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

Abstract

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.

1 Introduction

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 φ:X×X be such that (1) φ(x,x)0 for all xX; (2) for each fixed xX, yφ(x,y) is lower semicontinuous; (3) for each fixed yX, xφ(x,y) is quasiconcave, then there exists y X such that φ(x, y )0 for all xX.

Tan, Yu and Yuan [3] defined the inequality above as the Ky Fan inequality and called such a point y 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 [48]). 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.

2 Preliminaries

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

Definition 2.1 Let X and Y be two Hausdorff topological spaces, and F:X 2 Y be a set-valued mapping.

  1. (1)

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

  2. (2)

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

  3. (3)

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

  4. (4)

    F is said to be closed, if Graph(F)={(x,y)X×YyF(x)} 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 C+C=C, and a cone C is said to be pointed, if CC={θ}, where {θ} denotes the zero element of H.

Remark 2.3 (see [6])

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

Definition 2.4 Let X and Y be two topological vector spaces, K be a nonempty convex subset of X, F:K 2 Y be a set-valued mapping, and C be a closed, convex, pointed cone with intC.

  1. (1)

    F is said to be C-concave, if for every x 1 ,, x n K and λ i [0,1], i = 1 n λ i =1 then F( i = 1 n λ i x i ) i = 1 n λ i F( x i )+C and C-convex if −F is C-concave.

  2. (2)

    F is said to be C-quasiconcave-like, if for every x 1 ,, x n K and λ i [0,1], i = 1 n λ i =1 there exists i 0 {1,2,,n} such that F( i = 1 n λ i x i )F( x i 0 )+C and C-quasiconvex-like if −F is C-quasiconcave-like.

Remark 2.5 C-concave and C-quasiconcave-like are two different notions which cannot deduce from each other. For example, let X=[0,1], R + 2 =[0,+)×[0,+), vector valued function f=( f 1 , f 2 )=(x,x), g=( g 1 , g 2 )=( x 2 , x 2 ). It is easy to prove that f is R + 2 -concave but f is not R + 2 -quasiconcave-like, inverse g is R + 2 -quasiconcave-like but is not R + 2 -concave.

3 Existence for weakly Ky Fan’s points of set-valued mappings

Lemma 3.1 (see [12])

Let X be a nonempty subset of a Hausdorff topological vector space E, F:X 2 X be a set-valued mapping. For each xX, F(x) is closed, and there exists some x 0 X such that F( x 0 ) is compact. If co{ x 1 , x 2 ,, x n } i = 1 n F( x i ), where co{ x 1 , x 2 ,, x n } is the convex hull of { x 1 , x 2 ,, x n }, then x X F(x).

Theorem 3.2 Let X be a nonempty convex compact subset of a Hausdorff topological vector space E, C is a closed, convex, pointed cone with intC. If φ:X×X 2 E satisfies the following conditions:

  1. (1)

    φ(x,x)intC for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is C-quasiconcave-like,

then there exists y X such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD (i.e., for each xX and a neighborhood N( y ) of y , there exists a net { y α }N( y ) such that φ(x, y α )intC).

Proof Define a set-valued mapping F:X 2 X as follows:

F(x)= { y X φ ( x , y ) int C } ,xX.

By (1), we can easily know that F(x) for each xX. Next, we prove that for each { x 1 , x 2 ,, x n }X, co{ x 1 , x 2 ,, x n } i = 1 n F( x i )(). Suppose () is not true, then there exist some { x 1 , x 2 ,, x n }X and t i [0,1], i = 1 n t i =1 such that x= i = 1 n t i x i i = 1 n F( x i ). By the definition of F(x), we can know that φ( x i ,x)intC for each i=1,2,,n. By Theorem 3.2(2), Remark 2.3, and Definition 2.4(2), we can obtain that

φ(x,x)=φ ( i = 1 n t i x i , x ) φ( x i 0 ,x)+CintC+CintC,

which contradicts the condition (1), thus co{ x 1 , x 2 ,, x n } i = 1 n F( x i ) for each { x 1 , x 2 ,, x n }X. Define a set-valued mapping Cl(F):X 2 X as follows,

Cl ( F ( x ) ) =Cl { y X φ ( x , y ) int C } ,xX,

where Cl(F(x)) denotes the closure of F(x). Clearly, for each xX, Cl(F(x))X, X is compact, so Cl(F(x)) is compact. By F(x)Cl(F(x)) and (), we know that Cl(F):X 2 X also satisfies (), thus by Lemma 3.1 we have x X Cl(F(x)). Take y x X Cl(F(x)), then y Cl(F(x)) for each xX. Therefore, there exists y X, such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD. The proof is finished. □

Corollary 3.3 Let X be a nonempty convex compact subset of a Hausdorff topological vector space E, C is a closed, convex, pointed cone with intC. If a vector-valued function φ:X×XH satisfies the following conditions:

  1. (1)

    φ(x,x)intC for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is C-quasiconcave-like,

then there exists y X such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD.

