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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

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

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 , 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

(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

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.,

(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 , -index set, be a family of disjoint metric spaces. If and, for any x, y X,

then (X, d) is a generalized metric space. Moreover, if for each , (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:

Therefore, X is decomposed uniquely into (disjoint) equivalence classes , 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 be the canonical decomposition and let . Then: (I) For each , (Xβ, dβ) is a metric space; (II) For any , with β1 β2, d(x, y) = +∞ for any and ; and (III) (X, d) is a generalized complete metric space iff, for each , (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),

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.

Then the following alternative holds: either

(A) ; or

(B) .

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.,

(3)

Then the following alternative holds: either

(A) ; or

(B) .

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.

(4)

Then .

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.

(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.

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)?

### Generalized uniform spaces (X, ) and the class 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

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

() ;

() ; and

If 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

() .

(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 . We say that (xm : m ∈ ℕ) is -convergent in X if there is an x X such that (, 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

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

() ;

() If 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

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

() If 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

() For any sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X such that

(6)

and

(7)

the following holds

(8)

(b) Let be a class defined as follows

Remark 2 Let (X, ) be a generalized uniform space. (i) since . (ii) ; see Sections 10-13.

Definition 6 Let (X, ) be a generalized uniform space, let and let (xm : m ∈ ℕ) be a sequence in X.

(a) We say that (xm : m ∈ ℕ) is -Cauchy in X if .

(b) We say that (xm : m ∈ ℕ) is -convergent in X if there exists x X such that .

(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 . (i) If , then is a -family on X; examples of 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 to be

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 .

(I) If x y, x, y X, then .

(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 . Then, , since, by using (), it follows that . Defining the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) in X by xm = x and ym = y for m ∈ ℕ, and observing that , this implies that (6) and (7) for these sequences hold. Then, by (), (8) holds, so it is . On the other hand, is separating, so, since x y, it is . This leads to a contradiction.

(II) Since , by Definition 6(c), this proves the existence of x X such that . We can apply () to sequences (xm : m ∈ ℕ) and (ym = x/ : m ∈ ℕ) and then we find that . The uniqueness of the point of x follows from the fact that is separating. □

### Partial quasiordered space

Proposition 2 Let be a set of elements defined by the formula

and let . The relation on defined by

is a partial quasiordered on and the pair is a partial quasiordered space.

Proof. For all the condition holds. For all , the conditions and imply . For all , the conditions and imply Θ = Ω. □

Notation. The following notation is fixed throughout the article:

;

;

;

.

In the sequel, if , then will stand for and Θ ≠ Ω.

Definition 7 Let be a nonempty subset of . We say that is a infimum of if the following two conditions hold:

() ;

() .

Example 2 Let and let . If then and does not exist since (3, 5, 7) and (4, 1, 8) are not comparable. If then and .

### -distances on 2X, i ∈ {1, 2}

Definition 8 Let (X, ) be a Hausdorff generalized uniform space and let .

(a) For C ∈ 2X and , let us denote

(9)

(b) For A, B ∈ 2X let us denote:

(10)

(11)

(c) Let i ∈ {1, 2}. The map of the form

A, B ∈ 2X, is called a -distance on 2X generated by (-distance on 2X, for short).

Remark 4 For each A, B ∈ 2X, .

### -uniformly locally contractive and -contractive set-valued dynamic systems (X, T), i ∈ {1, 2}

Definition 9 Let (X, ) be a Hausdorff generalized uniform space, let and let i ∈ {1,2}.

(a) Let be a -distance on 2X and let and be such that . We say that a set-valued dynamic system (X, T), T : X → 2X, is -uniformly locally contractive on X if

(12)

(b) Let be a -distance on 2X and let be such that . We say that a set-valued dynamic system (X, T), T : X → 2X, is - contractive on X if

(13)

Remark 5 Let (X, ) be a Hausdorff generalized uniform space, let and let and be such that .

(i) If (X, T), T : X → 2X, is -uniformly locally contractive on X then it is -uniformly locally contractive on X.

(ii) If (X, T), T : X → 2X, is -contractive on X then it is -contractive on X.

(iii) Let i ∈ {1, 2}. If (X, T), T : X → 2X, is -contractive on X then it is -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, and one of the following properties holds:

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

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

Let i ∈ {1, 2}, let be a -distance on 2X and assume that a set-valued dynamic system (X, T), T : X → 2X, has the property

(C) {limm→∞ wm = w T is closed at w}.

