields the result when (C) and (P2) hold.

The proof of (I) is complete.

(II) Let i ∈ {1, 2}. Let w0 X, let the condition (C) holds and suppose that the assertion (B1) does not hold, i.e. suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M193">View MathML</a>

This implies that there exists the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M194">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M195">View MathML</a>. Consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M196">View MathML</a>

Clearly, (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M32">View MathML</a>-uniformly locally contractive on X since (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M33">View MathML</a>-contractive on X. From the above and by similar argumentations as in Steps 1-3 of the proof of Theorem 8(I) we conclude that all assumptions of Theorem 8(I) hold and the assertion (A1) of Theorem 8(I) does not hold. Consequently, using Theorem 8(I), we get that the assertion (A2) of Theorem 8(I) holds in the case when the property either (P1) or (P2) holds. Hence, the assertion (B2) of Theorem 8(II) holds.

The proof of Theorem 8 is complete. □

Proof of Theorem 9

(I) Let i ∈ {1, 2}. Let w0 X be arbitrary and fixed and suppose that the assertion (F1) does not hold. That is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M197">View MathML</a>

(37)

But then, using analogous considerations as in the Step 1 of the proof of Theorem 8(I), we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M198">View MathML</a>

(38)

Consequently, the sequence (wm : m ∈ {0} ∪ ℕ) such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M176">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M177">View MathML</a> is a dynamic process of T starting at w0 and, additionally, this sequence is a -Cauchy sequence on X, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M199">View MathML</a>

(39)

It is clear that (39) implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M200">View MathML</a>

(40)

and, since (X, ) is a Hausdorff -sequentially complete generalized uniform space, there exists a unique w X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M201">View MathML</a>

(41)

If, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42">View MathML</a>, x X and B Cl(X), we denote

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M202">View MathML</a>

(42)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M203">View MathML</a>

(43)

then (42) and (40) implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M204">View MathML</a>

(44)

Let m ∈ ℕ, m > m0, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M42">View MathML</a> be arbitrary and fixed and let

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M205">View MathML</a>

here m0 is defined by (37). Then, by (9)-(11) and definition of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M206">View MathML</a>, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M207">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M208">View MathML</a>. Hence, in particular, if v T(wm) is arbitrary and fixed, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M209">View MathML</a>

This implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M210">View MathML</a>

(45)

Now, by (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37">View MathML</a>), (remember that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M34">View MathML</a>), for each u T(w) and v T(wm), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M211">View MathML</a>

Hence, by (42) and (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M37">View MathML</a>), for each v T(wm), it follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M212">View MathML</a>

Further, by (38), (43), (44), and (11), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M213">View MathML</a>

Hence, by (41) and (44), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M214">View MathML</a>. However, this property of w, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M215">View MathML</a>

and fact that T(w) is closed, gives w T(w). This and (41) yield that (F2) holds.

(II) Let i ∈ {1, 2}. Let w0 X and suppose that the assertion (G1) does not hold, i.e. suppose that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M216">View MathML</a>

This implies that there exists the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M120">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M194">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M217">View MathML</a>. Consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M218">View MathML</a>

Clearly, (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M136">View MathML</a>- uniformly locally contractive on X since (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M135">View MathML</a>-contractive on X. Using now similar argumentation as in the proof of Theorem 8(II), we obtain that (G2) holds.

The proof of Theorem 9 is complete. □

Generalized locally convex spaces (X, )

We want to show an immediate consequence of the Section "Generalized uniform spaces (X, ) and the class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a> of -families of generalized pseu-dodistances on (X, )".

Definition 12 Let X be a vector space over ℝ.

(i) The family

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M220">View MathML</a>

is said to be a -family of generalized seminorms on X (-family, for short) if the following three conditions hold:

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M221">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M222">View MathML</a>;

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M223">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M224">View MathML</a>; and

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M225">View MathML</a>) If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a> and x, y X and if pα(x) and pα(y) are finite, then pα(x + y) is finite and pα(x + y) ≤ pα(x) + pα(y).

(ii) If is -family, then the pair (X, ) is called a generalized locally convex space.

(iii) A -family is said to be separating if

(<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M227">View MathML</a>) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M228">View MathML</a>.

(iv) If a -family is separating, then the pair (X, ) is called a Hausdorff generalized locally convex space.

Remark 6 It is clear that each generalized locally convex space is an generalized uniform space. Indeed, if X is a vector space over ℝ and (X, ) is a generalized locally convex space, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229">View MathML</a> where dα(x,y) = pα(x - y), (x,y) ∈ X × X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, is -family and (X, ) is a generalized uniform space.

Examples of the decompositions of the generalized uniform spaces

Example 3 For each n ∈ ℕ, let Zn = [2n - 2, 2n - 1] and let qn : Zn × Zn → [0, +∞) where qn(x,y) = |x - y| for x,y Zn. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M230">View MathML</a> and define q : Z × Z → [0, +∞] by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M231">View MathML</a>

(46)

Then (Z, q) is a complete generalized metric space.

