Abstract
Motivated by classical Banach contraction principle, Nadler investigated setvalued contractions with respect to Hausdorff distances h in complete metric spaces, Covitz and Nadler (Jr.) investigated setvalued maps which are uniformly locally contractive or contractive with respect to generalized Hausdorff distances H in complete generalized metric spaces and Suzuki investigated setvalued maps which are contractive with respect to distances Q_{p }in complete metric spaces with τdistances 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 setvalued 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 setvalued 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 wellknown 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 spaceIntroduction
Let 2^{X }denotes the family of all nonempty subsets of a space X. Recall that a setvalued dynamic system is defined as a pair (X, T), where X is a certain space and T is a setvalued map T : X → 2^{X}; in particular, a setvalued dynamic system includes the usual dynamic system where T is a singlevalued map.
Let (X, T) be a setvalued 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 w^{0 }∈ X or a motion of the system (X, T) at w^{0 }is a sequence (w^{m }: m ∈ {0} ∪ ℕ) defined by w^{m }∈ T(w^{m1}) for m ∈ ℕ (see, [1,2]).
If (X, T) is a dynamic system and w^{0 }∈ X then, by , we denote the set of all dynamic processes of the system (X, T) starting at w^{0}.
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 singlevalued map satisfying the condition
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) = inf_{c∈C }d(x, c), x ∈ X, C ∈ CB(X).
A natural question to ask is whether the singlevalued dynamic system in this principle can be replaced by the setvalued 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 setvalued dynamic system (X, T) satisfying T : X → CB(X) is (h, λ)contractive, i.e.,
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 setvalued 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 : X^{2 }→ [0, ∞] satisfies: (a) ∀_{x,y∈X }{d(x, y) = 0 iff x = y}; (b) ∀_{x,y∈X }{d(x, y) = d(y, x); (c) ∀_{x,y,z∈X }{[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 , (X_{β}, d_{β}) 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 , (X_{β}, d_{β}) is a metric space; (II) For any , with β_{1 }≠ β_{2}, d(x, y) = +∞ for any and ; and (III) (X, d) is a generalized complete metric space iff, for each , (X_{β}, d_{β}) 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 (x_{m }: m ∈ ℕ) in Y which is dconvergent to x.
(b) The class of all nonempty closed subsets of X is denoted by C(X), i.e. C(X) = {Y : Y ∈ 2^{X }∧ 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 : ∃_{e∈E }{d(x, e) < ε}}.
Theorem 4 [[5], Theorem 1] Let (X, d) be a generalized complete metric space and let w^{0 }∈ X. Assume that a setvalued dynamic system (X, T) satisfying T : X → C(X) is (H, ε, λ)uniformly locally contractive, i.e.
Then the following alternative holds: either
It is not hard to see that each (H, λ)contractive setvalued 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 w^{0 }∈ X. Assume that the setvalued dynamic system (X, T) satisfying T : X → C(X) is (H, λ)contractive, i.e.,
Then the following alternative holds: either
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 w^{0 }∈ X. Assume that a setvalued dynamic system (X, T) satisfying T : X → C(X) is (h, λ)contractive, i.e.
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 setvalued 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 τdistance on X if there exists a map η : X × [0, ∞) → [0, ∞) and the following conditions hold: (S1) ∀_{x,y,z∈X }{p(x, z) ≤ p(x, y) + p(y, z)}; (S2) ∀_{x∈X }∀_{t>0}{η(x, 0) = 0 ⋀ η(x, t) ≥ t} and η is concave and continuous in its second variable; (S3) lim_{n→∞ }x_{n }= x and lim_{n→∞ }sup_{m≥n }η(z_{n}, p(z_{n}, x_{m})) = 0 imply that ∀_{w∈X }{p(w, x) ≤ lim inf_{n→∞ }p(w, x_{n})}; (S4) lim_{n→∞ }sup_{m≥n }p(x_{n}, y_{m})) = 0 and lim_{n→∞ }η(x_{n}, t_{n}) = 0 imply that lim_{n→∞ }η(y_{n}, t_{n}) = 0; and (S5) lim_{n→∞ }η(z_{n}, p(z_{n}, x_{n})) = 0 and lim_{n→∞ }η(z_{n}, p(z_{n}, y_{n})) = 0 imply that lim_{n→∞ }d(x_{n}, y_{n}) = 0.
