Continuity of the Solution Maps for Generalized Parametric Set-Valued Ky Fan Inequality Problems

Under new assumptions, we provide suffcient conditions for the upper and lower semicontinuity and continuity of the solution mappings to a class of generalized parametric set-valued Ky Fan inequality problems in linear metric space. These results extend and improve some known results in the literature e.g., Gong, 2008; Gong and Yoa, 2008; Chen and Gong, 2010; Li and Fang, 2010 . Some examples are given to illustrate our results.


Introduction
The Ky Fan inequality is a very general mathematical format, which embraces the formats of several disciplines, as those for equilibrium problems of mathematical physics, those from game theory, those from vector optimization and vector variational inequalities, and so on see 1, 2 .Since Ky Fan inequality was introduced in 1, 2 , it has been extended and generalized to vector or set-valued mappings.The Ky Fan Inequality for a set-/vector-valued mapping is known as the weak generalized Ky Fan inequality W GKFI, in short .In the literature, existing results for various types of generalized Ky Fan inequalities have been investigated intensively, see 3-5 and the references therein.
It is well known that the stability analysis of solution maps for parametric Ky Fan inequality PKFI, in short is an important topic in optimization theory and applications.There are some papers to discuss the upper and/or lower semicontinuity of solution maps.Cheng and Zhu 6 discussed the upper semicontinuity and the lower semicontinuity of the solution map for a PKFI in finite-dimensional spaces.Anh and Khanh 7, 8 studied the stability of solution sets for two classes of parametric quasi-KFIs.Huang et al. 9 discussed the upper semicontinuity and lower semicontinuity of the solution map for a parametric implicit KFI.By virtue of a density result and scalarization technique, Gong 10 first discussed the lower semicontinuity of the set of efficient solutions for a parametric KFI with vector-valued maps.By using the ideas of Cheng and Zhu 6 , Gong and Yao 11 studied the continuity of the solution map for a class of weak parametric KFI in topological vector spaces.Then, Kimura and Yao 12 discussed the semicontinuity of solution maps for parametric quasi-KFIs.Based on the work of 6, 10 , the continuity of solution sets for PKFIs was discussed in 13 without the uniform compactness assumption.Recently, Li and Fang 14 obtained a new sufficient condition for the lower semicontinuity of the solution maps to a generalized PKFI with vector-valued mappings, where their key assumption is different from the ones in 11, 13 .
Motivated by the work reported in 10, 11, 14, 15 , this paper aims at studying the stability of the solution maps for a class of generalized PKFI with set-valued mappings.We obtain some new sufficient conditions for the semicontinuity of the solution sets to the generalized PKFI.Our results are new and different from the corresponding ones in 6, 10, 11, 13-17 .
The rest of the paper is organized as follows.In Section 2, we introduce a class of generalized set-valued PKFI and recall some concepts and their properties which are needed in the sequel.In Section 3, we discuss the upper semicontinuity and lower semicontinuity of the solution mappings for the class of generalized PKFI and compare our main results with the corresponding ones in the recent literature 10, 11, 13-15 .We also give two examples to illustrate that our main results are applicable.

