Abstract and Applied Analysis

Volume 2014 (2014), Article ID 238191, 10 pages

http://dx.doi.org/10.1155/2014/238191

Research Article

Class 𝔄-, General Type Theorems, and Their Applications in Topological Vector Space

Gusheng Tang¹ and Qingbang Zhang²

¹School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China

²College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China

Received 12 December 2013; Accepted 14 April 2014; Published 6 May 2014

Academic Editor: Fu-quan Xia

Copyright © 2014 Gusheng Tang and Qingbang Zhang. 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.

Abstract.

The class 𝔄- and generalized mapping are introduced, and some generalized theorems are proved. As applications, Ky Fan’s matching theorem and Fan-Browder fixed-point theorem are extended, and some existence theorems of solutions for the generalized vector equilibrium problems are established under noncompact setting, which improve and generalize some known results.

1. Introduction and Preliminaries

In 1929, Knaster, Kurnatoaski, and Mazurkiewicz proved the well-known theorem on -simplex. In 1961, Fan [1] generalized the theorem from Euclid space to infinite dimensional topological vector spaces by introducing mapping. In 1989, Park [2] introduced - mapping which is a generalized form of the mapping and obtained some new theorems. In 1991, Chang and Zhang [3] improved fundamentally Ky Fan’s mapping that makes theory have great development. Since then, many results related to principle were obtained and applied universally in the fields of nonlinear analysis (see [4–17]).

Let and be topological spaces, let and denote the nonempty finite subset of and the set of the nonempty subsets of , respectively, let,   denote the interior of in , and let be the closure of in (when , , and ). is said to be compactly closed (resp., compactly open) in if, for every nonempty compact subset of ,   is closed (resp., open) in . The compact closure of and the compact interior of (see [10]) are defined, respectively, by

It is easy to see that ,  ,  . For every nonempty subset of , the subset is closed in and is open in . The multivalued mapping is said to be transfer compactly open valued (resp., transfer compactly closed valued) on , if, for every and for each nonempty compact subset of ,  (resp., implies that there exists such that (resp., (see [10]). The mappings and are defined as and , respectively.

Recently, many scholars (see [18–21]) not only studied further the theorem involving mapping, but also established some new theorem, fixed-point theorems, and coincidence theorems and utilized them to research the existence of solution to generalized vector equilibria, which makes the theory more perfect and rich.

In this paper, we first introduce the new generalized class consisting of all multifunctions that have the generalized property and prove some theorems for - mapping. Applying these theorems, the Ky Fan matching theorems and the Fan-Browder fixed-point theorems are generalized. Finally, we establish some new existence theorems of solutions for generalized vector equilibrium problems under noncompact setting. These theorems improve and generalize many known results in the literature.

Definition 1. Let be a nonempty set, a nonempty convex subset of a linear space, and a topological space, and let ,  , and be three multivalued mappings. is said to be a - mapping with respect to if, for any , there exists , such that, for any , one has

The multivalued mapping is said to have the - property, if, for any -KKM mapping with respect to , the family has the finite intersection property. Let the set be denoted by -.

Remark 2. mapping with respect to is strictly weaker than the generalized - mapping with respect to in [5]. It is easy to see that Definition 1 is not the degenerate form of Definition 1 in [11] and Definition 3 in [6] and not a special case of Definition 1 in [6] and Definition 2 in [7].

Example 3. Let ,   be three mappings defined as follows:

Then is - mapping with respect to . In fact, for any , take . For any , it is easy to know that

Therefore, by the definition of and , we have

It follows that

However, for ,  ,  . Hence

Hence, is not a generalized - mapping with respect to .

Lemma 4 (see [12]). Let and be topological spaces and a set-valued mapping with . Then the following conditions are equivalent:

(i)is transfer compactly open valued,(ii)for each compact subset of and for each , there exists such that and .

Lemma 5 (see [13]). Let and be topological spaces and a set-valued mapping with for each . Then the following conditions are equivalent:

(i) is transfer compactly closed valued,(ii)the mapping defined by for each is transfer compactly open valued,(iii)for each compact subset of , ,(iv)for each compact subset of , .

