Some New Coincidence Theorems in Product GFC-Spaces with Applications

Zhao, Jianrong

doi:https://doi.org/10.1155/2014/392018

Abstract and Applied Analysis

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Volume 2014 | Article ID 392018 | https://doi.org/10.1155/2014/392018

Some New Coincidence Theorems in Product GFC-Spaces with Applications

Jianrong Zhao¹

Academic Editor: Xie-ping Ding

Received28 Feb 2014

Accepted12 Apr 2014

Published06 May 2014

Abstract

We first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.

1. Introduction and Preliminaries

Recently, Ding [1, 2], X. P. Ding and T. M. Ding [3], Tang et al. [4], Fang and Huang [5] established some collective fixed point theorems, coincidence theorems, and KKM-type theorems for the families of set-valued mappings, which are defined on FC-spaces or product FC-spaces without any convexity structure.

Very recently, Khanh and Quan [6], Khanh et al. [7] defined a new notion by GFC-space which is a generalization of FC-space and proved some continuous selections theorems, collectively fixed point theorems, coincidence theorems, and KKM-type theorems in this new GFC-space.

Motivated and inspired by the work mentioned above, we will give some new collective fixed point and coincidence theorems in product GFC-spaces in the present paper. For this purpose, we first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.

For our purpose, first, we present some known definitions and preliminary results. For a nonempty set denotes the family of all nonempty finite subsets and denotes the family of all subsets of . We denote the standard -simplex with vertices by . The following notion was introduced by Khanh and Quan [6].

Definition 1 (see [6]). (a) A generalized finitely continuous topological space or a GFC-space consists of a topological space and a nonempty set such that for each finite subset , one has a continuous mapping .
(b) Let and be given. Then is called an -subset of with respect to if for any and for any , one has , where .

Now we give a new concept as follows.

Definition 2. Assume that is a GFC-space, , and . Then is said to be a GFC-subspace of with respect to if for each and for any , one has , where .

Remark 3. By the Definition 2, we know that is also a GFC-space. Clearly, if is an -subset of with respect to , then is a GFC-subspace of with respect to . In addition, if , then a GFC-subspace of with respect to coincides with an FC-subspace of with respect to (see [1]).

Assume that is a family of GFC-subspace of with respect to and , where is an index set. It then follows from Definition 2 that is also a GFC-subspace of with respect to . For any given subset of , we define GFC-hull of with respect to by

We simply write instead of when . We can easily show that is also a GFC-subspace of with respect to .

Let be a topological space. A set is called compactly closed (resp., compactly open) if for each nonempty compact set such that is closed (resp., open) in . And the compact interior and the compact closure of (see [8]) are defined by

Clearly if is a nonempty compact subset of , then we deriver that , and .

Assume that and are nonempty sets and is a topological space. Define two set-valued mappings and . is called transfer compactly upper semicontinuous in with respect to (see [9]) if for any nonempty compact subset of and any , the set means that there is a relatively open neighborhood of in and a point such that for all .

A mapping is called transfer compactly open-valued (resp., transfer compactly closed-valued) on (see [8]) if for each and each nonempty compact subset of , (resp., ) means that there is such that (resp., ).

We need the following two results. The first statement was proved by Ding and Park [9].

Lemma 4 (see [9]). Let and be two nonempty sets and a topological space. Let and be two set-valued mappings. Then is transfer compactly upper semicontinuous in with respect to if and only if the mapping defined by is transfer compactly closed-valued on .

The second statement is Lemma 1.1 of Ding [8].

Lemma 5. Assume that and are two topological spaces and is a set-valued mapping with nonempty values. Then the following statements are equivalent: (i) is transfer compactly open-valued;(ii)for any compact subset of and any , one has satisfying that and .

Throughout this paper, we always let and be any given index set. Now we give the following statement, which generalizes Lemma 1.1 of Ding [1].

Lemma 6. Suppose that is a GFC-space for each . If , and , then is also a GFC-space.

Proof. Let be the projective mapping from to for each . For any given , we denote . Note that is a GFC-space. Then we have a continuous mapping for each . So we may let a mapping by , for any . It follows that is a continuous mapping, which means that is a GFC-space. So Lemma 6 is proved.

2. Continuous Selection and Collective Fixed Points

Theorem 7. Assume that is a GFC-space and is a compact topological space. Suppose that and are such that (i);(ii)for any given , is a GFC-subspace of with respect to .Then one has a continuous selection of satisfying , where and are continuous for some .

Proof. By condition (i), we know that there exists such that since is compact. Assume that is the continuous partition of unity subordinated to the open covering ; then for any and , one has Let be a mapping with . Clearly is continuous and for any , one has where . So by (3), we get . It then follows from condition (ii) that for , . It is easy to see that , which implies that is a continuous selection of . The proof of Theorem 7 is completed.