Preliminaries
Throughout this paper, if not, otherwise, specified, d •, • denotes the metric in any metric space.Let B 0, δ denote the closed ball with radius δ ≥ 0 and center 0 in any metric linear spaces.Let X and Y be two real linear metric spaces.Let Z be a linear metric space and let Λ be a nonempty subset of Z.Let Y * be the topological dual space of Y , and let C be a closed, convex, and pointed cone in Y with int C / ∅, where int C denotes the interior of C. Let be the dual cone of C.
Let A be a nonempty subset of X, and let F : A × A ⇒ Y \ {∅} be a set-valued mapping.We consider the following generalized KFI which consist in finding x ∈ A λ such that When the set A and the function F are perturbed by a parameter λ which varies over a set Λ of Z, we consider the following weak generalized PKFI which consist in finding x ∈ A λ such that where A : Λ ⇒ X \ {∅} is a set-valued mapping and For each λ ∈ Λ, the solution set of PKFI is defined by For each f ∈ C * \ {0} and for each λ ∈ Λ, the f-solution set of PKFI is defined by
ii When F is a vector-valued mapping, that is, PKFI reduces to the parametric Ky Fan inequality in 14 .
iii If for any λ ∈ Λ, x, y ∈ A λ , F x, y, λ : ϕ x, y, λ ψ y, λ − ψ x, λ , where ϕ : A μ × A μ × Λ → Y and ψ : A μ × Λ → Y are two vector-valued maps, the PKFI reduces to the parametric weak vector equilibrium problem PVEP considered in 10, 11, 13, 16 .Throughout this paper, we always assume V F, λ / ∅ for all λ ∈ Λ.This paper aims at investigating the semicontinuity and continuity of the solution mapping V F, λ as set-valued map from the set Λ into X.Now, we recall some basic definitions and their properties which are needed in this paper.
Definition 2.2 see 18 .Let X and Y be topological spaces, T : X ⇒ Y \ {∅} be a set-valued mapping.
i T is said to be upper semicontinuous u.s.c., for short at x 0 ∈ X if and only if for any open set V containing T x 0 , there exists an open set U containing x 0 such that T x ⊆ V for all x ∈ U.
ii T is said to be lower semicontinuous l.s.c., for short at x 0 ∈ X if and only if for any open set V with T x 0 ∩ V / ∅, there exists an open set U containing x 0 such that T x ∩ V / ∅ for all x ∈ U.
iii T is said to be continuous at x 0 ∈ X, if it is both l.s.c. and u.s.c. at x 0 ∈ X. T is said to be l.s.c.resp.u.s.c. on X, if and only if it is l.s.c.resp., u.s.c. at each x ∈ X.
From 19, 20 , we have the following properties for Definition 2.2.
Proposition 2.3.Let X and Y be topological spaces, let T : X ⇒ Y \ {∅} be a set-valued mapping.
i T is l.s.c. at x 0 ∈ X if and only if for any net {x α } ⊂ X with x α → x 0 and any y 0 ∈ T x 0 , there exists y α ∈ T x α such that y α → y 0 .
ii If T has compact values (i.e., T x is a compact set for each x ∈ X), then T is u.s.c. at x 0 if and only if for any net {x α } ⊂ X with x α → x 0 and for any y α ∈ T x α , there exist y 0 ∈ T x 0 and a subnet {y β } of {y α }, such that y β → y 0 .

Semicontinuity and Continuity of the Solution Map for PKFI
In this section, we obtain some new sufficient conditions for the semicontinuity and continuity of the solution maps to the PKFI .Firstly, we provide a new result of sufficient condition for the upper semicontinuity and closeness of the solution mapping to the PKFI .Theorem 3.1.For the problem PKFI , suppose that the following conditions are satisfied: Then, V F, • is u.s.c. and closed on Λ.
Proof.i Firstly, we prove V F, • is u.s.c. on Λ. Suppose to the contrary, there exists some μ 0 ∈ Λ such that V F, • is not u.s.c. at μ 0 .Then, there exist an open set V satisfying V F, μ 0 ⊂ V and sequences μ n → μ 0 and x n ∈ V F, μ n , such that x n / ∈ V, ∀n.