Example 4 Let Y = ℝ= ℝ × ℝ × ⋯ be a non-normable real Hausdorff and sequentially complete locally convex space with the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M232">View MathML</a> of calibrations cn,n ∈ ℕ, defined as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M233">View MathML</a>

For each s ∈ ℕ, let Ps = [2s - 2, 2s - 1]be a Hausdorff sequentially complete uniform space with uniformity defined by the saturated family {ps,n : n ∈ ℕ} of pseudometrics ps,n : Ps × Ps → [0, +∞), n ∈ ℕ, defined as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M234">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M235">View MathML</a> and define pn:P × P → [0, +∞], n ∈ ℕ, as follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M236">View MathML</a>

(47)

Then (P, {pn:P × P → [0, +∞], n ∈ ℕ}) is a Hausdorff {pn:P × P→ [0, +∞], n ∈ ℕ}-sequentially complete generalized uniform space.

Examples of elements of the class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a>

In this section we describe some elements of the class <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M28">View MathML</a>.

Example 5 Let (X, ) be a Hausdorff generalized uniform space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229">View MathML</a>, -index set, is a -family. Let the set E X, containing at least two different points, be arbitrary and fixed and, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, let La : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M237">View MathML</a>

(48)

We show that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M238">View MathML</a> is -family on (X, ).

First, we observe that the condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a>) holds. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a> and x, y, z X be arbitrary and fixed and such that Lα(x, z) < +∞ and Lα(z, y) < + ∞. By (48), this implies that: x, y, z E; dα(x, z) = Lα(x, z) < +∞; and dα(z, y) = Lα(z,y) < +∞. Then, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get that dα(x,y) < +∞ and dα(x, y) ≤ dα(x,z) + da(z,y). Consequently, since x,y,z E, this mean that Lα(x,y) = da(x,y) < +∞ and La(x,y) ≤ La(x,z) + La(z,y). Therefore, the condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a>) holds.

To prove that (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a>) holds, we assume that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M239">View MathML</a>

By definition of , this implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M240">View MathML</a>

Therefore, we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M241">View MathML</a>

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Hence we conclude that the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a> is satisfied.

Example 6 Let (X, ) be a generalized metric space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242">View MathML</a> is a -family. Let the set E X, containing at least two different points, be arbitrary and fixed and let L : X × X → [0, +∞] be defined by the formula (see (48)):

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M243">View MathML</a>

(49)

By Example 5, the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

Example 7 Let (X, ) be a Hausdorff generalized uniform space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229">View MathML</a>, -index set, is a -family. Let the sets E and F satisfying E F X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let 0 < aα < bα < cα < +∞, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, and let, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, Lα : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M245">View MathML</a>

(50)

We show that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M246">View MathML</a> is -family on X.

First, we observe that the condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a> holds. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a> and x, y, z X satisfying Lα(x, z) < + ∞ and Lα(z, y) < + ∞ be arbitrary and fixed. Clearly, by definition of Lα, this implies that x, y, z F. We consider the following cases:

Case 1. If Lα(x, y) = dα(x, y) + bα, then by (50) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z E, then Lα(x, z) = dα(x, z) + ca; Lα(z, y) = dα(z, y); and consequently, since bα < cα, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M247">View MathML</a>

If z F \ E, then Lα(x, z) = dα(x, z) + bα; Lα(z, y) = dα(z, y) + bα; and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M248">View MathML</a>

Case 2. If Lα(x, y) = dα(x, y) + cα, then by (50) we conclude that, x F \ Ey E. Now, if z E then Lα(x, z) = dα(x, z) + cα; Lα(z, y) = dα(z, y) + αα; and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M249">View MathML</a>

If z F \ E, then Lα(x, z) = dα(x, z) + bα; Lα(z, y) = dα(z, y) + cα; and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M250">View MathML</a>

Case 3. If Lα(x, y) = dα(x, y), then by (50) we conclude that, x Ey F\E. Now, if z E then Lα(x, z) = dα(x, z) + aα; Lα(z, y) = dα(z, y); and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M251">View MathML</a>

If z F\E, then Lα(x, z) = dα(x, z); Lα(z, y) = dα(z, y) + bα; and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M252">View MathML</a>

Case 4. If Lα(x, y) = dα(x, y) + aα, then by (50) we conclude that, x Ey E. Now, if z E then Lα(x, z) = dα(x, z) + aα; Lα(z, y) = dα(z, y) + aα; and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M253">View MathML</a>

If z F\E, then Lα(x, z) = dα(x, z); Lα(z, y) = dα(z, y) + cα; and consequently, since aα < cα, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M254">View MathML</a>

Consequently, the condition <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a> holds.

To prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a> holds, we assume that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M255">View MathML</a>

By definition of , this implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M256">View MathML</a>

As a consequence of this, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M257">View MathML</a>

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Therefore, the property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a> holds.

