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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 978257, 5 pages
On Stability of Fixed Points for Multi-Valued Mappings with an Application
College of Science, Guilin University of Technology, Guilin 541004, China
Received 12 November 2013; Accepted 22 January 2014; Published 27 February 2014
Academic Editor: Salvador Romaguera
Copyright © 2014 Qi-Qing Song. 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.
This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems.
In , the conception of an essential fixed point was introduced by Fort for continuous functions on a compact subset of a metric space (Brouwer type fixed points); the idea was applied to analyze many stabilities of solutions for many kinds of nonlinear problems, such as vector quasi-equilibrium problems, coincident points, and Nash equilibrium problems (see [2–4]).
In fact, the stability of fixed points is connected with continuities of fixed points mappings. In , some generic continuities of semicontinuous functions were obtained, which have also been used to analyze many problems (see [6–9]). In hyperconvex metric spaces, the generic stability of fixed points for upper semicontinuous mappings (Fan-Glicksberg type fixed points) was obtained (see ).
Generally, the existence of essential Brouwer type fixed points cannot be guaranteed, which was used to prompt the study of essential components of these fixed points in , and the existence of such essential components was proved. Further, essential sets of Brouwer type fixed points were introduced (see ). For upper semicontinuous multi-valued mappings, the existence of essential connected components of fixed points (Kakutani type fixed points) was also obtained in .
These sets of many kinds of solutions abovementioned and others (e.g., ) are compact, which facilitates the analysis of generic stabilties, essential sets, and essential components. It is well known that every lower semicontinuous function (Michael type function) from a paracompact space to the nonempty, closed, and convex subsets of a Banach space admits a continuous selection (see [15, Theorem 1] or [16, Theorem 3.2]). Noting the Brouwer’s fixed point theorem, the result for Michael's continuous selection implies that each lower semi-continuous self mapping with nonempty closed convex values on a compact convex set of a normed linear space has a fixed point (Michael type fixed point). However, in this paper, some examples show that the set of Michael type fixed points need not be compact.
The aim of this paper is to study the stability of multi-valued mappings concerning selections. Firstly, the multi-valued mappings admitting a continuous selection (which includes Mychael type functions) are considered. The fixed point set of these mappings is not necessarily compact. The fixed point mapping for these mappings is neither upper semicontinuous nor almost lower semicontinuous. Secondly, the multi-valued mappings admitting Lipschitz selections are studied. The fixed point set of these mappings need not also be compact. By finding an upper continuous submapping with compact values of each such mapping, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least one essentially stable fixed point for most of these mappings. Finally, we deduce a stability result for differential inclusion problems as an application.
2. Preliminaries and Motivations
Let and be two metric spaces. Firstly, we recall some notions for multi-valued mappings. Let be a multi-valued mapping, where denotes the collection of all subsets of . (i) is said to be upper semicontinuous at , if and only if, for each open set with , there exists an open neighborhood of such that for any ; (ii) is lower semicontinuous at , if and only if, for each open set with , there exists an open neighborhood of such that for any ; (iii) is continuous at , if and only if it is both upper semicontinuous and lower semicontinuous at ; (iv) is almost lower semicontinuous at , if and only if there is at least one such that, for each open neighborhood of , there exists an open neighborhood of such that , for all ; (v), is said to be metric upper semicontinuous at , if for each , there exists an open neighborhood of such that for all , where denotes the -neighborhood of ; (vi) is said to be metric lower semicontinuous at if for each , there exists an open neighborhood of such that for all .
Generally, if is upper semicontinuous (metric lower semicontinuous) at , is metric upper semicontinuous (lower semicontinuous) at .
Let be a nonempty, compact, and convex subset of the metric linear space . Let be a multi-valued mapping which admits a continuous selection, where denotes the family of nonempty closed subsets of . Denote by all such multi-valued mappings on , clearly, includes Michael type fixed point mappings. Then for each , by the Brouwer’s fixed point theorem, there is a fixed point of ; that is, .
For each , let . Then, defines a multi-valued mapping from to . Clearly, for each , we have .
The following example shows that the fixed point set for each is not necessarily closed and not compact. In fact, the lack of compactness of may result in some difficulties in the study of stability of these fixed points.
Example 1. Let and such that
We can check that is lower semicontinuous and has nonempty closed convex values, then, has a continuous section, to be prise, , , hence, . Clearly, we have , which is open in , hence, is not compact.
We define the metric between any and in as where is the Hausdorff metric induced by . Then is a metric space. The following two examples reveal that the fixed points mapping lacks both upper and lower semicontinuities.
Example 2. Let . Let satisfying , . Then, . Define , , such that Clearly, , . Then, , , and we have as gets close to infinity. Let , then is open in and , where . However, if is small enough, we will obtain that , . Thus, is not lower semicontinuous at .
Remark 3. For the in Example 2, for each fixed point and small enough neighborhood of , we have , similarly, we can construct such that , . That is, is not also almost lower semicontinuous at .
Example 4. Let . Let such that Then, . For each , define a multi-valued mapping on as Clearly, we can check that , and . Take an open set , then . However, , . Therefore, is not upper semicontinuous at .
Noting that is neither upper semicontinuous nor lower semicontinuous on , in order to study the continuity of further, for each , we denote by the set of all continuous selections of . That is, defines a multi-valued mapping from to , where is the usual space of continuous functions from to , on which the uniform metric is adopted. Clearly, for each . Define a multi-valued mapping such that denotes the set of fixed points of on . Obviously, is nonempty and compact for each .
Lemma 5 (see ). Let be a complete metric space, be a metric space, and be metric upper semicontinuous. Then, there exists a dense residual set such that is metric lower semicontinuous at each .