(I) If and satisfy and (X, T) is -uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(A1) ; or

(A2) {w Fix(T) ⋀ limm→∞ wm = w ⋀ (wm : m ∈ {0} ∪ ℕ) is -Cauchy}.

(II) If satisfies and (X, T) is -contractive on X then, for each w0 X, the following alternative holds: either

(B1) ; or

(B2) {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 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 is a -distance on C(X).

(I) If and satisfy and if a set-valued dynamic system (X, T) satisfying T : X C(X) is -uniformly locally contractive on X then, for each w0 X, the following alternative holds: either

(F1) ; or

(F2) .

(II) If satisfies and a set-valued dynamic system (X, T) satisfying T : X C(X) is -contractive on X then, for each w0 X, the following alternative holds: either

(G1) ; or

(G2) .

### 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,

(14)

Then there exists which is -Cauchy sequence on X; that is,

(15)

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

(16)

It follows from (16) and Definition 8(c), that there exists and

(17)

From this, denoting , we deduce that . Consequently, by (), there exists

(18)

such that which implies

(19)

If i = 1, then we note that, by (18), (9), and (10), . Clearly, . Thus, and the conclusion

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

If i = 2, then we also note that, by (18), (9) and (11), and . Clearly, . Thus, and the conclusion

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

This proves

(20)

Since, by (20), , it follows, using (12) and (20), that

That is,

(21)

Denoting , we see that condition (21) implies . Hence, by (), there exists

(22)

such that . This means

(23)

Let i = 1. Clearly, by (9), (10), and (22), . Moreover, by (20), . Therefore . This, by (9), (10) and (21)-(23), implies

Let i = 2. Clearly, by (9)-(11) and (22), and . Moreover, . Therefore . This, by (9)-(11) and (21)-(23), implies

That is,

(24)

By (24), we have and, using (12) and (24), we get

This means

(25)

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

(26)

It is clear that (26) implies that where and . 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

(27)

Applying (15), (27), and () (where (xm = wm : m ∈ ℕ) and (ym = w : m ∈ ℕ)), we find that

(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 and, additionally, this sequence is a -Cauchy sequence on X, i.e.

(29)

We prove that is a -Cauchy sequence on X, i.e. that

(30)

Indeed, by (29), we claim that

Hence, in particular,

(31)

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

(32)

then (31) implies that

(33)

Therefore, by (29), (33), and (), we get

(34)

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

(35)

and

(36)

Let now 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

Hence, we conclude that

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 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

This implies that there exists the family such that and . Consequently,

Clearly, (X, T) is -uniformly locally contractive on X since (X, T) is -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

(37)

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

(38)

Consequently, the sequence (wm : m ∈ {0} ∪ ℕ) such that and is a dynamic process of T starting at w0 and, additionally, this sequence is a -Cauchy sequence on X, i.e.

(39)

It is clear that (39) implies

(40)

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

(41)

If, for each , x X and B Cl(X), we denote

(42)

and

(43)

then (42) and (40) implies

(44)

Let m ∈ ℕ, m > m0, and be arbitrary and fixed and let

here m0 is defined by (37). Then, by (9)-(11) and definition of , we get that and . Hence, in particular, if v T(wm) is arbitrary and fixed, then

This implies

(45)

Now, by (), (remember that ), for each u T(w) and v T(wm), we have

Hence, by (42) and (), for each v T(wm), it follows

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

Hence, by (41) and (44), . However, this property of w, i.e.

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

This implies that there exists the family such that and . Consequently,

Clearly, (X, T) is - uniformly locally contractive on X since (X, T) is -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 of -families of generalized pseu-dodistances on (X, )".

Definition 12 Let X be a vector space over ℝ.

(i) The family

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

() ;

() ; and

() If 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

() .

(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 where dα(x,y) = pα(x - y), (x,y) ∈ X × X, , 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 and define q : Z × Z → [0, +∞] by the formula

(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 of calibrations cn,n ∈ ℕ, defined as follows:

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:

Let and define pn:P × P → [0, +∞], n ∈ ℕ, as follows

(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

In this section we describe some elements of the class .