Lemma 6. Let -,  and let be a nonempty subset of and a nonempty convex subset of ,  ; then -.

Proof. Suppose that is a - mapping with respect to . Then, for each , there exists , such that, for any , we have . We define a set-valued mapping by

Obviously, is also a - mapping with respect to . Since -, the family has the finite intersection property which implies that the family has the finite intersection property.

2. General Theorems

Theorem 7. Let be a topological space, a convex space, and a Hausdorff space. Suppose that , and are three multifunctions satisfying the following:

(1)- such that is compact in ,(2) is transfer compactly closed values,(3) is a - mapping with respect to .

Then .

Proof. Define by

By (3), for any , there exists , such that, for any , we have . It follows from that we have

This shows that is a - mapping with respect to , and so has finite intersection property. Since is transfer compactly closed by (1) and so is compactly closed and is compact in by (1), consequently is closed in compact subset of Hausdorff space . Therefore,

By Lemma 5, we have that holds.

Remark 8. Theorem 7 improves Theorem 4.3 of Chang et al. [5] in the following two aspects: (1) the generalized - mapping is generalized to - mapping with respect to ; (2) the compactly closed values property is replaced by the transfer compactly closed values property.

Theorem 9. Let be a topological space, a topological vector space, and a Hausdorff space. Suppose that , and are three multivalued mappings satisfying the following:

(1)-,(2) is transfer compactly closed values,(3) is a - mapping with respect to ,(4)for each compact subset of ,   is compact in , and for each convex subset of , is convex,(5)there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that . Then .

Proof. Suppose that the conclusion is not true; then . Define by for each ; then is nonempty for all . From (2) and Lemma 5, it follows that is transfer compactly open mapping on . Since is compact in , by Lemma 4, we have . Hence there exists such that . By (5), there exists compact convex subset of including such that

and . Since , we have , so . Therefore,

Since is a convex subset of , by Lemma 6, -. Define set-valued mapping by and for each . Then we have , and

By Lemma 4, we have that is transfer compactly open valued on . Hence it follows from Lemma 5 that is transfer compactly closed valued on .

We claim that is a - mapping with respect to . Since is a - mapping with respect to , for any , there exists such that, for any ,

Since is convex in and for each , we have

Hence . Therefore, is a - mapping with respect to . By Theorem 7, we have , which is a contradiction. Therefore, .

Remark 10. Theorem 9 generalizes Theorem 3.3 of Lin and Wan [14] in the following two aspects: (1) from transfer closed values to transfer compactly closed values; (2) from generalized mapping to - mapping with respect to .

Theorem 11. Let be two convex spaces and a Hausdorff space. Suppose that , and are three multivalued mappings and - satisfying the following:

(1) is a compact convex subset of if is a compact convex subset of ,(2)for any compact subset of , is compact in ,(3) is a - mapping with respect to such that is transfer compactly closed,(4)there exist a nonempty compact convex subset of and a compact subset of such that

Then .

Proof. Assume that ; then we have , and . It follows from Lemma 5 that

Since is transfer compactly open (by condition (3)), , where is open in for each and is open in . Therefore, there exists a finite subset of such that

By (4), we have

Let ; then is a compact convex subset of such that

that is,

Define by

Since is a - mapping with respect to , for any , there exists such that, for any . It follows that

which shows that is a - mapping with respect to for the triple . Since -, it follows easily that -. Moreover, since is compact convex, we have , and . Conditions (1) and (2) imply that is compact in . Then the compactness of implies that is compact in . Applying Theorem 7 to , , and , we obtain that

which contradicts . This completes the proof.

Remark 12. Theorem 11 improves Theorem 5.1 of Chang et al. [5] in the following two aspects: (1) the generalized - mapping is generalized to - mapping with respect to ; (2) the compactly closed values property is replaced by the transfer compactly closed values property.

3. Matching Theorems and Fixed-Point Theorems

In order to apply the above theorem to show the fixed-point theorems, we first establish the following generalization of the Ky Fan’s matching theorem.