Theorem 7 [[10], Theorem 3.7] Let (X, d) be a complete metric space and let p be a τdistance on X. Let a setvalued dynamic system (X, T) satisfying T : X → C(X) be (Q_{p}, λ)contractive, i.e.
where Q_{p}(A, B) = sup_{a∈A }inf_{b∈B }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 CovitzNadler's Theorem 6. Indeed, first we see that each metric d is τdistance (cf. [11]) and next we see that each (h, λ)contractive setvalued dynamic system (X, T) satisfying T : X → C(X) is (Q_{d}, λ)contractive; in fact, Q_{d }≤ h on C(X) (cf. [12]). Moreover, there exist (Q_{d}, 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 setvalued 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], AlHomidan et al. [24], Latif and AlMezel [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 setvalued 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 [329], 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 47, 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), Q_{p}) 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), Q_{p}), respectively, are continuous; (O3) By Theorems 1, 2 and 47, for each w ∈ Fix(T) the following equalities d(w, w) = 0, h(T(w), T(w)) = 0, H(T(w), T(w)) = 0, Q_{p}(T(w), T(w)) = 0 and p(w, w) = 0 hold, respectively; (O4) The distances h, H, and Q_{p }are defined only on the spaces CB(X) or C(X), respectively.
Also, let us observe that in [3036] we studied some families of generalized pseudodistances in uniform spaces and generalized quasipseudodistances in quasigauge spaces which generalize: metrics, distances of Tataru [37], wdistances of Kada et al. [38], τ distances of Suzuki [11] and τfunctions 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 Q_{p}, and setvalued 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 47 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 Q_{p }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 2^{X}, 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 (index set) of generalized pseudometrics on a nonempty set X and the generalized uniform spaces (X, ) are introduced, the classes 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 is defined (see Section "Partial quasiordered space ") and, using , distances on 2^{X }(i ∈ {1, 2}) with respect to families are introduced (see Section "distances on 2^{X}, i ∈ {1,2}"). Also, we introduce the definitions of uniformly locally contractive and contractive setvalued dynamic systems (X, T) (i ∈ {1, 2}) satisfying T : X → 2^{X }(see Section "uniformly locally contractive and contractive setvalued dynamic systems (X, T), i ∈ {1, 2}") and, for w^{0 }∈ X, we establish the conditions guaranteeing the convergence of dynamic processes and the existence of fixed points for such contractions and, additionally, a special case when T : X → C(X) and is studied (see Sections 68). 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 "). By generality of spaces and families, our results, in particular, include and essentially generalize Theorems 1, 2 and 47. 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 1013). 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 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:
If and x, y, z ∈ X and if d_{α}(x, z) and d_{α}(y, z) are finite, then d_{α}(x, y) is finite and d_{α}(x, y) ≤ d_{α}(x, z) + d_{α}(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 (x_{m }: m ∈ ℕ) be a sequence in X. We say that (x_{m }: m ∈ ℕ) is Cauchy sequence in X if . We say that (x_{m }: 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 q_{α}(x, z) and q_{α}(z, y) are finite, then q_{α}(x, y) is finite and q_{α}(x, y) ≤ q_{α}(x, z) + q_{α}(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 L_{α}(x, z) and L_{α}(z, y) are finite, then L_{α}(x, y) is finite and L_{α}(x, y) ≤ L_{α}(x, z) + L_{α}(z, y); and
() For any sequences (x_{m }: m ∈ ℕ) and (y_{m }: m ∈ ℕ) in X such that
and
the following holds
(b) Let be a class defined as follows
Remark 2 Let (X, ) be a generalized uniform space. (i) since . (ii) ; see Sections 1013.
Definition 6 Let (X, ) be a generalized uniform space, let and let (x_{m }: m ∈ ℕ) be a sequence in X.
(a) We say that (x_{m }: m ∈ ℕ) is Cauchy in X if .