Proof In Theorem 3.2, let φ(x,y)H, xX, yX. □

Corollary 3.4 Let X be a nonempty convex compact subset of a Hausdorff topological vector space E. If a function φ:X×X satisfies the following conditions:

  1. (1)

    φ(x,x)0 for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is quasiconcave,

then there exists y X such that for each xX and a net { y α } with { y α } y , φ(x, y α )0 for any αD.

Proof In Corollary 3.3, let H=, C=[0,+). □

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 3.6 Let X be a nonempty convex subset of a Hausdorff topological vector space E, C is a closed, convex, pointed cone with intC. If φ:X×X 2 H satisfies the following conditions:

  1. (1)

    φ(x,x)intC for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is C-quasiconcave-like,

  3. (3)

    Cl(F( x 0 ))=Cl{yXφ( x 0 ,y)intC} is compact,

then there exists y X such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD.

Proof Define a set-valued mapping F:X 2 X as follows:

F(x)= { y X φ ( x , y ) int C } ,xX.

From the proof of Theorem 3.2, we can know that for each { x 1 , x 2 ,, x n }X, co{ x 1 , x 2 ,, x n } i = 1 n F( x i )().

Define a set-valued mapping Cl(F):X 2 X as follows:

Cl ( F ( x ) ) =Cl { y X φ ( x , y ) int C } ,xX,

where Cl(F(x)) denotes the closure of F(x). Clearly, for each xX, Cl(F(x)) is closed. By Theorem 3.6(3), there exists x 0 such that Cl(F( x 0 ))=Cl{yXφ( x 0 ,y)intC} is compact. Thus the conditions of Lemma 3.1 are satisfied. So we have x X Cl(F(x)). Take y x X Cl(F(x)), then y Cl(F(x)) for each xX. Therefore, there exists y X, such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD. The proof is finished. □

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

Corollary 3.7 Let X be a nonempty convex subset of a Hausdorff topological vector space E, C is a closed, convex, pointed cone with intC. If a vector-valued function φ:X×XH satisfies the following conditions:

  1. (1)

    φ(x,x)intC for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is C-quasiconcave-like,

  3. (3)

    Cl(F( x 0 ))=Cl{yXφ( x 0 ,y)intC} is compact,

then there exists y X such that for each xX and a net { y α } with { y α } y , φ(x, y α )intC for any αD.

Corollary 3.8 Let X be a nonempty convex compact subset of a Hausdorff topological vector space E. If a function φ:X×X satisfies the following conditions:

  1. (1)

    φ(x,x)0 for all xX,

  2. (2)

    for each fixed yX, xφ(x,y) is quasiconcave,

  3. (3)

    Cl(F( x 0 ))=Cl{yXφ( x 0 ,y)0} is compact,

then there exists y X such that for each xX and any net { y α } of F(x) with { y α } y , φ(x, y α )0 for any αD.

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 yX, xφ(x,y) is C-concave in a similar way.

Remark 3.10 We call such points y 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.

4 Generic stability of the set for weakly Ky Fan’s points of set-valued mappings

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 intC, K(E) be the set of all nonempty compact subsets of E. M 1 ={φ:X×XK(E)φ satisfies the conditions (1), (2) in Theorem 3.2}.

φ 1 , φ 2 M 1 , define

ρ( φ 1 , φ 2 )= Sup ( x , y ) X × X h ( φ 1 ( x , y ) , φ 2 ( x , y ) ) ,

where h( φ 1 (x,y), φ 2 (x,y)) denotes the Hausdorff distance between φ 1 (x,y) and φ 2 (x,y) on X×X.

Clearly ( M 1 ,ρ) is a metric space, (K(E),h) is complete metric space (see [11]). For any φ M 1 , by Theorem 3.2, there exists y a weakly Ky Fan’s point of set-valued mappings. Let F(φ) be the set of all weakly Ky Fan’s points of φ, then F(φ), and thus define a set-valued mapping from M 1 into X, F: M 1 2 X , where F(φ)={yXfor each xX and a net { y α } with { y α }y we have φ(x, y α )intC for any αD}.

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])

Let X be a complete metric space, Y is a metric space, F:X 2 Y is an usco mapping. Then there is a dense G δ subset Q of X such that F is lower semicontinuous on Q.

Lemma 4.2 (see [11])

Let X and Y be two topological spaces with Y is compact. If F is a closed set-valued mapping from X to Y, then F is upper semi-continuous.