Theorem 13. Let be a nonempty set, a convex space, and a Hausdorff space. Suppose that ,  , and are three multivalued mappings satisfying the following:

(1)- such that is compact in ,(2) is transfer compactly open in ,(3).

Then there exists such that

Proof. Assume that, for any , . Then for any . Noting that is transfer compactly closed, we have that the conditions of Theorem 7 are satisfied for the mappings , , and . Thus

which implies that ; this contradicts with (3). So there exists a nonempty finite subset such that .

Remark 14. Theorem 13 implies that Theorem 7 holds and if not then , where is defined by . Theorem 13 shows that there exists such that

which implies that , contradicting the fact that is a - mapping with respect to .

Theorem 15. Let be a nonempty subset of a compact convex space and a Hausdorff space. Suppose that satisfies the following:

(1) is transfer compactly open in ,(2).

Then for any there exist a finite subset of and an such that .

Proof. Let be defined by for . Then -. Since and is compact, is compact in . Furthermore, by (2) we have that

So all of the conditions of Theorem 13 are satisfied for the mappings , , and . Thus, there exists a finite subset of such that

Remark 16. If is compactly open in for each ,   is a subset of a convex subset of a topological vector space , and is the inclusion mapping of into , then Theorem 15 reduces to Lemma 1 by Fan in [15].

In the sequel, we give the famous Fan-Browder type fixed-point theorem. We first give the following conclusion.

Theorem 17. Suppose that are two convex spaces and is a Hausdorff space. Assume that ,  , and - are three functions satisfying the following:

(1) is a compact convex subset of if is a compact convex subset of ,(2) is compact in if is compact in ,(3)for any ,   is a nonempty convex subset of ,(4)there exists a transfer compactly open values mapping such that any ,  , and ,(5)there exist a nonempty compact convex subset of and a compact subset of such that

Then there exists a finite subset of ; for any there exist such that, for any , there exists such that .

Proof. Define by for . Then is transfer compactly closed in .

(i) Suppose that . In this case, it is easy to know that . Then it follows from Theorem 11 that is not a - mapping with respect to ; that is, there exists a finite subset of ; for any , there exists such that . Choose such that . Then , and for any . Thus for any , there is such that . Since is convex, we see that .

(ii) Suppose that . Checking the proof of case (i), it suffices to show that is not a - mapping. On the contrary, assume that, for any , there exists such that, for any , we have . Then is a - mapping with respect to , and so by Theorem 11, we have

In particular, ; that is,

which contradicts the assumption that . This completes the proof.

Remark 18. Theorem 17 improves Corollary 5.2 of Chang et al. [5] in the following three aspects: (1) from generalized - mapping to - mapping with respect to ; (2) from compactly closed values to transfer compactly closed values; (3) from the single-valued mapping to the multivalued mapping .

For Theorem 17, if reduces to a single-value mapping , we have the following conclusion.

Theorem 19. Suppose that , are two convex spaces and is a Hausdorff space. Assume that , and - are three mappings satisfying the following conditions:

(1) is a compact convex subset of if is a compact convex subset of ,(2) is compact in if is compact in ,(3)for any ,  is a nonempty convex subset of ,(4)there exists a transfer compactly open values mapping such that any , and ,(5)there exist a nonempty compact convex subset of and a compact subset of such that

Then there exists a finite subset of ,  , such that, for any , there exists   such that .

Proof. The proof is similar to Theorem 17.

Corollary 20. Suppose that is a compact convex space. Assume that , and - are three functions satisfying the following:

(1) is a compact convex subset of if is a compact convex subset of ,(2) is compact in if is compact in ,(3)for any is a nonempty convex subset of ,(4)there exists a transfer compactly open values mapping such that any , and .

Then there exists a finite subset of , such that, for any , there exists   such that .

Corollary 21. Let be a compact convex space. Suppose that satisfies the following:

(1)for any ,   is a nonempty convex subset of ,(2) is transfer open in .

Then there is an such that .