It is worth noticing that, there exists x, y X such that, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, Lα(x, y) = Lα(y, x) does not hold. Indeed, if x E and y F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M258">View MathML</a>

Example 8 Let X, be a generalized metric space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242">View MathML</a> is a -family. Let the sets E and F satifying E F X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let L : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M259">View MathML</a>

(51)

By Example 7, the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

Example 9 Let (X, ) be a Hausdorff generalized uniform space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M229">View MathML</a>, -index set, is a -family. Let the sets E and F satisfying E F X be arbitrary and fixed and such that E contains at least two different points and F contains at least three different points. Let 0 < bα < cα < +∞, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, and let, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, Lα : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M260">View MathML</a>

(52)

We show that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M246">View MathML</a> is -family on X.

First, we observe that the condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a> holds. Indeed, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a> and x, y, z X satisfying Lα(x, z) < +∞ and Lα(z, y) < +∞ be arbitrary and fixed. Clearly, by definition of Lα, this implies that x, y, z F. We consider the following cases:

Case 1. If Lα(x, y) = dα(x, y) + bα, then by (52) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M261">View MathML</a>

and consequently, since bα < cα, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M262">View MathML</a>

If z F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M263">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M264">View MathML</a>

Case 2. If Lα(x, y) = dα(x, y) + cα, then by (52) we conclude that, x F\Ey E. Now, if z E then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M261">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M265">View MathML</a>

If z F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M266">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M267">View MathML</a>

Case 3. If Lα(x, y) = dα(x, y), then by (52) we conclude that, x E y E or x E y F\E. First, assume that x Ey E. Now, if z E then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M268">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M269">View MathML</a>

If z F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M270">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M271">View MathML</a>

Next, we assume that x E y F\E. Now, if z E then Lα(x, z) = dα(x, z); Lα(z, y) = dα(z, y); and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M272">View MathML</a>

If z F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M273">View MathML</a>

and consequently, by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M41">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M274">View MathML</a>

Consequently, the condition (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M54">View MathML</a> holds.

To prove that (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a> holds, we assume that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X satisfy (6) and (7). Then, in particular, (7) is of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M275">View MathML</a>

By definition of , this implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M276">View MathML</a>

As a consequence of this, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M277">View MathML</a>

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Therefore, the property (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M55">View MathML</a> holds.

It is worth noticing that, there exists x, y X such that, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M226">View MathML</a>, Lα(x, y) = Lα(y, x) does not hold. Indeed, if x E and y F \ E, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M278">View MathML</a>

Example 10 Let (X, ) be a generalized metric space where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M242">View MathML</a> is a -family. Let the sets E and F satisfying E F X be arbitrary and fixed, and such that E contains at least two different points and F contains at least three different points. Let L : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M279">View MathML</a>

(53)

By Example 9, the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

Examples which illustrate our theorems

The following example illustrates the Theorem 8(I) in the case when (X, ) is -sequentially complete and (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280">View MathML</a>-uniformly locally contractive on X where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M282">View MathML</a>.

Example 11 Let P and {pn : P × P → [0,+∞], n ∈ ℕ} be as in Example 4. Let X = P ⋂ [0,9]and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M283">View MathML</a>, dn : X × X → [0, +∞], n ∈ ℕ, where, for each n ∈ ℕ, we define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M284">View MathML</a>. Then (X, ) is a Hausdorff -sequentially complete generalized uniform space. This gives that the property (P2) of Theorem 8 holds.

The elements of ℝwe denote by x = (x1,x2,...). In particular, the element (x,x,...) ∈ ℝwe denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M285">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M286">View MathML</a> and let a set-valued dynamic system (X, T) be given by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M287">View MathML</a>

(54)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M288">View MathML</a> and let be a family of the maps given by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M289">View MathML</a>

(55)

By Example 4, the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M290">View MathML</a> is -family on X.

Now, we show that, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M291">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M292">View MathML</a>, (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M293">View MathML</a>-uniformly locally contractive on X, i.e. that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M294">View MathML</a>

(56)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M295">View MathML</a>

(57)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M296">View MathML</a>

(58)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M297">View MathML</a>

(59)

Indeed, let x, y X be arbitrary and fixed. Since, by (55), this family is symmetric on X, we may consider only the following four cases:

Case 1. Let x F and let y X\F.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M298">View MathML</a>, then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M299">View MathML</a>, by (55), for each n ∈ ℕ, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M300">View MathML</a>

By (47), from this, for each n ∈ ℕ, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M301">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M302">View MathML</a>, then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M303">View MathML</a>, by (55), we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M304">View MathML</a> for each y X \ F. Consequently, for each n ∈ ℕ, x F and y X\F, inequality Ln(x, y) < 1/2 in (56) does not hold and this case we do not have to consider this case.