(b) We say that (x_{m }: 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 (x_{m }: 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 (x_{m }: m ∈ ℕ) is Cauchy sequence in X, then (x_{m }: 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 (x_{m }: m ∈ ℕ) and (y_{m }: m ∈ ℕ) in X by x_{m }= x and y_{m }= 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 (x_{m }: m ∈ ℕ) and (y_{m }= 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 2^{X}, i ∈ {1, 2}
uniformly locally contractive and contractive setvalued 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 2^{X }and let and be such that . We say that a setvalued dynamic system (X, T), T : X → 2^{X}, is uniformly locally contractive on X if
(b) Let be a distance on 2^{X }and let be such that . We say that a setvalued dynamic system (X, T), T : X → 2^{X}, is  contractive on X if
Remark 5 Let (X, ) be a Hausdorff generalized uniform space, let and let and be such that .
(i) If (X, T), T : X → 2^{X}, is uniformly locally contractive on X then it is uniformly locally contractive on X.
(ii) If (X, T), T : X → 2^{X}, is contractive on X then it is contractive on X.
(iii) Let i ∈ {1, 2}. If (X, T), T : X → 2^{X}, 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 setvalued dynamic system (X, T), T : X → 2^{X}, is closed at x if whenever (x_{m }: m ∈ ℕ) is a sequence converging to x in X and (y_{m }: m ∈ ℕ) is a sequence converging to y in X such that y_{m }∈ T(x_{m}) 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 2^{X }and assume that a setvalued dynamic system (X, T), T : X → 2^{X}, has the property
(C) {lim_{m→∞ }w^{m }= w ⇒ T is closed at w}.
(I) If and satisfy and (X, T) is uniformly locally contractive on X then, for each w^{0 }∈ X, the following alternative holds: either
(A2) {w ∈ Fix(T) ⋀ lim_{m→∞ }w^{m }= w ⋀ (w^{m }: m ∈ {0} ∪ ℕ) is Cauchy}.
(II) If satisfies and (X, T) is contractive on X then, for each w^{0 }∈ X, the following alternative holds: either
(B2) {w ∈ Fix(T) ⋀ lim_{m→∞ }w^{m }= w ⋀ (w^{m }: 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 (x_{m }: 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 ∈ 2^{X }∧ 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 setvalued dynamic system (X, T) satisfying T : X → C(X) is uniformly locally contractive on X then, for each w^{0 }∈ X, the following alternative holds: either
(II) If satisfies and a setvalued dynamic system (X, T) satisfying T : X → C(X) is contractive on X then, for each w^{0 }∈ X, the following alternative holds: either
Proof of Theorem 8
(I) Let i ∈ {1, 2}. The proof is divided into three steps.
Step 1. Assume that w^{0 }∈ X and suppose that the assertion (A1) does not hold; that is,
Then there exists which is Cauchy sequence on X; that is,
Indeed, since (14) holds, thus, by (12), we get
It follows from (16) and Definition 8(c), that there exists and
From this, denoting , we deduce that . Consequently, by (), there exists
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
Since, by (20), , it follows, using (12) and (20), that
That is,
Denoting , we see that condition (21) implies . Hence, by (), there exists
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,
By (24), we have and, using (12) and (24), we get
This means
By induction, a similar argument as in the proofs of (17)(25) shows that
It is clear that (26) implies that where and . Additionally, this sequence (w^{m }: 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 w^{0 }∈ 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
Applying (15), (27), and () (where (x_{m }= w^{m }: m ∈ ℕ) and (y_{m }= w : m ∈ ℕ)), we find that
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 (w^{m }: m ∈ {0} ∪ ℕ) satisfies (28). Hence, by (C), T is closed at w and, since ∀_{m∈ℕ}{w^{m }∈ T(w^{m1})}, 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 w^{0 }∈ X and the assertion (A1) does not hold, then (A2) holds.
If (A1) does not hold, then, by Step 1, there exists a sequence (w^{m }: m ∈ {0} ⋂ ℕ) which satisfies and, additionally, this sequence is a Cauchy sequence on X, i.e.