Lemma 4.3 ( M 1 ,ρ) is a complete metric space.

Proof Let { φ n } n = 1 be any Cauchy sequence in M 1 , then for any ε>0, there exists N such that ρ( φ n , φ m ) for any n,mN, i.e., Sup ( x , y ) X × X h( φ n (x,y), φ m (x,y))<ε for any n,mN. It follows that for each (x,y)X×X, { φ n ( x , y ) } n = 1 is a Cauchy sequence in K(E). Since K(E) is a complete metric space, there exists a compact set φ(x,y)K(E) such that h( φ n (x,y),φ(x,y))ε() for any (x,y)X×X. Next, we prove that φ M 1 .

By (), we can obtain φ n (x,y)U(φ(x,y),ε) and φ(x,y)U( φ n (x,y),ε) for any nN, then we can obtain that φ( i = 1 n λ i x i ,y)U( φ n ( i = 1 n λ i x i ,y),ε). As φ n M 1 , and x φ n (x,y) is C-quasiconcave-like, we have φ n ( i = 1 n λ i x i ,y) φ n ( x i 0 ,y)+C where i 0 {1,,n}. Thus we have φ( i = 1 n λ i x i ,y)U( φ n ( x i 0 ,y)+C,ε)U(φ( x i 0 ,y)+C,2ε). Since ε is arbitrary, φ( i = 1 n λ i x i ,y)φ( x i 0 ,y)+C, then xφ(x,y) is C-quasiconcave-like. Now we suppose that φ(x,x)intC, then by () we have φ n (x,x)U(φ(x,x),ε). Since ε is arbitrary, we can obtain that φ n (x,x)φ(x,x), then we have φ n (x,x)φ(x,x)intC which contradicts the assumption that φ n (x,x)intC. Thus φ(x,x)intC. Hence, φ M 1 , ( M 1 ,ρ) is a complete metric space. □

Lemma 4.4 F: M 1 2 X is 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 φ n M 1 , φ n φ M 1 , z n F( φ n ), z n z, then zF(φ).

By z n F( φ n ), there exists a net y α n z n and φ n (x, y α n )intC for any αD. Next, we suppose that zF(φ). Then there exists some x, and for each y α n z, we have φ(x, y α n )intC. As φ n φ, we have φ n (x, y α n )U(φ(x, y α n ),ε) when nN. Since ε is arbitrary, we can obtain that φ n (x, y α n )φ(x, y α n )intC which contradicts the assumption that φ n (x, y α n )intC. Thus, zF(φ), i.e. F is a closed mapping. Therefore, by Lemma 4.2, F: M 1 2 X is a usco mapping. □

Definition 4.5 Let φ M 1 (1) y F(φ) is essential if for any ε>0, there exists δ>0 such that for each φ M 1 with ρ(φ, φ )<δ, there exists y F( φ ) with y y <ε. (2) φ is essential if every yF(φ) is essential.

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

Lemma 4.6 φ is essential if and only if the set-valued mapping F is lower semicontinuous on φ.

Theorem 4.7 There exists a dense G δ subset Q of M 1 such that each φQ, φ is essential.

Proof By Lemma 4.4, F: M 1 2 X is a usco mapping. By Lemma 4.1, there exists a dense G δ subset Q such that each φQ, φ is lower semicontinuous on Q. By Lemma 4.6, for each φQ, φ is essential. □

Remark 4.8 (1) Let φQ. By Lemma 4.4 and Lemma 4.6, F is continuous on Q. Then for any ε>0, there exists δ>0 such that for any φ M, with ρ(φ, φ )<δ, h(F(φ),F( φ ))<ε. Thus φ is stable.

  1. (2)

    Since Q is a dense residual subset, it is the second category set, therefore most of φ M 1 have stable solution sets in the sense of Baire category.

Theorem 4.9 If φ M 1 is such that F(φ) is a singleton set, then φ is essential.

Proof For any open set G of X, F(φ)G, by F(φ)={y}, then yG, and GF(φ). By Lemma 4.4, F: M 1 2 X is upper semicontinuous. There exists an open neighborhood O(φ) of φ such that GF( φ ) for any φ O(φ), thus GF( φ ), then F is lower semicontinuous on φ. By Lemma 4.6, φ must be essential. □

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Acknowledgements

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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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.

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Jia, W., Xiang, S., He, J. et al. The existence and stability for weakly Ky Fan’s points of set-valued mappings. J Inequal Appl 2012, 199 (2012). https://doi.org/10.1186/1029-242X-2012-199

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Keywords

  • weakly Ky Fan’s points
  • set-valued mappings
  • C-concave
  • C-quasiconcave-like
  • essential solution