Case 2. Let x, y F be such that x y or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M305">View MathML</a>. Then, by definition of F, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M302">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M306">View MathML</a>. But, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M303">View MathML</a>, therefore, by (55), we get ∀n∈ℕ{Ln(x,y) = +∞}. Therefore, by (56), this case we can also be omitted.

Case 3. Let x, y F be such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M307">View MathML</a>. Then, since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M308">View MathML</a>, by (55) and (47), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M309">View MathML</a>

(60)

and, consequently, for each n ∈ ℕ, the inequality Ln(x, y) < 1/2 holds. In virtue ofthis, we show that the inequalities <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M310">View MathML</a> in (56) hold. With this aim, we see that:

(3i) By (54), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M311">View MathML</a>;

(3ii) Next, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M312">View MathML</a>, then, by (3i),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M313">View MathML</a>

(3iii) Now, by (3i), (3ii), (58), and (59), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M314">View MathML</a>

(3iv) Therefore, by (3iii), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M315">View MathML</a>;

(3v) The consequence of (57) and (3iv) is

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M316">View MathML</a>

Hence, by (60), we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M317">View MathML</a>

holds.

Case 4. Let x, y F. Then we see that:

(4i) By (54), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M318">View MathML</a>;

(4ii) Next, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M312">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M319">View MathML</a>

(4iii) Now, by (4i) and (4ii), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M320">View MathML</a>

(4iv) Therefore, by (4iii), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M315">View MathML</a>;

(4v) According to (57) and (4iv), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M321">View MathML</a>

Consequently, by (60),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M322">View MathML</a>

We proved that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280">View MathML</a>-uniformly locally contractive on X. We see also that (C) holds.

Finally, we see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M323">View MathML</a>. Hence, for each w0 X, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that: (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M324">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M325">View MathML</a>; and (iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M326">View MathML</a>.

The following example illustrates the Theorem 8(I) in the case when (X, ) is -sequentially complete for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M63">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281">View MathML</a>, but not -sequentially complete and (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M280">View MathML</a>-uniformly locally contractive on X.

Example 12 Let X and {pn : P × P → [0, +∞], n ∈ ℕ} be as in Example 4. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M327">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M328">View MathML</a>, dk : X × X → [0,∞], k ∈ ℕ, where, for each k ∈ ℕ, we define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M329">View MathML</a>. Then (X, ) is a Hausdorff generalized uniform space.

We observe that (X, ) is not a -sequentially complete space. Indeed, we consider the sequence (xm : m ∈ ℕ) defined as follows: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M330">View MathML</a>, m ∈ ℕ. Of course, the sequence (xm : m ∈ ℕ) is -Cauchy sequence on X. Indeed, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M331">View MathML</a> which implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M332">View MathML</a>

Consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M333">View MathML</a>

However, there does not exist x X such that limm→∞ xm = x. Therefore, X is not -sequentially complete.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M334">View MathML</a> and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M335">View MathML</a> be a family of the maps given by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M336">View MathML</a>

By (47), this gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M337">View MathML</a>

(61)

where N = {0,1, 2, 3, 4, 5}, x, y X and k ∈ ℕ.

By Example 4, the familly <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M338">View MathML</a> is -family on X.

We show that X is -sequentially complete space. Indeed, let (xm : m ∈ ℕ) be arbitrary and fixed -Cauchy sequence in X, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M339">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M340">View MathML</a>

(62)

Hence, in particular, we conclude that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M341">View MathML</a>

(63)

Now, (63) and (61) gives that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M342">View MathML</a>

Of course, since (xm : m ∈ ℕ) is arbitrary nad fixed, then there exists a unique s0 N for all k ∈ ℕ. Now, putting l0 = mink∈ℕ{n0(k)} we obtain that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M343">View MathML</a>

(64)

The property (61) and (64) gives that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M344">View MathML</a>

(65)

Using (65), (64) and definition of E, we may consider only the following two cases:

Case 1. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M345">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M346">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M347">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M348">View MathML</a>, then in each of these situations the sequence, as a constant sequence, is, by (61), -convergent to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M349">View MathML</a>, respectively.

Case 2. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M350">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M351">View MathML</a>

so by (65) and (62), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M352">View MathML</a>

This gives that (xm : m ∈ ℕ) is a -Cauchy sequence in X, so also the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M353">View MathML</a> is a -Cauchy sequence in [4,5]. Since [4,5]is a -complete uniform space, so there exists x X such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M354">View MathML</a>

i.e (xm : m ∈ ℕ) is -convergent. In consequence, X is -sequentially complete generalized uniform space.

Now, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M355">View MathML</a> and let (X, T) be given by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M356">View MathML</a>

By the same reasoning as in Example 11, we obtain that, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M357">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M358">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M359">View MathML</a>-uniformly locally contractive on X, for each w0 X there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M360">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M361">View MathML</a>.