We prove that is a Cauchy sequence on X, i.e. that
Indeed, by (29), we claim that
Hence, in particular,
Let now r_{0}, j_{0 }∈ ℕ, r_{0 }> j_{0}, be arbitrary and fixed. If we define
then (31) implies that
Therefore, by (29), (33), and (), we get
From (32)(34), we then claim that
and
Let now and ε_{0 }> 0 be arbitrary and fixed, let n_{0 }= max{n_{2}(α_{0}, ε_{0}), n_{3}(α_{0}, ε_{0})} + 1 and let s, l ∈ ℕ be arbitrary and fixed such that s > l > n_{0}. Then s = r_{0 }+ n_{0 }and l = j_{0 }+ n_{0 }for some r_{0}, j_{0 }∈ ℕ such that r_{0 }> j_{0 }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 lim_{m}_{→∞ }w^{m }= 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 lim_{m→∞ }w^{m }= w.
Moreover, we observe that w ∈ Fix(T). Indeed, we have that a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) satisfies lim_{m→∞ }w^{m }= w. Hence, by (C), T is closed at w and, since ∀_{m∈ℕ}{w^{m }∈ T(w^{m1})}, 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 w^{0 }∈ 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 13 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 w^{0 }∈ X be arbitrary and fixed and suppose that the assertion (F1) does not hold. That is
But then, using analogous considerations as in the Step 1 of the proof of Theorem 8(I), we obtain that
Consequently, the sequence (w^{m }: m ∈ {0} ∪ ℕ) such that and is a dynamic process of T starting at w^{0 }and, additionally, this sequence is a Cauchy sequence on X, i.e.
It is clear that (39) implies
and, since (X, ) is a Hausdorff sequentially complete generalized uniform space, there exists a unique w ∈ X such that
If, for each , x ∈ X and B ⊂ Cl(X), we denote
and
then (42) and (40) implies
Let m ∈ ℕ, m > m_{0}, and be arbitrary and fixed and let
here m_{0 }is defined by (37). Then, by (9)(11) and definition of , we get that and . Hence, in particular, if v ∈ T(w^{m}) is arbitrary and fixed, then
This implies
Now, by (), (remember that ), for each u ∈ T(w) and v ∈ T(w^{m}), we have
Hence, by (42) and (), for each v ∈ T(w^{m}), 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 w^{0 }∈ 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 pseudodistances 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:
() If and x, y ∈ X and if p_{α}(x) and p_{α}(y) are finite, then p_{α}(x + y) is finite and p_{α}(x + y) ≤ p_{α}(x) + p_{α}(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 d_{α}(x,y) = p_{α}(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 Z_{n }= [2n  2, 2n  1] and let q_{n }: Z_{n }× Z_{n }→ [0, +∞) where q_{n}(x,y) = x  y for x,y ∈ Z_{n}. Let and define q : Z × Z → [0, +∞] by the formula
Then (Z, q) is a complete generalized metric space.
Example 4 Let Y = ℝ^{ℕ }= ℝ × ℝ × ⋯ be a nonnormable real Hausdorff and sequentially complete locally convex space with the family of calibrations c_{n},n ∈ ℕ, defined as follows:
For each s ∈ ℕ, let P_{s }= [2s  2, 2s  1]^{ℕ }be a Hausdorff sequentially complete uniform space with uniformity defined by the saturated family {p_{s,n }: n ∈ ℕ} of pseudometrics p_{s,n }: P_{s }× P_{s }→ [0, +∞), n ∈ ℕ, defined as follows:
Let and define p_{n}:P × P → [0, +∞], n ∈ ℕ, as follows
Then (P, {p_{n}:P × P → [0, +∞], n ∈ ℕ}) is a Hausdorff {p_{n}: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 L_{a }: X × X → [0, +∞] be defined by the formula:
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 L_{α}(x, z) < +∞ and L_{α}(z, y) < + ∞. By (48), this implies that: x, y, z ∈ E; d_{α}(x, z) = L_{α}(x, z) < +∞; and d_{α}(z, y) = L_{α}(z,y) < +∞. Then, by , we get that d_{α}(x,y) < +∞ and d_{α}(x, y) ≤ d_{α}(x,z) + d_{a}(z,y). Consequently, since x,y,z ∈ E, this mean that L_{α}(x,y) = d_{a}(x,y) < +∞ and L_{a}(x,y) ≤ L_{a}(x,z) + L_{a}(z,y). Therefore, the condition () holds.