3.1
Since x n ∈ A μ n and A • are u.s.c. at μ 0 with compact values by Proposition 2.3, there is an x 0 ∈ A μ 0 such that x n → x 0 here, we can take a subsequence {x n k } of {x n } if necessary .Now, we need to show that x 0 ∈ V F, μ 0 .By contradiction, assume that x 0 / ∈ V F, μ 0 .Then, there exists y 0 ∈ A μ 0 such that By the lower semicontinuity of A • at μ 0 , for y 0 ∈ A μ 0 , there exists y n ∈ A μ n such that y n → y 0 .
It follows from x n ∈ V F, μ n and y n ∈ A μ n that Since F •, •, • is l.s.c. at x 0 , y 0 , λ 0 , for z 0 ∈ F x 0 , y 0 , μ 0 , there exists z n ∈ F x n , y n , μ n such that 3.5 From 3.3 , 3.5 , and the openness of int C, there exists a positive integer N sufficiently large such that for all n ≥ N, which contradicts 3.4 .So, we have Since x n → x 0 here we can take a subsequence {x n k } of {x n } if necessary , we can find 3.7 contradicts 3.1 .Consequently, V F, • is u.s.c. on Λ.
ii In a similar way to the proof of i , we can easily obtain the closeness of V F, • on Λ.This completes the proof.ii the vector-valued mapping F •, •, • is extended to set-valued mapping, and the condition that C-monotone of mapping is removed; iii the assumption iii of Theorem 3.1 in 10 is removed; iv the condition that A • is uniformly compact near μ ∈ Λ is not required.
Moreover, we also can see that the obtained result extends Theorem 2.1 of 15 .
Now, we give an example to illustrate that Theorem 3.1 is applicable.
Then, we can verify that all assumptions of Theorem 3.1 are satisfied.By Theorem 3.1, V F, • is u.s.c. and closed on Λ.Therefore, Theorem 3.1 is applicable.
When F : X × X × Z → Y is a vector-valued mapping, one can get the following corollary.
Corollary 3.4.For the problem PKFI , suppose that F : X × X × Z → Y is a vector-valued mapping and the following conditions are satisfied: Then, V F, • is u.s.c. and closed on Λ.Now, we give a sufficient condition for the lower semicontinuity of the solution maps to the PKFI .Theorem 3.5.Let f ∈ C * \ {0}.Suppose that the following conditions are satisfied: where γ > 0 is a positive constant.
Then, V f F, • is l.s.c. on Λ.
Proof.By the contrary, assume that there exists λ 0 ∈ Λ, such that V f F, • is not l.s.c. at λ 0 .Then, there exist λ α with λ α → λ 0 and x 0 ∈ V f F, λ 0 , such that for any x α ∈ V f F, λ α with x α x 0 .Since x 0 ∈ A λ 0 and A • are l.s.c. at λ 0 , there exists x α ∈ A λ α such that x α → x 0 .We claim that x α ∈ A λ \ V f F, λ α .If not, for x α ∈ V f F, λ α , it follows from above-mentioned assumption that x α x 0 , which is a contradiction.By iii , there exists y α ∈ V f F, λ α , such that

3.10
For . at λ 0 with compact values by Proposition 2.3, there exist y 0 ∈ A λ 0 and a subsequence {y α k } of {y α } such that y α k → y 0 .In particular, for 3.10 , we have

3.11
Then, there exist

3.13
It follows from x 0 ∈ V f F, λ 0 and y 0 ∈ A λ 0 that inf z∈F x 0 ,y 0 ,λ 0 f z ≥ 0. Particularly, we have On the other hand, since Also, we have f z α k ≥ 0. It follows from the continuity of f that we have By 3.14 , 3.15 , and the linearity of f, we get For the above x 0 and y 0 , we consider two cases: Case i.If x 0 / y 0 , by 3.13 , we can obtain that z 0 z 0 ∈ − int C.

3.17
Then, it follows from f ∈ C * \ {0} that which is a contradiction to 3.16 .
Case ii.If x 0 y 0 , since y α ∈ V f F, λ α , y α → y 0 x 0 , this contradicts that for any x α ∈ V f F, λ α , x α do not converge to x 0 .Thus, V f F, • is l.s.c. on Λ.The proof is completed.Remark 3.6.Theorem 3.5 generalizes and improves the corresponding results of 14, Lemma 3.1 in the following three aspects: i the condition that A • is convex values is removed; ii the vector-valued mapping F •, •, • is extended to set-valued map; iii the constant γ can be any positive constant γ > 0 in Theorem 3.5, while it should be strictly restricted to γ 1 in Lemma 3.1 of 14 .
Moreover, we also can see that the obtained result extends the ones of Gong and Yao 11, Theorem 2.1 , where a strong assumption that C-strict/strong monotonicity of the mappings is required.
The following example illustrates that the assumption iii of Theorem 3.5 is essential.
Obviously, assumptions i and ii of Theorem 3.5 are satisfied, and A λ 0, 2 , for all λ ∈ Λ.For any given λ ∈ Λ, let f F x, y, λ z/3, for all z ∈ F x, y, λ .Then, it follows from a direct computation that