3. Stability of Fixed Points for Multi-valued Mappings
Theorem 6. The graph of , , is closed.
Proof. Let , , with . Then, is continuous on , and for each and . We need to show that . For each , we can obtain that where . As get close to infinity, we have , because and . Since is closed, we have . Additionally, is also continuous, which follows from . Therefore, is a continuous selection of . The proof is completed.
Theorem 7. The graph of , , is closed.
Proof. For , , with , clearly, it holds that . Noting the following inequality: and since with the continuity of , we have . Then, . Therefore, .
Remark 8. For the multi-valued mapping in Example 4, we can check that . For each multi-valued mapping with in Example 4, we have . Therefore, is not upper semicontinuous on , though the graphs of and are closed.
For each , though admits a continuous selection, there is no semicontinuities for fixed point mappings from examples in Preliminaries, we consider a kind of space for which each admits a Lipschitz selection. That is, where a -Lipschitz selection of means that , such that there exists a with satisfying that for any . Obviously, each Lipschitz selection of is a continuous selection.
For the space , we can also use to measure the metric between any two . Then is also a metric space. For each , let denote the set of fixed points, obviously, is also not necessarily compact; denote by the set of Lipschitz selections of ; that is, defines a multi-valued mapping from to . By the construction of , we know that for each .
Definition 9. For each , a set is called an essential fixed point set of with respect to if and only if it satisfies the following conditions.(1) is nonempty closed subset of ;(2)For any open set with , there exists an open neighborhood in such that , for any .
If is a singleton set , is called an essential fixed point of with respect to . An essential fixed point set is said to be minimal if it is a minimal element in the family of essential sets of ordered by set inclusion in .
Remark 10. An essential fixed point of means that for each multi-valued mapping near there is a fixed point near the point . Let be two nonempty sets with , if is essential, then follows.
Theorem 11. The metric space is complete.
Proof. Let be a Cauchy sequence in . Then, for any , there exists a number such that for any . That is,
Therefore, for each , is a Cauchy sequences in , where is the collection of nonempty and compact subsets in . Clearly, is complete. Then for each , there is a such that . That is, there exists a mapping such that .
For each , let be a -Lipschitz selection of . Since the collection of Lipschitz functions on bounded with is compact, there is a convergent subsequence of with . Noting , similar to the proof of Theorem 6, we can prove that , . Additionally, for any two points , since and , we have . That is, is a -Lipschitz selection of . Then, . The space is complete.
Theorem 12. The multi-valued mapping is upper semicontinuous with compact values.
Proof. Following the proof of Theorems 6 and 7, both and are closed. Noting that is compact and the collection of Lipschitz functions bounded with on is also compact, we have and upper semicontinuous with compact values. It follows that is upper semicontinuous and has compact values on .
By Theorem 12, in fact we find a special submapping of the fixed point mapping on . A result can be obtained as follows.
Theorem 13. For almost all , there exists at least one essential fixed point of with respect to .
Proof. Noting that each upper semicontinuous mapping on with compact values is also metric upper semicontinuous, by Lemma 5, Theorems 11 and 12, there exists a dense residual set in such that is metric lower semicontinuous on it; definitely, we have as also lower semicontinuous on . Then for each , and any open neighborhood of in , there is an open such that for all . Noting that , we have , . Then, is definitely an essential fixed point of with respect to .
Theorem 14. For each , there exists at least one minimal essential set of with respect to .
Proof. From Theorem 12, for each , is upper semicontinuous with compact values. Then, the set is closed. Hence, is essential with respect to from Definition 9. For each essential set , if , then is a minimal essential set of .
If is not a minimal essential set of . Let denote the collection of all essential subsets in . Then each decreasing chain of elements in has a lower bound, which is the intersection of the chain and is compact. Therefore, there is a minimal element in by Zorn’s lemma. For each essential set in with , if , we can obtain that since is minimal; clearly, if , we also have . Then, the set is minimally essential. Therefore, itself is either a minimally essential set or a set which includes at least one.
Remark 15. (a) By Theorem 13, a result for generic stabilities for fixed points of mappings on is obtained. (b) For the multi-valued mapping in Example 4, clearly, , is a -Lipschitz selection of , hence, and , a noncompact set.
Remark 16. For the space of upper semicontinuous multi-valued mapping with closed convex values on (see ), we know that the upper semicontinuity of the fixed point mapping results in the compactness of fixed point set and the existence of essential sets of fixed points. However, for the space (including Michael type mappings, lower semicontinuous multi-valued mappings with closed convex values), the fixed point mapping is neither upper semicontinuous nor lower semicontinuous. In the space , Theorem 14 shows the existence of essential sets of , and each closed set including a minimal essential set of is essentially stable.
Finally, as an application, we consider a kind of differential inclusion as follows: where the values of , is defined in a closed region , and is a multi-valued mapping. Let denote the solution set of this differential inclusion for each . Then, if is a -Lipschitz selection of , each initial value problem, must have a unique solution on , . Noting this fact, for each , if we also use to denote all Lipschitz selections of , then we now denote the solution set of the above differential inclusion with respect to all . Then, we have , . If fixed points is instead by solutions in Definition 9, we can define essential solution sets , minimal essential solution sets , and essential solutions of above differential inclusion in with respect to .
Theorem 17. For almost all , there has been at least an essential solution in ; for each , there exists a minimal essentially stable set which can resist the perturbation of in .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This project is supported by the Natural Science Foundation of Guangxi (no. 2012GXNSFBA053013 and no. 2013GXNSFBA19004) and Doctoral Research Fund of Guilin University of Technology.
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