To prove that () holds, we assume that the sequences (x_{m }: m ∈ ℕ) and (y_{m }: 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 (x_{m }: m ∈ ℕ) and (y_{m }: 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)):
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 < a_{α }< b_{α }< c_{α }< +∞, , and let, for each , L_{α }: X × X → [0, +∞] be defined by the formula:
We show that the family is family on X.
First, we observe that the condition ( holds. Indeed, let and x, y, z ∈ X satisfying L_{α}(x, z) < + ∞ and L_{α}(z, y) < + ∞ be arbitrary and fixed. Clearly, by definition of L_{α}, this implies that x, y, z ∈ F. We consider the following cases:
Case 1. If L_{α}(x, y) = d_{α}(x, y) + b_{α}, then by (50) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z ∈ E, then L_{α}(x, z) = d_{α}(x, z) + c_{a}; L_{α}(z, y) = d_{α}(z, y); and consequently, since b_{α }< c_{α}, by , we get
If z ∈ F \ E, then L_{α}(x, z) = d_{α}(x, z) + b_{α}; L_{α}(z, y) = d_{α}(z, y) + b_{α}; and consequently, by , we get
Case 2. If L_{α}(x, y) = d_{α}(x, y) + c_{α}, then by (50) we conclude that, x ∈ F \ E∧y ∈ E. Now, if z ∈ E then L_{α}(x, z) = d_{α}(x, z) + c_{α}; L_{α}(z, y) = d_{α}(z, y) + α_{α}; and consequently, by , we get
If z ∈ F \ E, then L_{α}(x, z) = d_{α}(x, z) + b_{α}; L_{α}(z, y) = d_{α}(z, y) + c_{α}; and consequently, by , we get
Case 3. If L_{α}(x, y) = d_{α}(x, y), then by (50) we conclude that, x ∈ E∧y ∈ F\E. Now, if z ∈ E then L_{α}(x, z) = d_{α}(x, z) + a_{α}; L_{α}(z, y) = d_{α}(z, y); and consequently, by , we get
If z ∈ F\E, then L_{α}(x, z) = d_{α}(x, z); L_{α}(z, y) = d_{α}(z, y) + b_{α}; and consequently, by , we get
Case 4. If L_{α}(x, y) = d_{α}(x, y) + a_{α}, then by (50) we conclude that, x ∈ E∧y ∈ E. Now, if z ∈ E then L_{α}(x, z) = d_{α}(x, z) + a_{α}; L_{α}(z, y) = d_{α}(z, y) + a_{α}; and consequently, by , we get
If z ∈ F\E, then L_{α}(x, z) = d_{α}(x, z); L_{α}(z, y) = d_{α}(z, y) + c_{α}; and consequently, since a_{α }< c_{α}, by , we get
Consequently, the condition holds.
To prove that holds, we assume that the sequences (x_{m }: m ∈ ℕ) and (y_{m }: 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 (x_{m }: m ∈ ℕ) and (y_{m }: m ∈ ℕ) satisfy (8). Therefore, the property ( holds.
It is worth noticing that, there exists x, y ∈ X such that, for each , L_{α}(x, y) = L_{α}(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:
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 < b_{α }< c_{α }< +∞, , and let, for each , L_{α }: X × X → [0, +∞] be defined by the formula:
We show that the family is family on X.
First, we observe that the condition ( holds. Indeed, let and x, y, z ∈ X satisfying L_{α}(x, z) < +∞ and L_{α}(z, y) < +∞ be arbitrary and fixed. Clearly, by definition of L_{α}, this implies that x, y, z ∈ F. We consider the following cases:
Case 1. If L_{α}(x, y) = d_{α}(x, y) + b_{α}, then by (52) we conclude that, {x, y} ⋂ F\E = {x, y}. Now, if z ∈ E, then
and consequently, since b_{α }< c_{α}, by , we get
If z ∈ F \ E, then
Case 2. If L_{α}(x, y) = d_{α}(x, y) + c_{α}, then by (52) we conclude that, x ∈ F\E∧y ∈ E. Now, if z ∈ E then
If z ∈ F \ E, then
Case 3. If L_{α}(x, y) = d_{α}(x, y), then by (52) we conclude that, x ∈ E ∧ y ∈ E or x ∈ E ∧ y ∈ F\E. First, assume that x ∈ E∧y ∈ E. Now, if z ∈ E then
If z ∈ F \ E, then
Next, we assume that x ∈ E ∧ y ∈ F\E. Now, if z ∈ E then L_{α}(x, z) = d_{α}(x, z); L_{α}(z, y) = d_{α}(z, y); and consequently, by , we get
If z ∈ F \ E, then
Consequently, the condition ( holds.
