Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Włodarczyk, Kazimierz; Plebaniak, Robert

doi:https://doi.org/10.1155/2015/942814

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2015 | Article ID 942814 | https://doi.org/10.1155/2015/942814

Dynamic Processes, Fixed Points, Endpoints, Asymmetric Structures, and Investigations Related to Caristi, Nadler, and Banach in Uniform Spaces

Kazimierz Włodarczyk¹and Robert Plebaniak¹

Academic Editor: Ademir F. Pazoto

Received30 Oct 2014

Accepted15 Jan 2015

Published18 Feb 2015

Abstract

In uniform spaces with symmetric structures determined by the -families of pseudometrics which define uniformity in these spaces, the new symmetric and asymmetric structures determined by the -families of generalized pseudodistances on are constructed; using these structures the set-valued contractions of two kinds of Nadler type are defined and the new and general theorems concerning the existence of fixed points and endpoints for such contractions are proved. Moreover, using these new structures, the single-valued contractions of two kinds of Banach type are defined and the new and general versions of the Banach uniqueness and iterate approximation of fixed point theorem for uniform spaces are established. Contractions defined and studied here are not necessarily continuous. One of the main key ideas in this paper is the application of our fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by -families. Results are new also in locally convex and metric spaces. Examples are provided.

1. Introduction

The concepts of the symmetric and asymmetric structures became established and investigated in mathematics and in theoretical computer science and are some creative ideas in fixed point theory by which some fascinating results have been achieved. In the proofs of these results, some deep methods based on those symmetric and asymmetric structures do play very important roles. The range of important applications of these results is enormous.

Let be a uniform space with uniformity defined by a saturated family of pseudometrics , , uniformly continuous on (-family, for short); here is a nonempty index set.

It was discovered that the -families of generalized pseudodistances defined below generalize: metrics , distances of Tataru [1], -distances of Kada et al. [2], -distances of Suzuki [3], and -functions of Lin and Du [4] in metric spaces and also -families of pseudometrics and distances of Vályi [5] in uniform spaces .

Definition 1 (see [6]). Let be a Hausdorff uniform space.(a)The family of maps , , is said to be a -family of generalized pseudodistances on (-family, for short) if the following two conditions hold:).()For any sequences and in such that and , the following holds .(b)Define

Definition 2. Let be a metric space.(a)Then is said to be a -family on if is a generalized pseudodistance on ; that is, the following two conditions hold:().()For any sequences and in such that and ,, the following holds .(b)Define is a -family on .

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

Remark 3. Let be a Hausdorff uniform space.(a) and .(b)([7, Remark 1.1]) Let . If , , , then .(c)Let . If , then, for each , is quasipseudometric; examples of such that the maps , , are not quasipseudometrics are given in Section 4.

Definition 4. Let be a Hausdorff uniform space. is said to be admissible if where

Remark 5. It is a remarkable fact that -family is admissible and . Indeed, we have that , where . Therefore, by Definition 4, we get the following .

Let denote the family of all nonempty subsets of a space . A set-valued dynamic system is defined as a pair , where is a certain space and is a set-valued map ; in particular, a set-valued dynamic system includes the usual dynamic system where is a single-valued map.

Let be a set-valued dynamic system. By and we denote the sets of all fixed points and endpoints of , respectively; that is, and . A dynamic process or a trajectory starting at or a motion of the system at is a sequence defined by for (see, Aubin and Siegel [8], Aubin and Ekeland [9], Aubin and Frankowska [10], and Yuan [11]).

Recall that a map is proper if its effective domain, , is nonempty.

Caristi’s fixed point theorem [12] concerning dissipative single-valued dynamic systems in metric spaces with lower semicontinuous entropies is equivalent to Ekeland’s variational principle [13–15] providing approximate solutions of nonconvex minimization problems concerning lower semicontinuous maps .