3.20
However, V f F, λ is even not l.c.s. at λ 3. The reason is that the assumption iii is violated.Indeed, if x 0 ∈ V f F, λ , for λ 3 and for all γ > 0, there exist y , for λ 3 and for all γ > 0, there exist y 1/2 ∈ A λ \ V f F, λ , using a similar method, we have F x, y, λ F y, x, λ B 0, d γ x, y / ⊆−C.Therefore, iii is violated.Now, we show that V f F, • is not l.s.c. at λ 3. Indeed, there exists 0 ∈ V f F, 3 and there exists a neighborhood −2/9, 2/9 of 0, for any neighborhood N 3 of 3, there exists 3 < λ < 5 such that λ ∈ N 3 and V f F, λ .

3.24
Theorem 3.9.For the problem PKFI , suppose that the following conditions are satisfied: where γ > 0 is a positive constant.
iv for each λ ∈ Λ and for each Then, V F, • is closed and continuous i.e., both l.s.c. and u.s.c. on Λ.
Proof.From Theorem 3.1, it is easy to see that V F, • is u.s.c. and closed on Λ.Now, we will only prove that V F, • is l.s.c. on Λ.For each λ ∈ Λ and for each x ∈ A λ , since F x, •, λ is C-like-convex on A λ , F x, A λ , λ C is convex.Thus, by virtue of Proposition 3.8, for each λ ∈ Λ, it holds 3.26 By Theorem 3.5, for each f ∈ C * \ {0}, V f F, • is l.s.c. on Λ.Therefore, in view of Theorem 2 in 20, page 114 , we have V F, • is l.s.c. on Λ.This completes the proof.
Remark 3.10.Theorem 3.9 generalizes and improves the work in 15, Theorems 3.4-3.5 .Our approach on the semi continuity of the solution mapping V F, • is totally different from the ones by Chen and Gong 15 .In 15 , the V f F, λ is strictly to be a singleton, while it may be a set-valued one in our paper.In addition, the assumption that C-strictly monotonicity of the mapping F is not required and the C-convexity of F is generalized to the C-like-convexity.
When the mapping F is vector-valued, we obtain the following corollary.
Corollary 3.11.For the problem PKFI , suppose that F : X×X×Z → Y is a vector-valued mapping and the following conditions are satisfied: where γ > 0 is a positive constant.
iv for each λ ∈ Λ and for each x ∈ A λ , F x, •, λ is C-like-convex on A λ .
Remark 3.12.Corollary 3.11 generalizes and improves 10, Theorem 4.2 and 13, Theorem 4.2 , respectively, because the assumption that C-strict monotonicity of the mapping F is not required.
Next,we give the following example to illustrate the case.Let f 0, 2 ∈ C * \ {0}, it follows from a direct computation that V f F, λ 0, 1 for any λ ∈ Λ.Hence, for any x ∈ A λ \ V f F, λ , there exists y 0 ∈ V f F, λ , such that,

3.29
Thus, the condition iii of Corollary 3.11 is also satisfied.By Corollary 3.11, V F, • is closed and continuous i.e., both l.s.c. and u.s.c. on Λ.However, the condition that F is a C-strictly monotone mapping is violated.Indeed, for any λ ∈ Λ −1, 1 and x ∈ A λ \ V f F, • , there exist y −x ∈ V f F, • with y −x, such that F x, y, λ F y, x, λ −3 − 2λ

Remark 3 . 2 .
Theorem 3.1 generalizes and improves the corresponding results of Gong 10, Theorem 3.1 in the following four aspects: i the condition that A • is convex values is removed;

Example 3 . 13 . 2 − λ 2
Let X Z R, Y R 2 , C R 2 , Λ −1, 1 be a subset of Z.Let F : X × X × Λ → Y be a mapping defined by F x, y, λ − 3 , λ 2 1 x 3.28 and define A : Λ → 2 Y by A λ −1, 1 .Obviously, A • is a continuous set-valued mapping from Λ to R with nonempty compact convex values, and conditions ii and iv of Corollary 3.11 are satisfied.
By virtue of Theorem 1.1 in 15 or Lemma 2.1 in 16 , we can get the following proposition.