To prove that ( holds, we assume that the sequences (x_{m }: m ∈ ℕ) and (y_{m }: 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 (x_{m }: m ∈ ℕ) and (y_{m }: m ∈ ℕ) satisfy (8). Therefore, the property ( holds.
It is worth noticing that, there exists x, y ∈ X such that, for each , L_{α}(x, y) = L_{α}(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:
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 {p_{n }: P × P → [0,+∞], n ∈ ℕ} be as in Example 4. Let X = P ⋂ [0,9]^{ℕ }and let , d_{n }: 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 ℝ^{ℕ }we denote by x = (x_{1},x_{2},...). In particular, the element (x,x,...) ∈ ℝ^{ℕ }we denote by .
Let and let a setvalued dynamic system (X, T) be given by the formula
Let and let be a family of the maps given by the formula:
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
where
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 L_{n}(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}{L_{n}(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
and, consequently, for each n ∈ ℕ, the inequality L_{n}(x, y) < 1/2 holds. In virtue ofthis, we show that the inequalities in (56) hold. With this aim, we see that:
(3_{ii}) Next, if , then, by (3_{i}),
(3_{iii}) Now, by (3i), (3ii), (58), and (59), we get
(3_{iv}) Therefore, by (3_{iii}), we have ;
(3_{v}) The consequence of (57) and (3_{iv}) is
Hence, by (60), we conclude that
holds.
Case 4. Let x, y ∉ F. Then we see that:
(4_{iii}) Now, by (4_{i}) and (4_{ii}), we get
(4_{iv}) Therefore, by (4_{iii}), ;
(4_{v}) According to (57) and (4_{iv}), 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 w^{0 }∈ X, there exists a dynamic process (w^{m }: 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 {p_{n }: P × P → [0, +∞], n ∈ ℕ} be as in Example 4. Let and let , d_{k }: 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 (x_{m }: m ∈ ℕ) defined as follows: , m ∈ ℕ. Of course, the sequence (x_{m }: m ∈ ℕ) is Cauchy sequence on X. Indeed, we have which implies that
Consequently,
However, there does not exist x ∈ X such that lim_{m→∞ }x_{m }= x. Therefore, X is not sequentially complete.
Let and let be a family of the maps given by the formula:
By (47), this gives
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 (x_{m }: m ∈ ℕ) be arbitrary and fixed Cauchy sequence in X, i.e.
This implies that
Hence, in particular, we conclude that
Now, (63) and (61) gives that
Of course, since (x_{m }: m ∈ ℕ) is arbitrary nad fixed, then there exists a unique s_{0 }∈ N for all k ∈ ℕ. Now, putting l_{0 }= min_{k∈ℕ}{n_{0}(k)} we obtain that
The property (61) and (64) gives that
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.
so by (65) and (62), we obtain
This gives that (x_{m }: m ∈ ℕ) is a Cauchy sequence in X, so also the sequence is a Cauchy sequence in [4,5]^{ℕ}. Since [4,5]^{ℕ }is a complete uniform space, so there exists x ∈ X such that
i.e (x_{m }: 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 w^{0 }∈ X there exists a dynamic process (w^{m }: 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 w^{0 }∈ X.
Example 13 Let (X, ) be a complete metric space where X = [0,1] and let , d:X × X→ [0,∞), d(x, y) = xy, x, y ∈ X. Let a dynamic system (X, T) be given by the formula:
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 w^{0 }∈ 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 w^{0 }∈ 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
implies, in particular,
and, for η > 0, then the following hold
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 w^{0 }∈ X, the assertion (A1) holds. Indeed, we have:
Case 1. Let w^{0 }∈ [0,1/4). Then, for each dynamic process (w^{m }: m ∈ {0}∪ ℕ) of (X, T) starting at w^{0}, by (66), we have: (i) if w^{1 }≠ 1, then ∀_{m∈ℕ}{w^{m }∈ E} and, by (51), L(w^{0},w^{1}) = +∞ > 1/2 and
or (ii) if w^{1 }= 1, then ∀_{m∈ℕ}{w^{m }= 1 ∈ F\E} and, by (51), L(w^{0}, w^{1}) = +∞ > 1/2 and
Consequently, for each w^{0 }∈ [0,1/4), each a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }satisfies ∀_{m∈ℕ}{L(w^{m1}, w^{m}) > 1/2}, i.e. for each w^{0 }∈ [0,1/4), the assertion (A1) holds.