Theorem 6. Let be a complete metric space.(I)(Caristi [12]) Let be a map which is proper lower semicontinuous and let be a single-valued dynamic system satisfying the condition Then (i.e., there exists such that ).(II)(Ekeland [13–15]) Let be a map which is proper lower semicontinuous. For every and for every , there exists such that and .

Let be a sequentially complete uniform space. We say that a set is closed in if , where , the closure of in , denotes the set of all for which there exists a sequence in which converges to . If a set is closed in , then is a sequentially complete uniform space.

Define ; that is, denotes the class of all nonempty closed subsets of .

The following fixed point and endpoint version of Caristi type theorem for dissipative set-valued dynamic systems without lower semicontinuous entropies in uniform spaces with structures determined by is included in a more general result [6, Theorem 4.5].

Theorem 7. Let be a Hausdorff sequentially complete uniform space and let be admissible.(I)(Fixed point theorem) Assume the following.(A1) The family satisfies .(A2) is a family of finite positive numbers.(A3) is a set-valued dynamic system.(A4)For each , is a set defined by (A5) For each , the set is nonempty.(A6)For each , the set is a closed subset in .
Then, there exists such that (i.e., and ).(II)(Endpoint theorem) Assume, in addition, that(A7)for each , each dynamic process starting at and satisfying satisfies .
Then, there exists such that (i.e., and ).

It is known that a weaker condition than continuity is lower semicontinuity.

Definition 8. Let be a Hausdorff sequentially complete uniform space. Let , and let . The map is lower semicontinuous on with respect to (written: is -lsc when and is lsc when ) if the set is a closed subset in for each .

The following alternative characterizations of lower semicontinuity hold.

Theorem 9. Let be a Hausdorff sequentially complete uniform space. Let , and let . The following conditions are equivalent.(Z1) The map is lower semicontinuous on with respect to .(Z2) For each , here (Z3) The map is sequentially lower semicontinuous on with respect to ; that is, for each , for any sequence in such that ; here

Remark 10 (see [6, Remark 4.6]). The following hold.(a)A special case of condition (A6) is a condition () defined by()for each , the map is -lsc.(b)If , then a special case of condition (A6) is a condition (A6′′) defined by()for each , the map is -lsc.(c)Theorem 7(I) esentially generalizes Theorem 6(I) even in metric spaces.
A classic result of Banach [16], from 1922, is the milestone in the history of fixed point theory and its applications.

Theorem 11 (Banach [16]). Let be a complete metric space. Assume that the single-valued dynamic system is -contraction; that is, Then has a unique fixed point in (i.e., and ) and, for each , the sequence satisfies .

In a slightly different direction is the following elegant result of Nadler on set-valued dynamic systems.

Theorem 12 (Nadler [17, Theorem 5]). Let be a complete metric space, let denotes the class of all nonempty closed and bounded subsets of , and let be defined by where . Assume that the set-valued dynamic system satisfying is -contraction; that is, Then (i.e., there exists such that ).

Remark 13. Let be a complete metric space.(a)It is well known that Caristi’s fixed point theorem [12] yields Banach’s [16] and Nadler’s [17, Theorem 5] results.(b)Maps satisfying (3) are not necessarily continuous.(c)It is well known that is a complete metric space and that the continuity of maps and satisfying conditions (9) and (11) plays an important role in the proofs of Theorems 11 and 12, respectively.

Contractions (3) of Caristi, (9) of Banach, (11) of Nadler, and others are among the most important notions in fixed point theory, as well as in its numerous applications. As one will see from the literature, the known results about them have been achieved by employing complicated machineries from various branches of mathematics and the answers for many basic problems about them are still missing. Moreover, examples show that these fundamental results are not optimal even in metric spaces.

The several authors have made essential progress in these problems and have solved many cases, and similar methods and ideas have since been applied in greater generality; see for example [1–67] and the references cited therein. However, the complete solutions of some key open problems are still missing.

In this paper we show that there are complementary approaches to generalize the Nadler and Banach statements concerning uniform, locally convex, and metric spaces. They involve mixed properties of asymmetric structures and fixed point theory. One of the key ideas in this paper is that in the families construct the symmetric and asymmetric structures on which generalize the symmetric structure determined by on and then, by subtle techniques, we may use stated above Theorem 7.