Proof. Let and be the identity mapping ; it follows from Corollary 20 that there exist a finite subset of and such that . Furthermore, . This completes the proof.

Remark 22. If is a nonempty compact convex subset of a topological vector space and for any ,   is open in , the above corollary is just the Fan-Browder type fixed-point theorem 1 in [16].

4. Generalized Vector Equilibrium Problems

In this section, we will introduce some definitions and conclusions and show the existence of solutions to the generalized vector equilibrium problems.

Definition 23. Let , be two topological spaces and nonempty sets. Let ,  ,   be multivalued mapping. A generalized vector equilibrium problem is to find such that for all . A generalized vector equilibrium problem is to find such that, for each , there exists satisfying .

Lemma 24. Let be a topological vector space and nonempty sets. Let ,  ,  ,  ,  ,  . Suppose that the following conditions are satisfied:

(1)for each , there exists such that ,(2)for each , there exists such that implies that whenever and ,(3)for each ,   is convex.

Then defined by is a - mapping with respect to .

Proof. If the conclusion does not hold, then there exists ,   such that

Therefore, there exist and such that, for each ,  . By the definition of , we have

By condition (2), for all and , we have

Thus , and . Therefore, ; this contradicts with (1). The proof is completed.

Lemma 25. Let be a topological vector space and nonempty sets. Let ,  ,  ,  ,  . Suppose that the following conditions are satisfied:

(1)for each and ,  ,(2)for each ,  , there exists such that implies that for all and some ,(3)for each ,   such that is convex.

Then defined by is a - mapping with respect to .

Proof. There exists ,   such that

Therefore, there exist and such that, for each ,  . By the Definition of , we have

By the condition (2), there exists such that, for each , we have that   and

Thus , and . Therefore, for some ; this contradicts with (i). The proof is completed.

Remark 26. To avoid the structure of the space, Lemmas 24 and 25 generalize Lemmas 2.4 and 2.5 of X. P. Ding and T. M. Ding [13] from the following two aspects: (1) from generalized mapping with respect to to - mapping with respect to ; (2) condition (2) in our results is obviously weaker than that in [13].

Definition 27 (see [13]). Let and be topology spaces and nonempty set. Let and be set-valued mappings. is said to be a -transfer compactly continuous mapping of the generalized vector equilibrium problem in first argument if, for any compact subset of and any , there exists such that ; then there is a point such that .

Proposition 28 (see [13]). Let and be topological spaces and nonempty set. Let and be set-valued mappings. is said to be a -transfer compactly continuous mapping of the generalized vector equilibrium problem in first argument if and only if the mapping defined by is a transfer compactly closed-valued mapping.

Proposition 29 (see [13]). Let , and be topological spaces. Let and be set-valued mappings such that

(i) has closed (resp., open) graph;(ii)for each is upper semicontinuous on each compact subset of .

Then the mapping defined by has compactly closed values.

Theorem 30. Let be a topological space, a topological vector space, and a Hausdorff space; let ,  -,  ,  ,   be multivalued mappings. Suppose that the following conditions are satisfied:

(1) is a -transfer compactly continuous mapping of the generalized vector equilibrium problem ,(2)for each , there exists such that ,(3)for each , there exists such that which implies that whenever and ,(4)for each , the set is a convex subset of ,(5)for each compact subset of ,   is compact in , and for each convex subset of ,   is convex,(6)setting by , there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that .

Then there exists such that ; that is, is a solution to the generalized vector equilibrium problem .

Proof. By condition (1), is transfer compactly closed-valued mapping from Proposition 28. By conditions (2)–(4) and Lemma 24, we know that is a - mapping with respect to . Conditions (5) and (6) imply that conditions (4) and (5) of Theorem 9 hold. From Theorem 9, it follows that and for all .