Now, in Example 13, for given (X, ) and (X, T), we study the assertions of Theorem 8(I) with respect to changing of the family of and of the point w0 X.

Example 13 Let (X, ) be a complete metric space where X = [0,1] and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M362">View MathML</a>, d:X × X→ [0,∞), d(x, y) = |x-y|, x, y X. Let a dynamic system (X, T) be given by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M363">View MathML</a>

(66)

Question 2 For these (X, D) and (X, T) and for ε = 1/2 and λ = 1/2, what are the assertions of our theorems with respect to changing of the family L and of the point w0 X?

Answer 1 We show that there exists -family on X such that: (a) (X,T) is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X; and (b) (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365">View MathML</a>-uniformly locally contractive on X and for each w0 X the assertion (A1) holds.

(a) Let E = (1/2,1) and F = (1/2,1] ⊂ X (we see that E F X)and let L : X × X → [0,+∞] be defined by (51). It follows from Example 8 that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

We see that (X, T) is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X. Otherwise, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M367">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M368">View MathML</a>

We note, by (51), (66) and definitions of E and F, that the condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M369">View MathML</a>

(67)

implies, in particular,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M370">View MathML</a>

(68)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M371">View MathML</a>

(69)

and, for η > 0, then the following hold

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M372">View MathML</a>

(70)

Indeed, if x, y X satisfying (67) are arbitrary and fixed, then from (51) we conclude that (67) holds only if x E and y F \ E. Hence, we get that x ∈ (1/2,1), y = 1 and d(x,y) < 1/2, which, by (66), gives (68). Of course, by (51), the equality (69) holds. Now, if η > 0, then, by (68),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M373">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M374">View MathML</a>

Thus, (70) holds.

Now, by (67)-(70), we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M375">View MathML</a>

that is, for x ∈ (1/2,1) and y = 1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M376">View MathML</a>.

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M377">View MathML</a>

Consequently, we proved that (X, T) is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for i = 2 and for defined by (51) where X = [0,1], E = (1/2,1) and F = (1/2,1] does not hold.

(b) However, by (67)-(70), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M378">View MathML</a>

that is, for x ∈ (1/2,1) and y = 1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M379">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M380">View MathML</a>

Consequently, we proved that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365">View MathML</a>-uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for i = 1 and for defined by (51) where X = [0,1], E = (1/2, 1) and F = (1/2, 1] hold.

Now, we see that, for each w0 X, the assertion (A1) holds. Indeed, we have:

Case 1. Let w0 ∈ [0,1/4). Then, for each dynamic process (wm : m ∈ {0}∪ ℕ) of (X, T) starting at w0, by (66), we have: (i) if w1 ≠ 1, then ∀m∈ℕ{wm E} and, by (51), L(w0,w1) = +∞ > 1/2 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M381">View MathML</a>

or (ii) if w1 = 1, then ∀m∈ℕ{wm = 1 ∈ F\E} and, by (51), L(w0, w1) = +∞ > 1/2 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M382">View MathML</a>

Consequently, for each w0 ∈ [0,1/4), each a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 satisfies ∀m∈ℕ{L(wm-1, wm) > 1/2}, i.e. for each w0 ∈ [0,1/4), the assertion (A1) holds.

Case 2. Let w0 ∈ [1/4,1). Then, for each dynamic process (wm : m ∈ {0} ∪ ℕ) of (X,T) starting at w0, by (66), we have that ∀m∈ℕ{wm E} and, by (51),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M383">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M384">View MathML</a>

Consequently, for each w0 ∈ [1/4,1), each a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X,T) starting at w0 satisfies ∀m∈ℕ{L(wm-1, wm) > 1/2}, i.e. for each w0 ∈ [1/4,1), the assertion (A1) holds.

Case 3. Let w0 = 1. Then, for a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0, by (66), we have that ∀m∈ℕ{wm = 1 ∈ F\E} and, by (51),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M385">View MathML</a>

Consequently, if w0 = 1, a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 satisfies ∀m∈ℕ{L(wm-1, wm) > 1/2}, i.e. for w0 = 1, the assertion (A1) holds.

Remark 7 Let us observe that, for each w0 X, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) starting at w0 such that limm→∞ wm = 1, limm→∞ L(wm, 1) = limm→∞ d(wm, 1) = 0 and 1 ∈ Fix(T). However, assertion (A2) does not hold since from Cases 1-3 it follows that, for each w0 X, each dynamic process (wm : m ∈ {0} ∪ ℕ) starting at w0 is not -Cauchy.

Answer 2 We show that there exists -family on X such that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X and, for each w0 X, the assertion (A2) holds.