More precisely, let be a Hausdorff uniform space. For and , let the distance on be defined as in Definitions 15 and 23, and let the distance on be defined as in Definitions 29 and 33.

This paper has two aims.(1)To determine , various classes of not necessarily continuous set-valued dynamic systems satisfying , , and the conditions guaranteeing that the maps , , attains its global optimal minimum at a point (not necessarily unique) satisfying and or .(2)To determine , various classes of not necessarily continuous single-valued dynamic systems satisfying , , and the conditions guaranteeing that the maps , , attains its unique global optimal approximate minimum at satisfying , , and , where and is arbitrary.

Remark 14. (a) The methods of this paper provide a way to compute the fixed point and endpoint theorems in uniform, locally convex and metric spaces with structures determined by .
(b) Theorems 17, 20, 21, 22, 25, 26, 27, 31, 34 and Examples 1–4 and 5–7 shows that our fixed point and endpoint results are new in uniform and locally convex spaces and even in metric spaces, are different from fixed point and endpoint results given in the literature, and their proofs are simpler.

2. Fixed Point and Endpoint Theorems for Set-Valued Contractions (of Nadler Type) in Uniform and Metric Spaces

The following definitions will be much used in the sequel.

Definition 15. Let be a Hausdorff sequentially complete uniform space, assume that , let and let .(a)Define on the distance , , as follows: (b)Let a set-valued dynamic system satisfy . If satisfies then we say that is a -contraction on for .

Remark 16. Each -contraction on is -contraction on but converse does not hold.

One can prove the following characterizations of -contractions :

Theorem 17. Let be a Hausdorff sequentially complete uniform space, and . Suppose also the following.(I)A set-valued dynamic system satisfies .(II)There exists a family such that is a -contraction on .(III)The family satisfies .
The following hold.(B1).(B2) where (B3) where (B4).(B5).

Proof. Let the family be defined by
Proof of (B1). By assumption (II) and Definitions 15(a) and 15(b), Using this, we may thus conclude that and hence On the other hand it is clear that By applying (20) and (21), we obtain (B1).
Proof of (B2). By (12), we have Further, by assumption (III), . Hence, for arbitrary and fixed and , by (22) and definition of infimum, we obtain that Consequently, So we have proved (B2).
Proof of (B3). Let , , and be arbitrary and fixed. Then, by (B2), we have and Clearly, by (B1), property implies . Thus Using (25) and (26) we obtain We proved that Therefore, (B3) holds.
Proof of (B4). Let , , and be arbitrary and fixed. Then, by (B1), since , we obtain . This and (22) imply Therefore, holds. Next, it follows from (22) and (B1) that This shows that (B4) holds.
Proof of (B5). By (30) and (22), Therefore, (B5) holds.

Definition 18. Let be a Hausdorff sequentially complete uniform space and let . We say that the family is continuous in if, for each and for each sequence in such that we have

Remark 19. The family is continuous in .
Assertion (B5) says that, for each , the set has the property Let be a family of positive numbers satisfying and, for each , let the the set be defined by
Now, for -contractions , we can give the following characterizations of the sets , , defined in (37).

Theorem 20. Let be a Hausdorff sequentially complete uniform space, and . Suppose also the following.(I) is admissible.(II)A set-valued dynamic system satisfies .(III)There exists a family such that is a -contraction on .(IV)For each family satisfying and for each , let the set be defined by
The following hold.(C1)If there exists a family satisfying and such that then .(C2)If there exists a family satisfying and such that, for each , the map is -lsc, then, for each , is a closed subset in .(C3)Let the family be continuous in . Then, for each family satisfying and for each , is a closed subset in .(C4)Let . If there exists a family satisfying and such that then .(C5)Let . Then, for each family satisfying and for each , is a closed subset in .