Theorem 31. Let be a topological space, a topological vector space, and a Hausdorff space, and let ,  -,  ,  ,  ,   be multivalued mappings. Suppose that the following conditions are satisfied:

(1) is a -transfer compactly continuous mapping of the generalized vector equilibrium problem ,(2)for each and ,  ,(3)for each , there exists such that implies that for all and some ,(4)for each ,   such that is convex,(5)for each compact subset of ,   is compact in , and for each convex subset of ,   is convex,(6)setting by , there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that .

Then there exists such that ; that is, is a solution to the generalized vector equilibrium problem .

Proof. By condition (1), is transfer compactly closed-valued mapping from Proposition 29. By conditions (2)–(4) and Lemma 25, we have that is a generalized - mapping with respect to . Conditions (5) and (6) imply that conditions (4) and (5) of Theorem 9 hold. From Theorem 9, it follows that and for all .

Theorem 32. Let be a topological space, a topological vector space, and a Hausdorff space, and let ,  -,  ,  ,  ,   be multivalued mappings. Suppose that the following conditions are satisfied:

(1) has closed graph,(2)for each ,   is upper semicontinuous on each compact subset of with nonempty compact values on ,(3)for each , there exists such that ,(4)for each , there exists such that implies that whenever and ,(5)for each ,   such that is convex,(6)for each compact subset of ,   is compact in , and for each convex subset of ,   is convex,(7)setting by , there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that .

Then there exists such that ; that is, is a solution to the generalized vector equilibrium problem .

Proof. By condition (1) and Proposition 29, has compactly closed values and so it is a transfer compactly closed-valued mapping. Hence the conclusion of Theorem 32 holds from Theorem 30.

Theorem 33. Let be a topological space, a topological vector space, and a Hausdorff space, and let ,  -,  ,  ,  ,   be multivalued mappings. Suppose that the following conditions are satisfied:

(1) has closed graph,(2)for each ,   is upper semicontinuous on each compact subset of with nonempty compact values on ,(3)for each , there exists such that ,(4)for each , there exists such that implies that for all and some ,(5)for each ,   such that is convex,(6)for each compact subset of is compact in , and for each convex subset of ,   is convex,(7)setting by , there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that .

Then there exists such that .

Proof. By condition (1) and Proposition 29, has compactly closed values and so it is a transfer compactly closed-valued mapping. Hence the conclusion of Theorem 33 holds from Theorem 31.

The following result is a simplicity version of Theorem 1 in [17].

Lemma 34. Let , , , and be topological spaces. Let and be set-valued mappings.

(1)If, for each fixed and are both lower semicontinuous, then the mapping defined by satisfies that, for each is lower semicontinuous on .(2)If, for each fixed ,   and are both upper semicontinuous with compact values, then, for each ,   is upper semicontinuous on with nonempty compact values.

Theorem 35. Let be a topological space, a topological vector space, and a Hausdorff space, and let ,  -,  ,  ,  ,  ,   be multivalued mappings. Suppose that the following conditions are satisfied:

(1) has closed graph,(2)for each ,   and are both upper semicontinuous compact values on ,(3)for each , there exists such that ,(4)for each , , there exists such that implies that whenever and ,(5)for each ,   is convex,(6)for each compact subset of ,   is compact in , and for each convex subset of ,   is convex,(7)setting by , there exists a nonempty compact subset of such that, for each , there exists compact convex subset of including such that .

Then there exists such that, for each , there exists satisfying ; that is, is a solution to the generalized vector equilibrium problem .

Proof. Define set-valued mappings by and for each and , respectively. By Lemma 34 and Theorem 32, the rest is similar to the proof of Theorem 4.7 in [13].

Remark 36. For the above results, Theorems 30 and 31 generalize Propositions 4.3 and 4.4 of Lin and Wan in [14] in the following two aspects: (1) from transfer closed values to transfer compactly closed values; (2) from generalized mapping to - mapping with respect to . To avoid the structure of the space, condition (3) in our results is more general than that in Theorems 4.1, 4.2, 4.3, 4.5, and 4.7 of [13].

Conflict of Interests

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

Acknowledgments

The authors are grateful to the referees’ suggestion for improving the paper. The research is supported by the Natural Science Foundation of Hunan Province (no. 2014JJ4044).

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