Let E = [1/2,1] ⊂ X and let L : X × X → [0, +∞] be defined by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M386">View MathML</a>

(71)

It follows, from Example 6, that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

We see that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X, i.e.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M387">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M388">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M389">View MathML</a>

Indeed, first, we see that, by (66) and (71),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M390">View MathML</a>

(72)

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M391">View MathML</a>

(73)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M392">View MathML</a>

(74)

and, for η > 0,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M393">View MathML</a>

(75)

Indeed, if x, y X satisfying (72) are arbitrary and fixed, then from (71) we conclude that (72) holds only if x, y E. Hence, we get that x, y ∈ [1/2,1] and d(x, y) < 1/2, which, by (66), gives (73). Of course, by (49), (74) holds. Now, if η > 0, then, by (73),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M394">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M395">View MathML</a>

Thus, (75) holds.

Now, by (72)-(75), we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M396">View MathML</a>

that is, for x, y ∈ [1/2,1], we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M397">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M398">View MathML</a>

Consequently, we proved that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X.

This gives that the assumptions of Theorem 8(I) for defined by (71) and for i = 2 hold.

We see that, for each w0 X, the assertion (A2) holds. Indeed, we have that: 1 ∈ Fix(T); for each w0 X and for each dynamic processes (wm : m ∈ {0} ∪ ℕ) of (X,T) starting at w0, by (66), we have that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M399">View MathML</a>, so limm→∞ L(wm, 1) = limm→∞ d(wm, 1) = 0 and limn→∞ supm>n L(wn,wm) = limn→∞ supm>n d(wn, wm) = 0. Therefore, the sequence (wm : m ∈ {0} ∪ ℕ) is -Cauchy.

Remark 8 We see that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M400">View MathML</a>. Indeed, by (71), L(1,1) = d(1,1) = 0 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M401">View MathML</a>

Answer 3 We show that there exists -family on X such that: (i) (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M365">View MathML</a>-uniformly locally contractive on X; (ii) There exists w X such that End(T) = {w}; (iii) For each w0 X\End(T) the assertion (A2) holds; and (iv) For w0 = w the assertion (A1) holds (since L(w, w) = 3 where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a>).

Define E = (1/2,1) and F = (1/2,1] ⊂ X (we see that E F X) and let L:X × X→[0, +∞] be defined by (53). It follows from Example 10 that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X.

First, we show that (X,T) is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X. Otherwise,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M402">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M403">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M404">View MathML</a>

Let us notice that, by (53) and (66),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M405">View MathML</a>

(76)

implies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M406">View MathML</a>

(77)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M407">View MathML</a>

(78)

and, for η > 0,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M408">View MathML</a>

(79)

Indeed, if x, y X satisfying (76) are arbitrary and fixed, then from (53) we conclude that (76) holds only in two following cases: (i) (x, y) ∈ E × (F\E) or (ii) (x,y) ∈ E × E.

Now we see that, in particular, if x E and y F \ E, then we get that x ∈ (1/2,1), y = 1 and d(x, y) < 1/2, which, by (66), gives (77). Of course, by (53), (78) holds. Now, if η > 0, then, by (77),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M409">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M410">View MathML</a>

Thus, (79) holds. Further, by (76)-(79), we see that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M411">View MathML</a>

that is, for x ∈ (1/2,1) and y = 1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M412">View MathML</a>.

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M413">View MathML</a>

Consequently, we proved that (X,T) is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M364">View MathML</a>-uniformly locally contractive on X. This gives that the assumptions of Theorem 8(I) for such and for i = 2 do not hold.

Next, to prove that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>-uniformly locally contractive on X, we assume that x, y X satisfying (76) are arbitrary and fixed. Then, by (53), we conclude that (76) holds only in the following two cases:

Case 1. Let x E and let y F\E. By (76)-(79), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M414">View MathML</a>

that is, for x ∈ (1/2,1) and y = 1, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M379">View MathML</a>.

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M415">View MathML</a>

Case 2. Let x, y E. By (66), T(x) = {x/2 + 1/2}, T(y) = {y/2 + 1/2}, and, consequently, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M416">View MathML</a>

that is, for x, y ∈ (1/2,1), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M412">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M417">View MathML</a>

From Cases 1 and 2 it follows that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>-uniformly locally contractive on X.

It is clear that the assumptions of Theorem 8(I) for such and for i = 1 hold.

Now we prove that if w0 ∈ [0,1), then the assertion (A2) holds and if w0 = 1 then the assertion (A1) holds. Indeed, we have the following three cases:

Case 1. Let w0 ∈ [0,1/4). Then, by (66), there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 of the form: w1 ≠ 1 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M418">View MathML</a>. Then, by (53), L(w0, w1) = +∞ and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M419">View MathML</a>. Consequently, a dynamic process (wm : m ∈ {0} ∪ ℕ) is -Cauchy on X, limm→∞ wm = 1 and 1 ∈ Fix(T), i.e. for each w0 ∈ [0,1/4), the assertion (A2) holds.