Proof. Let the family be defined by
Proof of (C1). Denote Then, by (B2), (B3), and (IV), Hence, we conclude that, for each , the set is nonempty whenever .
Proof of (C2). The assertion follows immediately from Remark 10(a).
Proof of (C3). The assertion also follows from Remark 10(a). Indeed, let be arbitrary and fixed and let a sequence in be convergent to ; that is, let .
If , and are arbitrary and fixed, then, by (1), This gives Hence Furthermore, this holds for each and, thus, by (12), However, is -contraction on . Therefore, Consequently, we obtain that Since the family is continuous, this implies Therefore, for each , is lsc in .
Moreover, if , , and are arbitrary and fixed, then, by (1), Since is continuous, this gives that is, for each , the map is lsc in .
Using these two facts, in particular, we have that, for each , the map is -lsc; that is, (A) holds.
Proof of (C4). This follows from (C1).
Proof of (C5). This follows from (C3) and Remarks 3(a) and 19.

We use notations and auxiliary Theorems 17 and 20 above in proving the following basic fixed point and endpoint theorem for set-valued contractions with respect to (of Nadler-type) in uniform spaces .

Theorem 21. Let be a Hausdorff sequentially complete uniform space, and . Suppose also the following.(I) is admissible.(II)A set-valued dynamic system satisfies .(III)There exists a family such that is a -contraction on .(IV)For each family satisfying and for each , let the set be defined by (V)There exists a family satisfying and such that, for each , is a nonempty closed subset in .The following hold,(E1) (Fixed point theorem) and there exists satisfying .(E2) (Endpoint theorem) If, for each , each dynamic process starting at and satisfying satisfies , then and .

Proof. The proof will be broken into five steps.
Step 1. Let the family be defined by The family satisfies the assumption (A1) of Theorem 7; that is, .
Indeed, by (B1), Also, by Definition 1, and, by definition of , . Hence we conclude that .
Step 2. The assumptions (A5) and (A6) of Theorem 7 hold where , and is defined in Step 1.
Indeed, by assumption (V) (i.e., by assumption ) it follows that
Step 3. There exists such that .
This is a consequence of (I)–(V), Steps 1 and 2, and Theorem 7.
Step 4. We now observe that .
Otherwise, . Consequently, for each , which is absurd.
Step 5. The assertions hold.
This follows from assumptions of Theorem 17, Steps 1–4, definition of , and Theorem 7.

As a corollary of the above Theorems 17, 20, and 21 we have the following new fixed point and endpoint theorem for set-valued contractions with respect to -families (of Nadle-type) in uniform spaces .

Theorem 22. Let be a Hausdorff sequentially complete uniform space and let . Suppose also the following.(I)A set-valued dynamic system satisfies .(II)There exists a family such that is a -contraction on .(III)For each family satisfying and for each , let the set be defined by (IV)There exists a family satisfying and such that .
The following hold.(F1) (Closedness property) For each , is a closed subset in .(F2) (Fixed point theorem) .(F3) (Endpoint theorem) If, for each , each dynamic process starting at and satisfying satisfies , then .

We now state consequences of the above in metric spaces.

Definition 23. Let be a complete metric space, let , and let .(a)Let Define as follows: (b)Let a set-valued dynamic system satisfy . If satisfies then we say that is a -contraction on .(c)Let . is said to be admissible if .(d)We say that is continuous in if, for each and for each sequence in such that , we have

Remark 24. Let . It is clear that is -family; is admissible; is continuous; and .

As corollaries from Theorems 17, 20, and 21 and their proofs we get the following three theorems concerning contractions with respect to (of Nadler-type) in metric spaces .

Theorem 25. Let be a complete metric space, and . Suppose also the following.(I) is admissible.(II)A set-valued dynamic system satisfies .(III)There exists such that is a -contraction on .(IV)For each satisfying and for each let the set be defined by The following hold.(G1)If there exists satisfying and such that then .(G2)If there exists satisfying and such that, for each , the map is -lsc, then, for each , is a closed subset in .(G3)Let be continuous in . Then, for each satisfying and for each , is a closed subset in .

