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A New Fixed Point Theorem and Applications
A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space) without the convexity structure. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are different from known results in the literature.
In the last decade, the theory of fixed points has been investigated by many authors; see, for example, [1–11] and references therein, which has been exploited in the existence study for almost all areas of mathematics, including optimization and applications in economics. Now, there have been a lot of generalizations of the fixed points theorem under different assumptions and different underlying space, and various applications have been given in different fields.
On the other hand, the weak KKM-type theorem introduced by Balaj  has attracted an increasing amount of attention and has been applied in many optimization problems so far; see [12–14] and references therein.
Inspired by the research works mentioned above, we establish a collectively fixed points theorem and a fixed point theorem. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are new and different from known results in the literature.
The rest of the paper is organized as follows. In Section 2, we first recall some definitions and theorems. Section 3 is devoted to a new collectively fixed points theorem under noncompact situation on GFC-space and a new fixed point theorem. In Section 4, we show a new weak KKM theorem in underlying GFC-space, and, by using the weak KKM theorem, a new minimax inequality of Ky Fan type is developed.
Let be a topological space and . Let and denote the interior of in and in , respectively. Let denote the set of all nonempty finite subsets of a set , and let denote the standard -dimensional simplex with vertices . Let and be two topological spaces. A mapping is said to be upper semicontinuous (u.s.c.) (resp., lower semicontinuous (l.s.c)) if for every closed subset of , the set (resp., ) is closed.
A subset of is said to be compactly open (resp., compactly closed) if for each nonempty compact subset of , is open (resp., closed) in .
These following notions were introduced by Hai et al. .
Definition 1. Let be a topological space, a nonempty set, and a family of continuous mappings , . A triple is said to be a generalized finitely continuous topological space (GFC-space) if and only if for each finite subset of , there is of the family .
In the sequel, we also use to denote .
Definition 2. Let be a multivalued mapping. A subset of is called an -subset of if and only if for each and each , one has , where is the face of corresponding to , that is, the simplex with vertices . Roughly speaking, if is an -subset of , then is a GFC-space.
The class of GFC-space contains a large number of spaces with various kinds of generalized convexity structures such as FC-space and G-convex space (see [15–17]).
Definition 3 (see ). Let be a GFC-space and a nonempty set. Let and be two set-valued mappings; is called a weak KKM mapping with respect to , shortly, weak T-KKM mapping if and only if for each , and , .
Definition 4 (see ). Let be a Hausdorff space, a GFC-space, a topological space, , , and . Let . is called -GFC quasiconvex if and only if for each , , , and , one has the implication , for all implies that for all .
For , define and by and , respectively.
Lemma 5 (see ). For , if is -GFC quasiconvex, then is a weak T-KKM mapping.
The following result is the obvious corollary of Theorem 3.1 of Khanh et al. .
Lemma 6. Let be a family of GFC-spaces and a compact Hausdorff space. For each , let and be such that the conditions hold as follows:(i)for each , each and each , one has for all , (ii) for all . Then, there exists such that for all .
3. Fixed Points Theorems
Let be an index set, topological spaces, , and . The collectively fixed points problem is to find such that , for all .
Theorem 7. Let be a family of GFC-spaces and a Hausdorff space. For each , let , , and with the following properties: (i)for each , , and , one has for all , (ii)for each compact subset of and each , ; (iii)there exists a nonempty compact subset of and for each , there exists an -subset of containing with being compact such that
where , , and .
Then, there exists such that for all .
Proof. As is a compact subset of , by the condition (ii), there exists a finite set , such that
By the condition (iii), there exists an -subset of containing such that
and it follows that
We observe that the family is a family of GFC-space and is compact for each , defining set-valued mapping and as follows:
We check assumptions (i) and (ii) of Lemma 6 for replaced and by and , respectively. By (i) and the definition of -subset, for each , each and each , we have
then assumption (i) of Lemma 6 is satisfied.
By (4), we have On the other hand, for all , Hence, Thus, (ii) of Lemma 6 is also satisfied. According to Lemma 6, there exists a point such that for all .
Remark 8. Theorem 7 generalizes Theorem 3.4 of Ding  from FC-space to GFC-space, and our condition (iii) is different from its condition (iii). Theorem 7 also extends Theorem 3 in . Note that Theorem 7 is the variation of Theorem 3.2 in .
As a special case of Theorem 7, we have the following fixed point theorem that will be used to prove a weak KKM theorem in Section 4.
Corollary 9. Let be the Hausdorff space, a GFC-space, , , and with the following properties: (i)for each , , and , one has , (ii)for each compact subset of , , (iii)for each , there exists an -subset of containing with being compact such that Then, there exists such that .
Theorem 10. Let be a Hausdorff space, a GFC-space, a nonempty set, , , and ; assume that (i)H is a weak T-KKM mapping, (ii)for each , the set is compactly closed, (iii)there exists a compact of , and, for any , there exists an -subset of containing with being compact such that Then, there exists a point such that for each .
Proof. Define and by
Suppose the conclusion does not hold. Then, for each , there exists a such that It is easy to see that has nonempty values. By (ii), for each , is compactly open. Then, Since is a compact subset of , then there exists such that Then, assumption (ii) of Corollary 9 is satisfied.
It follows from (iii) that there exists a compact of and for any , there exists a -subset of containing with being compact such that Therefore, assumption (iii) of Corollary 9 is also satisfied.
Furthermore, has no fixed point. Indeed, if , then there exists such that which contracts the definition of . Thus, assumption (i) of Corollary 9 must be violated; that is, there exist an , , and such that That is, for each , Hence,
On the other hand, since is a weak T-KKM mapping and , we have which is contradict. This completes the proof.
Remark 11. (1) Theorem 10 extends Theorem 1 in  from the G-convex space to GFC-space, and our proof techniques are different. Theorem 10 also generalizes Theorem 4.1 of  from the compactness assumption to noncompact situation.
(2) If is a topological space, condition (ii) in Theorem 10 is fulfilled in any of the following cases (see ): (i) has closed values, and is u.s.c, on each compact subset of . (ii) has compactly closed values, and is u.s.c, on each compact of subset of and its values are compact.
Theorem 12. Let be a Hausdorff space, a GFC-space, a topological space, u.s.c., . and ; assume that(i)for each , is u.s.c. on each compact subset of , (ii) is -GFC quasiconvex for all sufficiently close to , (iii) there exists a compact of , and, for any , there exists an -subset of containing with being compact such that Then,
Proof. Let be arbitrary. By Lemma 5 and condition (ii), is a weak T-KKM mapping. It follows from condition (i) that has closed values. Hence, the set is compactly closed for all (see Remark 11 (2)). Thus, all the conditions of Theorem 10 are satisfied, and so there exists an such that This implies that and so Since is arbitrary, we get the conclusion. This completes the proof.
Remark 13. Theorem 12 improves Theorem 4.2 of  from the compactness assumption to noncompact situation. Theorem 12 also extends Theorem 4 of  from compact G-convex space to noncompact GFC-space. Our result includes corresponding earlier Fan-type minimax inequalities due to Tan , Park , Liu , and Kim .
This work was supported by the University Research Foundation (JBK120926).
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Copyright © 2013 Min Fang and Xie Ping Ding. 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.