Case 2. Let w^{0 }∈ [1/4,1). Then, for each dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X,T) starting at w^{0}, by (66), we have that ∀_{m∈ℕ}{w^{m }∈ E} and, by (51),
and
Consequently, for each w^{0 }∈ [1/4,1), each a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X,T) starting at w^{0 }satisfies ∀_{m∈ℕ}{L(w^{m1}, w^{m}) > 1/2}, i.e. for each w^{0 }∈ [1/4,1), the assertion (A1) holds.
Case 3. Let w^{0 }= 1. Then, for a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0}, by (66), we have that ∀_{m∈ℕ}{w^{m }= 1 ∈ F\E} and, by (51),
Consequently, if w^{0 }= 1, a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }satisfies ∀_{m∈ℕ}{L(w^{m1}, w^{m}) > 1/2}, i.e. for w^{0 }= 1, the assertion (A1) holds.
Remark 7 Let us observe that, for each w^{0 }∈ X, there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) starting at w^{0 }such that lim_{m→∞ }w^{m }= 1, lim_{m→∞ }L(w^{m}, 1) = lim_{m→∞ }d(w^{m}, 1) = 0 and 1 ∈ Fix(T). However, assertion (A2) does not hold since from Cases 13 it follows that, for each w^{0 }∈ X, each dynamic process (w^{m }: m ∈ {0} ∪ ℕ) starting at w^{0 }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 w^{0 }∈ X, the assertion (A2) holds.
Let E = [1/2,1] ⊂ X and let L : X × X → [0, +∞] be defined by the formula:
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),
implies
and, for η > 0,
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 w^{0 }∈ X, the assertion (A2) holds. Indeed, we have that: 1 ∈ Fix(T); for each w^{0 }∈ X and for each dynamic processes (w^{m }: m ∈ {0} ∪ ℕ) of (X,T) starting at w^{0}, by (66), we have that , so lim_{m→∞ }L(w^{m}, 1) = lim_{m→∞ }d(w^{m}, 1) = 0 and lim_{n→∞ }sup_{m>n }L(w^{n},w^{m}) = lim_{n→∞ }sup_{m>n }d(w^{n}, w^{m}) = 0. Therefore, the sequence (w^{m }: 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 w^{0 }∈ X\End(T) the assertion (A2) holds; and (iv) For w^{0 }= 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),
implies
and, for η > 0,
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 w^{0 }∈ [0,1), then the assertion (A2) holds and if w^{0 }= 1 then the assertion (A1) holds. Indeed, we have the following three cases:
Case 1. Let w^{0 }∈ [0,1/4). Then, by (66), there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }of the form: w^{1 }≠ 1 and . Then, by (53), L(w^{0}, w^{1}) = +∞ and . Consequently, a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) is Cauchy on X, lim_{m→∞ }w^{m }= 1 and 1 ∈ Fix(T), i.e. for each w^{0 }∈ [0,1/4), the assertion (A2) holds.
Case 2. Let w^{0 }∈ [1/4,1). Then, for each a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X,T) starting at w^{0}, by (66), we have that ∀_{m∈ℕ}{w^{m }∈ E} and, by (53),
and . Consequently, for each w^{0 }∈ [1/4,1), each a dynamic process (w^{m }: m ∈ {0}∪ ℕ) of (X, T) starting at w^{0 }is Cauchy on X, lim_{m →∞ }w^{m }= 1 and 1 ∈ Fix(T), i.e., for each w^{0 }∈ [1/4,1), the assertion (A2) holds.