Theorem 26. Let be a complete metric space, and . Suppose also the following.(I) is admissible.(II)A set-valued dynamic system satisfies .(III)There exists such that is a -contraction on .(IV)For each satisfying and for each let the set be defined by (V)There exists satisfying such that, for each , is a nonempty closed subset in .The following hold.(K1) (Fixed point theorem) and there exists such that .(K2) (Endpoint theorem) If, for each , each dynamic process starting at and satisfying satisfies , then and .

Theorem 27. Let be a complete metric space, and . Suppose also the following.(I)A set-valued dynamic system satisfies .(II)There exists such that is a -contraction on .(III)For each satisfying and for each let the set be defined by
The following hold.(L1) (Nonemptness and closedness property) For each satisfying and for each , is a nonempty closed subset in .(L2) (Fixed point theorem) .(L3) (Endpoint theorem) If there exists satisfying and such that, for each , each dynamic process starting at and satisfying satisfies , then .

Remark 28. Theorem 27(L2) generalizes Theorem 12 (see Examples 5 and 6).

3. Fixed Point Theorems for Single-Valued Contractions (of Banach-Type) in Uniform and Metric Spaces

Definition 29. Let be a Hausdorff sequentially complete uniform space, assume that and let .(a)Define on the distance , , as follows: (b)Let be a single-valued dynamic system, . If satisfies then we say that is a -contraction on for .

Remark 30. Each -contraction on is -contraction on but converse does not hold.

We use notations above and Theorem 21 in proving the following new fixed point theorem for single-valued contractions with respect to (of Banach-type) in uniform spaces .

Theorem 31. Let be a Hausdorff sequentially complete uniform space, let , and let . Suppose also the following.(I) is admissible.(II) is a single-valued dynamic system, .(III)There exists a family such that is a -contraction on for .(IV).The following hold.(M1) has a unique fixed point in ; that is, and .(M2).(M3)For each , the sequence satisfies

Proofs of (M1) and (M2). By Remark 30, Definition 29, and the assumptions (I)–(IV) of Theorem 31, we see that Let now satisfying be arbitrary and fixed. One then immediately finds that or, equivalently, where . Consequently, for each , the singleton set is a nonempty closed subset in .
From the above and Theorem 21 it follows that has a fixed point in (i.e., ) and ).
It remains to verify that . Suppose that . By Definition 29 and assumptions of Theorem 31, we obtain that, if , then and, if , then Hence . From this information, by Remark 3(b), we deduce that .
Therefore, the assertions (M1) and (M2) hold.
Proof of (M3). Let now be arbitrary and fixed and put . By Definition 29, assumptions of Theorem 31 and the fact that for , we obtain that, if , then and, if , then Hence This gives the assertion (72), since, by Definition 1,
Finally, let be arbitrary and fixed and put , , and . Using assertion (M2), we then have and, using assertion (72), we get Hence, using Definition 1 , we find Thus (73) holds.

Remark 32. (a) Theorem 31 includes Theorem 11 [16] and the result of [52]. Theorem 31 is different from Theorem 11 [16] and the result of [52] even in metric spaces and in uniform spaces, respectively (see Examples 4 and 7).
(b) Let . Assumptions (III) and (IV) imply that is also a -contraction on . However, the dynamic systems and are not necessarily -contractions on or , respectively (see Examples 4 and 7).
(c) Assumptions (II) and (IV) and assertions (M1) and (M2) imply that is a unique fixed point of and . Assertion (M3) implies, in particular, that, for each starting point of the space , the dynamic process of the system converges to .

The above has interesting implications for metric spaces.

Definition 33. Let be a complete metric space, assume that and let .(a)Define on the distance as follows: (b)Let be a single-valued dynamic system, . If satisfies then we say that is a -contraction on for .

As a corollary from Theorem 31 and its proof we get the following fixed point theorem for single-valued contractions with respect to (of Banach-type) in metric spaces .

