The existence and stability for weakly Ky Fan’s points of set-valued mappings

In this paper, the notion of weakly Ky Fan’s points of set-valued mappings is established, and we prove some existence theorems of weakly Ky Fan’s points for functions with no continuity or space with no compactness. Then, from the viewpoint of the essential stability, we prove that most of problems in weakly Ky Fan’s points (in the sense of Baire category) are essential.

MSC: 26D20, 26E25.

Ky Fan [1] gave an inequality for real valued functions which plays a very important role in nonlinear analysis (e.g., see Lin and Simons [2]). Let X be a nonempty compact convex subset of a Hausdorff topological vector space, and be such that (1) for all ; (2) for each fixed , is lower semicontinuous; (3) for each fixed , is quasiconcave, then there exists such that for all .

Tan, Yu and Yuan [3] defined the inequality above as the Ky Fan inequality and called such a point Ky Fan’s point, which is fundamental in proving many theorems in nonlinear analysis such as optimization problem, Nash equilibrium problem, variational inequality problem. There have been numerous generalizations of the Ky Fan inequality (see [4-8]). In [4], Yu and Yuan studied the existence of weight Nash equilibria and Pareto equilibria for multiobjective games using the Ky Fan minimax inequality. In [5], Luo proved the existence of an essential component of the solution set for vector equilibrium problems. Yang and Yu [6] gave a generalization of the Ky Fan inequality to vector-valued functions. They proved that for every vector-valued function (satisfying some continuity and convexity condition), there exists at least one essential component of the set of its Ky Fan’s points. Yu and Xiang [8] proposed a notion of essential components of Ky Fan’s points and proved its existence under some conditions, the Ky Fan’s points have at least one essential component. Besides, they proved that for every n-persons noncooperative game, there exists at least one essential component of the set of its Nash equilibrium points. Zhou, Xiang and Yang [9] studied the stability of solutions for Ky Fan’s section theorem with some applications. For our purpose, we give the notion of weakly Ky Fan’s points of set-valued mappings and obtain some existence theorems of weakly Ky Fan’s points for functions with no continuity or space with no compactness. Then, we prove that most of problems in weakly Ky Fan’s points (in the sense of Baire category) are essential, thus they are stable. Our results include corresponding results in the literature as a special case.

Now we recall some definitions in [10,11].

Definition 2.1 Let X and Y be two Hausdorff topological spaces, and be a set-valued mapping.

(1) F is said to be upper semicontinuous at , if for any open subset O of Y with , there exists an open neighborhood of x such that for any and F is said to be upper semicontinuous on X, if F is upper semicontinuous at each .

(2) F is said to be lower semicontinuous at , if for any open subset O of Y with , there exists an open neighborhood of x such that for any and F is said to be lower semicontinuous on X, if F is lower semicontinuous at each .

(3) F is said to be a usco mapping, if F is upper semicontinuous on X and is compact for each .

(4) F is said to be closed, if is closed.

Definition 2.2 Let H be a topological vector space and C be a cone of H. A cone C is said to be convex, if , and a cone C is said to be pointed, if , where denotes the zero element of H.

Remark 2.3 (see [6])

If C is a closed, convex, pointed cone with , where intC denotes the interior of C in H, then we can easily obtain that .

Definition 2.4 Let X and Y be two topological vector spaces, K be a nonempty convex subset of X, be a set-valued mapping, and C be a closed, convex, pointed cone with .

(1) F is said to be C-concave, if for every and , then and C-convex if −F is C-concave.

(2) F is said to be C-quasiconcave-like, if for every and , there exists such that and C-quasiconvex-like if −F is C-quasiconcave-like.

Remark 2.5C-concave and C-quasiconcave-like are two different notions which cannot deduce from each other. For example, let , , vector valued function , . It is easy to prove that f is -concave but f is not -quasiconcave-like, inverse g is -quasiconcave-like but is not -concave.

Lemma 3.1 (see [12])

LetXbe a nonempty subset of a Hausdorff topological vector spaceE, be a set-valued mapping. For each, is closed, and there exists somesuch thatis compact. If, whereis the convex hull of, then.

Theorem 3.2LetXbe a nonempty convex compact subset of a Hausdorff topological vector spaceE, Cis a closed, convex, pointed cone with. Ifsatisfies the following conditions:

(1) for all,

(2) for each fixed, isC-quasiconcave-like,

then there existssuch that for eachand a netwith, for any (i.e., for eachand a neighborhoodof, there exists a netsuch that).

Proof Define a set-valued mapping as follows:

By (1), we can easily know that for each . Next, we prove that for each , . Suppose (∗) is not true, then there exist some and , such that . By the definition of , we can know that for each . By Theorem 3.2(2), Remark 2.3, and Definition 2.4(2), we can obtain that

which contradicts the condition (1), thus for each . Define a set-valued mapping as follows,

where denotes the closure of . Clearly, for each , , X is compact, so is compact. By and (∗), we know that also satisfies (∗), thus by Lemma 3.1 we have . Take , then for each . Therefore, there exists , such that for each and a net with , for any . The proof is finished. □

Corollary 3.3LetXbe a nonempty convex compact subset of a Hausdorff topological vector spaceE, Cis a closed, convex, pointed cone with. If a vector-valued functionsatisfies the following conditions:

(1) for all,

(2) for each fixed, isC-quasiconcave-like,

then there existssuch that for eachand a netwith, for any.

Proof In Theorem 3.2, let , , . □

Corollary 3.4LetXbe a nonempty convex compact subset of a Hausdorff topological vector spaceE. If a functionsatisfies the following conditions:

(1) for all,

(2) for each fixed, is quasiconcave,

then there existssuch that for eachand a netwith, for any.