Case 3. Let w^{0 }= 1. Then, for a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0}, by (66), we have that ∀_{m∈ℕ}{w^{m }= 1 ∈ F\E} and, by (53), ∀_{m∈ℕ}{L(w^{m1}, w^{m}) = d(w^{m1}, w^{m}) + 3 > 1/2}. Consequently, if w^{0 }= 1, a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }satisfies ∀_{m∈ℕ}{L(w^{m1}, w^{m}) > 1/2}, i.e. for w^{0 }= 1, the assertion (A1) holds.
Finally, we see that, for each w^{0 }∈ X, there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) such that lim_{m→∞ }w^{m }= 1, lim_{m→∞ }L(w^{m}, 1) = lim_{m→∞ }d(w^{m}, 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, CovitzNadler'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], CovitzNadler'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 setvalued 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 setvalued dynamic system (X, T) defined in Example 14 is not uniformly locally contractive on X and thus we cannot use Theorems 1, 2 and 47; (c) In Example 16 we construct a complete metric space which is family on X and uniformly locally contractive setvalued dynamic system (X, T) such that, for each w^{0 }∈ 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 w^{0 }∈ X there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) such that lim_{m→∞ }w^{m }= 2 and 2 ∈ Fix(T).
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 ε_{0 }∈ (0, ∞), λ_{0 }∈ [0, 1) and i ∈ {1, 2} such that
We consider the following three cases:
Case 1. If ε_{0 }= 1, then, in particular, for x_{0 }= 1 and y_{0 }= 1/2, since x_{0}, y_{0 }∈ [0,1], by formula (46), we get
However, T(x_{0}) = {4, 5}, T(y_{0}) = {1, 2}, and, by (46),
Hence
Consequently,
and (80) gives
This leads to a contradiction.
Case 2. If ε_{0 }∈ (1, ∞), then by a similar reasoning as in Case 1 we prove that (80) does not hold.
Case 3. If ε_{0 }∈ (0, 1), then, in particular, for x_{0 }= 1 and y_{0 }= ((1  ε_{0})/2), we obtain that x_{0}, y_{0 }∈ [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:
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 w^{0 }∈ X, the assertion (A2) holds. Indeed, we have the following three cases:
Case 1. Let w^{0 }∈ [0, 1/4). Then, there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }of the form: w^{1 }≠ 1, and ∀_{m∈ℕ}{w^{m }∈ [1/2, 1) = E}. Then, by (53), L(w^{0}, w^{1}) = +∞ and ∀_{m≥2}{L(w^{m1}, w^{m}) = d(w^{m1}, w^{m})}. Consequently, a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) is Cauchy on X, lim_{m→∞ }w^{m }= 1 and 1 ∈ Fix(T), i.e. for each w^{0 }∈ [0, 1/4), the assertion (A2) holds.
Case 2. Let w^{0 }∈ [1/4, 1). Then, for each a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0}, by (81), we have that ∀_{m∈ℕ}{w^{m }∈ E} and, by (53),
and ∀_{m≥2}{L(w^{m1}, w^{m}) = d(w^{m1}, w^{m})}. Consequently, for each w^{0 }∈ [1/4, 1), each a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0 }is Cauchy on X, lim_{m→∞ }w^{m }= 1 and 1 ∈ Fix(T), i.e. for each w^{0 }∈ [1/4, 1), the assertion (A2) holds.
Case 3. Let w^{0 }= 1. Then, there exists a dynamic process (w^{m }: m ∈ {0} ∪ ℕ) of (X, T) starting at w^{0}, of the form: w^{0 }= 1, w^{1 }= 1/2, ∀_{m≥2}{w^{m }∈ E}, and, by (53), L(w^{0}, w^{1}) = d(w^{0}, w^{1})+4 and ∀_{m≥2}{L(w^{m1}, w^{m}) = d(w^{m1}, w^{m})}. Consequently, this dynamic process (w^{m }: m ∈ {0} ∪ ℕ) is Cauchy on X, lim_{m→∞ }w^{m }= 1 and 1 ∈ Fix(T), i.e. for w^{0 }= 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 singlevalued 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 ∀_{x}_{∈}_{X}{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 x_{0 }∈ dom(φ), there exists w ∈ X such that: (i) φ(w)+εd(x_{0}, w) ≤ φ(x_{0}); and (ii) ∀_{x}_{∈}_{X}_{\{}_{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 47) 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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