Example 5 Let (X, ) be a Hausdorff generalized uniform space where , -index set, is a -family. Let the set E X, containing at least two different points, be arbitrary and fixed and, for each , let La : X × X → [0, +∞] be defined by the formula:

(48)

We show that the family is -family on (X, ).

First, we observe that the condition () holds. Indeed, let 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 , 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 () holds.

To prove that () 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

By definition of , this implies that

Therefore, we obtain that

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Hence we conclude that the condition is satisfied.

Example 6 Let (X, ) be a generalized metric space where 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)):

(49)

By Example 5, the family is -family on X.

Example 7 Let (X, ) be a Hausdorff generalized uniform space where , -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 < +∞, , and let, for each , Lα : X × X → [0, +∞] be defined by the formula:

(50)

We show that the family is -family on X.

First, we observe that the condition ( holds. Indeed, let 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 , we get

If z F \ E, then Lα(x, z) = dα(x, z) + bα; Lα(z, y) = dα(z, y) + bα; and consequently, by , we get

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 , we get

If z F \ E, then Lα(x, z) = dα(x, z) + bα; Lα(z, y) = dα(z, y) + cα; and consequently, by , we get

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 , we get

If z F\E, then Lα(x, z) = dα(x, z); Lα(z, y) = dα(z, y) + bα; and consequently, by , we get

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 , we get

If z F\E, then Lα(x, z) = dα(x, z); Lα(z, y) = dα(z, y) + cα; and consequently, since aα < cα, by , we get

Consequently, the condition holds.

To prove that 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

By definition of , this implies that

As a consequence of this, we get

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Therefore, the property ( holds.

It is worth noticing that, there exists x, y X such that, for each , Lα(x, y) = Lα(y, x) does not hold. Indeed, if x E and y F \ E, then

Example 8 Let X, be a generalized metric space where 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:

(51)

By Example 7, the family is -family on X.

Example 9 Let (X, ) be a Hausdorff generalized uniform space where , -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 < +∞, , and let, for each , Lα : X × X → [0, +∞] be defined by the formula:

(52)

We show that the family is -family on X.

First, we observe that the condition ( holds. Indeed, let 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

and consequently, since bα < cα, by , we get

If z F \ E, then

and consequently, by , we get

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

and consequently, by , we get

If z F \ E, then

and consequently, by , we get

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

and consequently, by , we get

If z F \ E, then

and consequently, by , we get

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 , we get

If z F \ E, then

and consequently, by , we get

Consequently, the condition ( holds.

To prove that ( 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

By definition of , this implies that

As a consequence of this, we get

This means that the sequences (xm : m ∈ ℕ) and (ym : m ∈ ℕ) satisfy (8). Therefore, the property ( holds.

It is worth noticing that, there exists x, y X such that, for each , Lα(x, y) = Lα(y, x) does not hold. Indeed, if x E and y F \ E, then

Example 10 Let (X, ) be a generalized metric space where 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:

(53)

By Example 9, the family 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 -uniformly locally contractive on X where and .

Example 11 Let P and {pn : P × P → [0,+∞], n ∈ ℕ} be as in Example 4. Let X = P ⋂ [0,9]and let , dn : X × X → [0, +∞], n ∈ ℕ, where, for each n ∈ ℕ, we define . 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 .

Let and let a set-valued dynamic system (X, T) be given by the formula

(54)

Let and let be a family of the maps given by the formula:

(55)

By Example 4, the family is -family on X.

Now, we show that, for and , (X, T) is -uniformly locally contractive on X, i.e. that

(56)

where

(57)

(58)

(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 , then, since , by (55), for each n ∈ ℕ, we have

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

If , then, since , by (55), we obtain that 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 . Then, by definition of F, or . But, , 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 . Then, since , by (55) and (47), we get

(60)

and, consequently, for each n ∈ ℕ, the inequality Ln(x, y) < 1/2 holds. In virtue ofthis, we show that the inequalities in (56) hold. With this aim, we see that:

(3i) By (54), we have ;

(3ii) Next, if , then, by (3i),

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

(3iv) Therefore, by (3iii), we have ;

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

Hence, by (60), we conclude that

holds.

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

(4i) By (54), we have ;

(4ii) Next, if , then

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

(4iv) Therefore, by (4iii), ;

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

Consequently, by (60),

We proved that (X, T) is -uniformly locally contractive on X. We see also that (C) holds.