Proof In Corollary 3.3, let , . □

Remark 3.5 From the proof process of Theorem 3.2, we can easily extend it to the case in which X is not compact.

Theorem 3.6LetXbe a nonempty convex subset of a Hausdorff topological vector spaceE, Cis a closed, convex, pointed cone with. Ifsatisfies the following conditions:

(1) for all,

(2) for each fixed, isC-quasiconcave-like,

(3) is compact,

then there existssuch that for eachand a netwith, for any.

Proof Define a set-valued mapping as follows:

From the proof of Theorem 3.2, we can know that for each , .

Define a set-valued mapping as follows:

where denotes the closure of . Clearly, for each , is closed. By Theorem 3.6(3), there exists such that is compact. Thus the conditions of Lemma 3.1 are satisfied. So we have . Take , then for each . Therefore, there exists , such that for each and a net with , for any . The proof is finished. □

In the same way, Corollary 3.3 and Corollary 3.4 can be promoted respectively as follows.

Corollary 3.7LetXbe a nonempty convex subset of a Hausdorff topological vector spaceE, Cis a closed, convex, pointed cone with. If a vector-valued functionsatisfies the following conditions:

(1) for all,

(2) for each fixed, isC-quasiconcave-like,

(3) is compact,

then there existssuch that for eachand a netwith, for any.

Corollary 3.8LetXbe a nonempty convex compact subset of a Hausdorff topological vector spaceE. If a functionsatisfies the following conditions:

(1) for all,

(2) for each fixed, is quasiconcave,

(3) is compact,

then there existssuch that for eachand any netofwith, for any.

Remark 3.9 By Remark 2.5, we know that C-concave and C-quasiconcave-like are two different notions which cannot deduce from each other. Then Theorem 3.2, Theorem 3.6 can easily extend the case in which for each fixed , is C-concave in a similar way.

Remark 3.10 We call such points the weakly Ky Fan’s points in Theorem 3.2, Theorem 3.6. It is obvious that Ky Fan’s points must be weakly Ky Fan’s points, inverse is not true.

In this section, we first give some lemmas and concepts, then we study the generic stability of the set for weakly Ky Fan’s points for set-valued mappings.

Let X be a nonempty convex compact subset of a Banach space E with norm , C be a closed, convex, pointed cone with , be the set of all nonempty compact subsets of E. .

, define

where denotes the Hausdorff distance between and on .

Clearly is a metric space, is complete metric space (see [11]). For any , by Theorem 3.2, there exists a weakly Ky Fan’s point of set-valued mappings. Let be the set of all weakly Ky Fan’s points of φ, then , and thus define a set-valued mapping from into X, , where .

Next, we give some important lemmas in proving the generic stability of weakly Ky Fan’s points for set-valued mappings.

Lemma 4.1 (see [13])

LetXbe a complete metric space, Yis a metric space, is an usco mapping. Then there is a densesubsetQofXsuch thatFis lower semicontinuous onQ.

Lemma 4.2 (see [11])

LetXandYbe two topological spaces withYis compact. IfFis a closed set-valued mapping fromXtoY, thenFis upper semi-continuous.

Lemma 4.3is a complete metric space.

Proof Let be any Cauchy sequence in , then for any , there exists N such that for any , i.e., for any . It follows that for each , is a Cauchy sequence in . Since is a complete metric space, there exists a compact set such that for any . Next, we prove that .

By (∗), we can obtain and for any , then we can obtain that . As , and is C-quasiconcave-like, we have where . Thus we have . Since ε is arbitrary, , then is C-quasiconcave-like. Now we suppose that , then by (∗) we have . Since ε is arbitrary, we can obtain that , then we have which contradicts the assumption that . Thus . Hence, , is a complete metric space. □

Lemma 4.4is a usco mapping.

Proof Since X is compact, by Lemma 4.2, it suffices to show that F is a closed mapping, i.e., if for any , , , , then .

By , there exists a net and for any . Next, we suppose that . Then there exists some x, and for each , we have . As , we have when . Since ε is arbitrary, we can obtain that which contradicts the assumption that . Thus, , i.e. F is a closed mapping. Therefore, by Lemma 4.2, is a usco mapping. □

Definition 4.5 Let (1) is essential if for any , there exists such that for each with , there exists with . (2) φ is essential if every is essential.

By Definition 2.1(2) and Definition 4.5, it is easy to obtain the following results.

Lemma 4.6φis essential if and only if the set-valued mappingFis lower semicontinuous onφ.

Theorem 4.7There exists a densesubsetQofsuch that each, φis essential.

Proof By Lemma 4.4, is a usco mapping. By Lemma 4.1, there exists a dense subset Q such that each , φ is lower semicontinuous on Q. By Lemma 4.6, for each , φ is essential. □

Remark 4.8 (1) Let . By Lemma 4.4 and Lemma 4.6, F is continuous on Q. Then for any , there exists such that for any , with , . Thus φ is stable.

(2) Since Q is a dense residual subset, it is the second category set, therefore most of have stable solution sets in the sense of Baire category.

Theorem 4.9Ifis such thatis a singleton set, thenφis essential.

Proof For any open set G of X, , by , then , and . By Lemma 4.4, is upper semicontinuous. There exists an open neighborhood of φ such that for any , thus , then F is lower semicontinuous on φ. By Lemma 4.6, φ must be essential. □

The authors declare that they have no competing interests.

WSJ and SWX carried out the design of the study and performed the analysis. JHH and YLY participated in its design and coordination. All authors read and approved the final manuscript.

This work is supported by National Natural Science Foundation of China (11161008), Doctoral Program Fund for Ministry of Education (20115201110002) and Natural Science Fund of Guizhou Province (20122139).

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