Case 2. Let w0 ∈ [1/4,1). Then, for each a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X,T) starting at w0, by (66), we have that ∀m∈ℕ{wm E} and, by (53),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M420">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M419">View MathML</a>. Consequently, for each w0 ∈ [1/4,1), each a dynamic process (wm : m ∈ {0}∪ ℕ) of (X, T) starting at w0 is -Cauchy on X, limm →∞ wm = 1 and 1 ∈ Fix(T), i.e., for each w0 ∈ [1/4,1), the assertion (A2) holds.

Case 3. Let w0 = 1. Then, for a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0, by (66), we have that ∀m∈ℕ{wm = 1 ∈ F\E} and, by (53), ∀m∈ℕ{L(wm-1, wm) = d(wm-1, wm) + 3 > 1/2}. Consequently, if w0 = 1, a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 satisfies ∀m∈ℕ{L(wm-1, wm) > 1/2}, i.e. for w0 = 1, the assertion (A1) holds.

Finally, we see that, for each w0 X, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that limm→∞ wm = 1, limm→∞ L(wm, 1) = limm→∞ d(wm, 1) = 0 and 1 ∈ Fix(T).

Remark 9 Let us point out that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M421">View MathML</a>. Indeed, by (53), L(1, 1) = d(1, 1) + 3 = 3 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M422">View MathML</a>

Examples and comparisons of our results with Banach's, Nadler's, Covitz-Nadler's and Suzuki's results

It is worth noticing that our results in metric spaces and in generalized metric spaces include Banach's [3], Nadler's [[4], Th. 5], Covitz-Nadler's [[5], Theorem 1] and Suzuki's [[10], Theorem 3.7] results.

Clearly, it is not otherwise. More precisely: (a) In Example 14 we construct -complete generalized metric space (X, ), a -family on X satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M281">View MathML</a> and a set-valued dynamic system (X, T) which is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M423">View MathML</a>-uniformly locally contractive on X and next we show that the assertion (A2) holds; (b) In Example 15 we show that, for each ε ∈ (0, ∞), λ ∈ [0, 1) and i ∈ {1, 2}, the set-valued dynamic system (X, T) defined in Example 14 is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M424">View MathML</a>-uniformly locally contractive on X and thus we cannot use Theorems 1, 2 and 4-7; (c) In Example 16 we construct a complete metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M425">View MathML</a> which is -family on X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>-uniformly locally contractive set-valued dynamic system (X, T) such that, for each w0 X, the assertion (A2) holds and, additionally, L(w, w) > 0 for w Fix(T) which gives that our theorems are different from Theorem 7.

Example 14 Let Z and q be as in Example 3. Let X = Z ⋂ [0,9] and let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426">View MathML</a> where d = q|[0,9]. Then (X, ) is a -complete generalized metric space. Let F = {1, 7} and let (X, T) be given by the formula

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M427">View MathML</a>

we see that T : X C(X). Let E = {0, 1, 2} ∪ [4,5] ∪ {6, 8} and let L be of the form

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M428">View MathML</a>

By Example 6, the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X. By the similar reasoning as in Example 11, we show that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M423">View MathML</a>-uniformly locally contractive on X. We see that for each w0 X there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that limm→∞ wm = 2 and 2 ∈ Fix(T).

Remark 10 We notice that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M429">View MathML</a>.

Example 15 Let X, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426">View MathML</a> and T be such as in Example 14. We show that, for any ε ∈ (0, ∞), λ ∈ [0, 1) and i ∈ {1, 2}, T is not <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M424">View MathML</a>-uniformly locally contractive on X.

Otherwise, there exist ε0 ∈ (0, ∞), λ0 ∈ [0, 1) and i ∈ {1, 2} such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M430">View MathML</a>

(80)

We consider the following three cases:

Case 1. If ε0 = 1, then, in particular, for x0 = 1 and y0 = 1/2, since x0, y0 ∈ [0,1], by formula (46), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M431">View MathML</a>

However, T(x0) = {4, 5}, T(y0) = {1, 2}, and, by (46),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M432">View MathML</a>

Hence

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M433">View MathML</a>

Consequently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M434">View MathML</a>

and (80) gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M435">View MathML</a>

This leads to a contradiction.

Case 2. If ε0 ∈ (1, ∞), then by a similar reasoning as in Case 1 we prove that (80) does not hold.

Case 3. If ε0 ∈ (0, 1), then, in particular, for x0 = 1 and y0 = ((1 - ε0)/2), we obtain that x0, y0 ∈ [0,1] and by a similar reasoning as in Case 1 we prove that (80) does not hold.