Remark 8. Applying the definition of GFC-subspace, we extend Theorem 2.2 of Tarafdar [10], Proposition 1 of Browder [11], and Theorem 2.1 of Ding [2] to GFC-spaces without any convexity.

Theorem 9. Assume that and is a compact GFC-space for each . Suppose that and are two set-valued mappings satisfying the following conditions: (i);(ii)for any given , is a GFC-subspace of with respect to . Then one has a point such that for .

Proof. For any given , using Theorem 7, we obtain that there are continuous mappings and such that is a continuous selection of for some positive integer . Assume that is the linear hull of the set for each . Clearly is a locally convex Hausdorff topological vector space as it is finite dimensional and is a compact convex subset of . Moreover, is a locally convex Hausdorff topological vector space and is a compact convex subset of . Now let and be two continuous mappings with where is the projection of onto for any given . It then follows from the Tychonoff fixed point theorem that the continuous mapping has a fixed point ; that is, . Let . It follows that This means that for each . So Theorem 9 is proved.

Theorem 10. Assume that is a topological space and is a GFC-space for each . Suppose that and and is a continuous mapping such that (i)for any compact subset of , ;(ii)for any is a GFC-subspace of with respect to ;(iii)there is a nonempty subset of such that the set is empty or compact in , and for any , there exists a compact GFC-subspace of with respect to containing .Then one has a point such that for each .

Proof. For any given , if is a nonempty compact subset of , then by (i) one has
Moreover, noticing that is compact, we can find that there is a finite set such that
It then follows from (7) that
If is empty in (iii), then we derive that
From condition (iii), we know that there exists a compact GFC-subspace of with respect to containing . So by (8) and (9), we get
Now let , , and . By condition (iii), we deduce that is a compact GFC-space. It then follows from Lemma 6 that is also a compact GFC-space. Let and be two set-valued mappings with
In order to show that the conditions (i) and (ii) of Theorem 9 hold, we only need to show that is a compact GFC-subspace of with respect to , and . By conditions (ii) and (iii), it is easy to see that is a compact GFC-subspace of with respect to . It remains to show that . On one hand, for each , we deduce that
On the other hand, by (10), we obtain
It then follows form (12), (13), and the continuity of that
Hence . Then Theorem 9 tells us that there exists a point such that for each . This completes the proof of Theorem 10.

Theorem 11. Let be a topological space and a GFC-space for each . Assume that , and are set-valued mappings and is a continuous mapping such that (i)for any compact subset of , ;(ii)for any ;(iii)if is not compact, then there is a nonempty subset of such that a compact subset of satisfying , and for any , there exists a compact GFC-subspace of with respect to containing .Then we have a point such that for each .

Proof. Using the similar argument of Theorem 10, we only need to show that the conditions (ii) and (iii) of Theorem 10 are satisfied. By the definition of , we know that is a GFC-subspace of with respect to . It then follows from condition (ii) that is a GFC-subspace of with respect to , which means that the condition (i) of Theorem 10 holds. It remains to deal with condition (iii). If is noncompact, by (iii), we get Clearly is a closed subset of compact set . If is nonempty, then is compact in . This means that the condition (iii) of Theorem 10 is satisfied. So the statement of Theorem 11 follows immediately from Theorem 10.

Remark 12. (a) Let be a topological spaces for each . By Lemma 5, if using assumption (i) in Lemma 5 for assumption (i) in Theorems 9 and 10, then the statements of Theorems 9 and 10 are still true.
(b) The results of Theorems 10 and 11 generalize Theorem 2.1 of Lan and Webb [12], Theorem 1 of Ansari and Yao [13], Theorem 2.2 of Ding and Park [14], Theorem 3.1 of Ding and Park [15], and Theorem 3.1 of Lin and Ansari [16] and Ding [2] to GFC-spaces.

3. Coincidence Theorems for Two Families of Set-Valued Mappings

Theorem 13. Let and be GFC-spaces for any and . Assume that and . Let , and be set-valued mappings satisfying the following conditions: (i)for any compact subset of ;(ii)for any , is a GFC-subspace of with respect to ;(iii)there exists a nonempty subset of of such that the set is empty or compact in and for each , there is a compact GFC-subspace of with respect to containing ;(iv)for any compact subset of ;(v)for any , is a GFC-subspace of with respect to .Then one has and such that and for any and .