Finally, we see that . Hence, for each w0 X, there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that: (i) ; (ii) ; and (iii) .

The following example illustrates the Theorem 8(I) in the case when (X, ) is -sequentially complete for some , , but not -sequentially complete and (X, T) is -uniformly locally contractive on X.

Example 12 Let X and {pn : P × P → [0, +∞], n ∈ ℕ} be as in Example 4. Let and let , dk : X × X → [0,∞], k ∈ ℕ, where, for each k ∈ ℕ, we define . 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: , m ∈ ℕ. Of course, the sequence (xm : m ∈ ℕ) is -Cauchy sequence on X. Indeed, we have which implies that

Consequently,

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

Let and let be a family of the maps given by the formula:

By (47), this gives

(61)

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

By Example 4, the familly 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.

This implies that

(62)

Hence, in particular, we conclude that

(63)

Now, (63) and (61) gives that

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

(64)

The property (61) and (64) gives that

(65)

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

Case 1. If or or or , then in each of these situations the sequence, as a constant sequence, is, by (61), -convergent to , respectively.

Case 2. If , then

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

This gives that (xm : m ∈ ℕ) is a -Cauchy sequence in X, so also the sequence is a -Cauchy sequence in [4,5]. Since [4,5]is a -complete uniform space, so there exists x X such that

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

Now, let and let (X, T) be given by the formula

By the same reasoning as in Example 11, we obtain that, for and is -uniformly locally contractive on X, for each w0 X there exists a dynamic process (wm : m ∈ {0} ∪ ℕ) such that and .

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 , d:X × X→ [0,∞), d(x, y) = |x-y|, x, y X. Let a dynamic system (X, T) be given by the formula:

(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 -uniformly locally contractive on X; and (b) (X, T) is -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 is -family on X.

We see that (X, T) is not -uniformly locally contractive on X. Otherwise, , where

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

(67)

implies, in particular,

(68)

(69)

and, for η > 0, then the following hold

(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),

and

Thus, (70) holds.

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

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

Consequently, we proved that (X, T) is not -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

that is, for x ∈ (1/2,1) and y = 1, we have . Therefore,

Consequently, we proved that (X, T) is -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

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

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),

and

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),

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 -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:

(71)

It follows, from Example 6, that the family is -family on X.

We see that (X, T) is -uniformly locally contractive on X, i.e.

where

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

(72)

implies

(73)

(74)

and, for η > 0,

(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),

and

Thus, (75) holds.

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

that is, for x, y ∈ [1/2,1], we have . Therefore,

Consequently, we proved that (X, T) is -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 , 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 . Indeed, by (71), L(1,1) = d(1,1) = 0 and

Answer 3 We show that there exists -family on X such that: (i) (X, T) is -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 ).

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 is -family on X.

First, we show that (X,T) is not -uniformly locally contractive on X. Otherwise,

where

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

(76)

implies

(77)

(78)

and, for η > 0,

(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),

and

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

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

Consequently, we proved that (X,T) is not -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 -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

that is, for x ∈ (1/2,1) and y = 1, we have .

Therefore,

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

that is, for x, y ∈ (1/2,1), we have . Therefore,

From Cases 1 and 2 it follows that (X, T) is -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 . Then, by (53), L(w0, w1) = +∞ and . 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),

and . 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 . Indeed, by (53), L(1, 1) = d(1, 1) + 3 = 3 and

### 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 and a set-valued dynamic system (X, T) which is -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 -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 which is -family on X and -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 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

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

By Example 6, the family is -family on X. By the similar reasoning as in Example 11, we show that (X, T) is -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 .

Example 15 Let X, 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 -uniformly locally contractive on X.

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

(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

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

Hence

Consequently,

and (80) gives

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 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 is -family on X. Let (X, T) be given by the formula:

(81)

First, we show that (X, T) is -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};

that is, for x ∈ [1/2, 1) and y = 1, we have 1/2 - x/2 ≤ 1/4 ≤ x/2 and . Therefore,

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

that is, for x, y ∈ (1/2, 1), we have . Therefore,

Consequently, we proved that (X, T) is -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),

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 . Indeed, we have L(1, 1) = d(1, 1) + 3 = 3 and

### 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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