Theorem 34. Let be a complete metric space, and . Suppose also the following.(I) is admissible.(II) is a single-valued dynamic system, .(III)There exists such that is a -contraction on for .(IV).
The following hold.(S1) has a unique fixed point in (i.e., and ).(S2).(S3)For each , the sequence satisfies and .

Remark 35. Theorem 34 generalizes Theorem 11 (see Example 7).

4. Examples Illustrating the Results

The following example describes some -family in metric spaces.

Example 1. Let be a metric space. Let the set , containing at least two different points, be arbitrary and fixed and let satisfy where . Let be defined by the formulae: . Then (see [6, Example 6.12]).

The following example illustrates the Theorem 26(K1) in the case when , .

Example 2. Let be a complete metric space with a metric , , . Let be of the form: Let and let be of the form: Clearly, (Example 1).
We observe that .
Let . We show that is a -contraction on . Indeed, let be arbitrary and fixed. We consider three cases.
Case 1. If , then we have that and .
Case 2. If and , then , , and . Hence, we calculate the following.(2.1)For , and, consequently, .(2.2)For , and, consequently, .(2.3)By (2.1) and (2.2), for and ,
Case 3. If and , then also (92) holds.
By Cases 1–3, is a -contraction on .
Now, let . We prove that, for each , is a nonempty closed subset in . Indeed, for each , we have , and This implies the following.
Case 1. If , then
Case 2. If , then
Case 3. If , then
Assumptions of Theorem 26(K1) hold for , , and, for each , .

The following example illustrates the Theorem 26(K2) in the case when , .

Example 3. Let , , , , and be such as in Example 2 and let be of the form: Then and, by analogous considerations as in Example 2, we obtain that is a -contraction on .
Next, let us observe that, for , Hence we have the following.
Case 1. If , then , for and, consequently,
Case 2. If , then , for and, consequently, Therefore, for each , each dynamic process starting at and satisfying satisfies .
Assumptions of Theorem 26(K2) hold, and .

The following example illustrates the Theorem 34 in the case when , .

Example 4. Let be a metric space with a metric , , . Let and let be of the form: Clearly, is a -family on (Example 1).
Let and let be of the form:
Then and . Thus assumption (IV) of Theorem 34 holds.
We see that that is, is a -contraction on . Indeed, we have the following.
Case 1. If and , then
Case 2. If , then
Case 3. If , then
Assumptions of Theorem 34 hold and the assertions (S1)–(S3) are as follows. , and, for each , the sequence satisfies

5. Comparisons of Our Results with Nadler’s and Banach’s Results

It is worth noticing that our results in metric spaces include Nadler’s and Banach’s results. Clearly, it is not otherwise. More precisely we have the following.(a)In Examples 5 and 6 below we show that, for each , the set-valued dynamic systems and defined in Examples 2 and 3, respectively, are not -contractions on and thus we cannot use Theorem 12.(b)In Example 7 we show that, for each , the single-valued dynamic system defined in Example 4 is not -contractions on and thus we cannot use Theorem 11.

Therefore, in our concepts of -contractive set-valued dynamic systems and -contractive single-valued dynamic systems, , the existence of -family such that is essential.

Example 5. Let and be such as in Example 2 and let . We observe that .
Next, we see that, for each , is not a -contraction on . Indeed, suppose that Then, in particular, for and , we obtain the following.(1) and .(2)For , and, consequently, (3)For , and, consequently, (4)By (2) and (3),

Hence, we get

which is absurd.

Example 6. Let and be such as in Example 3 and let . By similar argumentation as in Example 5, we observe that, for each , is not a -contraction on .

Example 7. Let and be such as in Example 4 and let . Clearly, .
We observe that, for each , is not a -contraction on . Otherwise, by Definition 29 for (or by (9)), the following holds: However, in particular, for and , we get and then which is absurd. This gives that the condition (113) does not hold.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2015 Kazimierz Włodarczyk and Robert Plebaniak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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