Proof. For any given , if is empty in (iii), it is easy to see that
If is nonempty compact set in , by (iv), we know that
Note that is compact. Then we can find a finite set such that
It then follows from (16) and (18) that if either is empty or compact in , we get
Furthermore, by (iii), there exists a compact GFC-subspace of with respect to containing . So by (19), we get Assume that , and . It then follows from Lemma 6 that is a compact GFC-subspace of with respect to . So is a compact GFC-space. We consider the restrictions and of and on , respectively, for each . Then by condition (ii), we have as a GFC-subspace of with respect to for any . From condition (i), we obtain
Moreover, by Theorem 7, there exists a continuous selection of for any . Let be a mapping with for any . Clearly is a continuous mapping. Define two set-valued mappings and by
Since is a compact GFC-subspace of relative to , by (v), we know that is also a compact GFC-subspace of with respect to .
Now we claim that . For any given , we deduce that
Then by (20) we obtain
It then follows form (23), (24), and the continuity of that
Hence . So the claim is proved.
Note that is a compact GFC-subspace of relative to . So by the above claim and Theorem 9, we know that there exists a point such that for any given . Assume that . Thus we derive that there exist and such that and for any and , which implies that Theorem 13 is true.

Theorem 14. Let and be GFC-spaces for any and . Assume that and . Let , , , , , and be set-valued mappings for each and . If the following conditions hold: (i)for any compact subset of , ,(ii)for any ,(iii)if is not compact, then there exists a nonempty subset of of as well as a compact subset of such that and for each , there is a compact GFC-subspace of with respect to containing ,(iv)for any compact subset of ,(v)for any ,then there exist and such that and for any and .

Proof. By the definition of the GFC-hull with respect to and condition (ii), it is easy to see that is a GFC-subspace of with respect to for any and . Similarly, by (v), we obtain that is a GFC-subspace of with respect to for any and . Then conditions (ii) and (v) of Theorem 13 hold. On the other hand, by (iii), if is not compact, we know that If is nonempty, then is a closed subset of compact set . This implies that is compact in . So condition (iii) tells us condition (iii) of Theorem 13 holds. Then the statement of Theorem 14 follows immediately from Theorem 13.

Remark 15. Theorems 13 and 14 improve Theorem 9 of Yu and Lin [17], Theorem 3.3 of Lin and Ansari [16], and Theorems 3.1 and 3.2 of Ding [2] to GFC-spaces without any convexity structure.

4. Applications

In the current section, we will give some applications of our theorems.

Theorem 16. Let be a GFC-space and a topological space for each . Assume that and are a subset of and , respectively. Let and for each , let be a continuous mapping and a set-valued mapping satisfying the following conditions: (i)for any compact subset of , ;(ii)for any ;(iii)if is not compact, then there exists a nonempty subset of of such that a compact subset of such that and for each , there is a compact GFC-subspace of with respect to containing .Then we have a point such that for each .

Proof. For any given , we define two set-valued mappings and by for all . It then follows from assumptions (i)–(iii) and Theorem 11 that for any , there exists a point such that , which implies that . So Theorem 16 is true.

Corollary 17. For each , let be a GFC-space, a set-valued mapping, , and . Assume that and are the families of subsets of and , respectively. If the following conditions hold: (i)for any compact subset of , ,(ii)for all ,(iii)if is not compact, then there exists a nonempty subset of of such that a compact subset of such that and for each , there is a compact GFC-subspace of relative to containing ,then .

Proof. Define a mapping as the projection of onto . Clearly is a continuous mapping. Using Theorem 16 with and , we find that there exists a point such that for any , which implies that . Hence .

In the rest of this section, for each and , we always let and be two nonempty set and and be two GFC-spaces. Assume that and . For any and , let , and be set-valued mappings. Then we have the follows results.

Theorem 18. Suppose that the following conditions hold: (i)for any compact subset of , ;(ii)for any ;(iii)if is not compact, then there exists a nonempty subset of of such that a compact subset of such that and for each , there is a compact GFC-subspace of with respect to containing ;(iv)for any compact subset of , ;(v)for any .Then there exist and such that and for each and .

Proof. Assume that , and are set-valued mappings as follows: It then follows from Theorem 14 that there exist and such that and for any and . By the above definition, we obtain that and for any and . This completes the proof.

Theorem 19. For each and , let and be topological spaces. Suppose that the following conditions hold: (i)for each , the set is nonempty and is transfer compactly upper semicontinuous in with respect to ;(ii)for any ;(iii)if is not compact, then there exists a nonempty subset of of as well as a compact subset of such that and for each , there is a compact GFC-subspace of with respect to containing ;(iv)for each , the set is nonempty and is transfer compactly upper semicontinuous in with respect to ;(v)for any . Then there exist and such that and for all and .

Proof. Let , and be set-valued mappings as defined in the proof of Theorem 18. Using the same argument of Theorem 18, we only need to show that the assumptions (i) and (iv) of Theorem 18 hold. From assumption (i) and Lemma 4, we know that for any is transfer compactly open-valued. Thus Lemma 5 tells us that the assumption (i) of Theorem 18 holds. In a similar way, from (iv) and Lemma 4, we obtain that the assumption (iv) of Theorem 18 is satisfied. So the statement of Theorem 19 follows immediately from Theorem 18.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the anonymous referees for careful reading of the paper and for helpful comments and suggestions which improved its presentation.

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Copyright © 2014 Jianrong Zhao. 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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