Example 16 Let X = [0,1] and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M426">View MathML</a> where d : X × X → [0, ∞) is defined by the formula d(x, y) = |x - y|, x, y X. Then (X, ) is a complete metric space. Let E = [1/2, 1) and F = [1/2, 1] ⊂ X (we see that E F X) and let L : X × X → [0, +∞] be defined by (53). It follows from Example 10 that the family <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M244">View MathML</a> is -family on X. Let (X, T) be given by the formula:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M436">View MathML</a>

(81)

First, we show that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>-uniformly locally contractive on X. Assume that x, y X satisfying L(x, y) < 1/2 are arbitrary and fixed. Then from (53) we conclude that L(x, y) < 1/2 implies (x, y) ∈ E × (F \ E) or (x, y) ∈ E × E. Consequently, the following two cases hold:

Case 1. Let x E and y F\E. Then, by (81) we get: T(x) = {x/2+1/2};

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M437">View MathML</a>

that is, for x ∈ [1/2, 1) and y = 1, we have 1/2 - x/2 ≤ 1/4 ≤ x/2 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M438">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M439">View MathML</a>

Case 2. Let x, y E. Then, by (81), T(x) = {x/2 + 1/2}, T(y) = {y/2 + 1/2} and, consequently, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M440">View MathML</a>

that is, for x, y ∈ (1/2, 1), we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M441">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M442">View MathML</a>

Consequently, we proved that (X, T) is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M366">View MathML</a>-uniformly locally contractive on X. We also see that all assumptions of Theorem 8(I) for this and for i = 1 hold.

Now, we show that, for each w0 X, the assertion (A2) holds. Indeed, we have the following three cases:

Case 1. Let w0 ∈ [0, 1/4). Then, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 of the form: w1 ≠ 1, and ∀m∈ℕ{wm ∈ [1/2, 1) = E}. Then, by (53), L(w0, w1) = +∞ and ∀m≥2{L(wm-1, wm) = d(wm-1, wm)}. Consequently, a dynamic process (wm : m ∈ {0} ∪ ℕ) is -Cauchy on X, limm→∞ wm = 1 and 1 ∈ Fix(T), i.e. for each w0 ∈ [0, 1/4), the assertion (A2) holds.

Case 2. Let w0 ∈ [1/4, 1). Then, for each a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0, by (81), we have that ∀m∈ℕ{wm E} and, by (53),

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M443">View MathML</a>

and ∀m≥2{L(wm-1, wm) = d(wm-1, wm)}. Consequently, for each w0 ∈ [1/4, 1), each a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0 is -Cauchy on X, limm→∞ wm = 1 and 1 ∈ Fix(T), i.e. for each w0 ∈ [1/4, 1), the assertion (A2) holds.

Case 3. Let w0 = 1. Then, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) of (X, T) starting at w0, of the form: w0 = 1, w1 = 1/2, ∀m≥2{wm E}, and, by (53), L(w0, w1) = d(w0, w1)+4 and ∀m≥2{L(wm-1, wm) = d(wm-1, wm)}. Consequently, this dynamic process (wm : m ∈ {0} ∪ ℕ) is -Cauchy on X, limm→∞ wm = 1 and 1 ∈ Fix(T), i.e. for w0 = 1, the assertion (A2) holds.

Remark 11 One can also notice that L(1, 1) = 3 > 0 and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M444">View MathML</a>. Indeed, we have L(1, 1) = d(1, 1) + 3 = 3 and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/104/mathml/M445">View MathML</a>

Concluding remarks

The Caristi [41] and Ekeland [42] results can be read, respectively, as follows.

Theorem 10 [41]Let (X, d) be a complete metric space. Let T : X X be a single-valued map. Let φ : X → (-∞, +∞] be a map which is proper lower semicontinuous and bounded from below; we say that a map φ : X → (-∞, +∞] is proper if its effective domain, dom(φ) = {x : φ(x) < +∞}, is nonempty. Assume xX{d(x, T(x)) ≤ φ(x) - φ(T(x))}. Then T has a fixed point w in X, i.e. w = T(w).

Theorem 11 [42]Let (X, d) be a complete metric space. Let φ : X → (-∞, +∞] be a proper lower semicontinuous and bounded from below. Then, for every ε > 0 and for every x0 dom(φ), there exists w X such that: (i) φ(w)+εd(x0, w) ≤ φ(x0); and (ii) xX\{w}{φ(w) < φ(x) +εd(x, w)}.

The Banach [3], Nadler [[4], Th. 5], Caristi [41], and Ekeland [42] results have extensive applications in many fields of mathematics and applied mathematics, they have been extended in many different directions and a number of authors have found their simpler proofs. Caristi's and Nadler's results yield Banach's result and Caristi's and Ekeland's results are equivalent. Jachymski [[14], Theorem 5], using a similar idea as in Takahashi [13], proved that Caristi's result yields Nadler's result.

Regarding this, we raise a question:

Question 3 Is it possible to find some analogons of Caristi's and Ekeland's theorems in generalized uniform spaces (or in generalized locally convex spaces or in generalized metric spaces) with generalized pseudodistances, and without lower semicontinuity assumptions as in [30]?

It is also natural to ask the following question:

Question 4 What additional assumptions in Theorems 8 and 9 (and thus also in Theorems 2 and 4-7) guarantee the uniqueness of fixed points?

Competing interests

The authors declare that they have no competing interests.

Authors' Contributions

The authors have equitably contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.

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