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A note on solution lower semicontinuity for parametric generalized vector equilibrium problems

Zai Yun Peng1*, Zhi Lin1, Kai Zhi Yu2 and Da Cheng Wang1

Correspondence:

pengzaiyun@126.com

'College of Science, Chongqing

JiaoTong University, Chongqing,

400074, P.R. China

Full list of author information is

available at the end of the article

Abstract

By using a density technique, sufficient conditions for lower semicontinuity of strong solutions to a parametric generalized vector equilibrium problem are established, where the monotonicity is not necessary. The obtained results are different from the corresponding ones in the literature (Gong and Yao in J. Optim. Theory Appl. '38:197-205,2008; Gong in J. Optim. Theory Appl. '39:35-46,2008; Chen and Li in Pac. J. Optim. 6:141-151,2010; Li and Fang in J. Optim. Theory Appl. '47:507-515,2010; Gong and Yao in J. Optim. Theory Appl. 138:189-196,2008). Some examples are given to illustrate the results. MSC: 49K40; 90C29; 90C31

Keywords: stability; strong solution; lower semicontinuity; parametric vector equilibrium problem; density

1 Introduction

It is well known that the vector equilibrium problem (VEP, in short) is a very general mathematical model, which embraces the formats of several disciplines, as those for Nash equilibria, those from Game Theory, those from (Vector) Optimization and (Vector) Variational Inequalities, and so on (see [1-4]).

The stability analysis of solution maps for parametric vector equilibrium problems (PVEP, in short) is an important topic in optimization theory and applications. There are some papers discussing the upper and/or lower semicontinuity of solution maps. Cheng and Zhu [5] obtained a result on the lower semicontinuity of the solution set map to a PVEP in finite-dimensional spaces by using a scalarization method. Huang etal. [6] used local existence results of the models considered and additional assumptions to establish the lower semicontinuity of solution mappings for parametric implicit vector equilibrium problems. Recently, by virtue of a density result and scalarization technique, Gong and Yao [7] have first discussed the lower semicontinuity of efficient solutions to parametric vector equilibrium problems, which are called generalized systems in their paper. By using the idea of Cheng and Zhu [5], Gong [8] has discussed the continuity of the solution maps to a weak PVEP in Hausdorff topological vector spaces. Kimura and Yao [9] discussed the semicontinuity of solution maps for parametric vector quasi-equilibrium problems by virtue of the closedness or openness assumptions for some certain sets. Xu and Li [10] proved the lower semicontinuity for PVEP by using a new proof method, which is different

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from the one used in Gong and Yao [7]. Chen and Li [11] studied the continuity of solution sets for parametric generalized systems without the uniform compactness assumption, which improves the corresponding results in [8]. We observe that the semicontinuity of solution maps for the PVEP has been discussed under the assumption of C-strict (strong) monotonicity, which implies that the f -solution set of the PVEP is a singleton for a linear continuous functionalf (see [5, 7, 8,10,11]). However, it is well known that the f -solution set of (weak) PVEPs should be general, but not a singleton. Moreover, to the best of our knowledge, there are few results of semicontinuity have been established for strong solution maps of PVEP in the literature. So, in this paper, by using a density skill, we aim at studying the lower semicontinuity of the strong solution map for a class of parametric generalized vector equilibrium problems (PGVEPs), when the f -solution set is a general set by removing the assumption of C-strict monotonicity.

The rest of the paper is organized as follows. In Section 2, we introduce a class of parametric generalized vector equilibrium problem, and recall some concepts and their properties. In Section 3, by the density and scalarization technique, we discuss the lower semi-continuity of strong solution mappings to the PGVEP, and compare our main results with the corresponding ones in the recent literature [7, 8,10,11]. We also give some examples to illustrate our results.

2 Preliminaries

Throughout this paper, unless specified otherwise, let X, Y be normed spaces and Z be Banach space. Let Y* be the topological dual space of Y, and C bea closed convex pointed cone in Y with nonempty topological interior int C.

C* := f e Y*:f(y) > 0, Vy e C} be the dual cone of C. Denote the quasi-interior of C* by C, i.e.,

C := f e Y* :f (y)>0,Vy e C \{0}}.

Let A bea nonempty subset of X and F: A x A ^ Y bea vector-valued mapping. We consider the following generalized vector equilibrium problem:

Find x e A such that F(x, y) e -K, Vy e A,

where K U {0} is a convex cone in Y.

When the subset A and the mapping F are perturbed by a parameter ¡x e A,in which A is a nonempty subset of Z, we consider the following parametric generalized vector equilibrium problem (PGVEP):

Find x e A(x) such that F(x,y, ¡¡) e -K, Vy e A(x),

where A : A ^ 2X \{0} is a set-valued mapping, F : B x B x A c X x X x Z ^ Y is a vector-valued mapping with A(A) = (J A A(x) c B.

Definition 2.1 [12] A vector x e A(x) is called a weak solution to the (PGVEP), iff F(x, y, ¡) e - int C, Vy e A(x).

The set of the weak solutions to the (PGVEP) is denoted by Vw(¡). Definition 2.2 A vector x e A(x) is called a strong solution to the (PGVEP), iff F (x, y, x)e-C \{0}, VyeA(x).

The set of the strong solutions to the (PGVEP) is denoted by Vs(x).

Definition 2.3 [7] Let f e C* \ {0}. A vector x e A(x) is called an f -solution to the (PGVEP), iff

f (F(x,y, ¡)) > 0, VyeA(x).

The set of the f-solution to the (PGVEP) is denoted by Vf (¡).

Definition 2.4 Let F: X x X x A ^ Y be a vector-valued mapping.

(i) F(■, ■, ■) is called C-monotone on A(x) x A(x) x A, iff for any given ¡x e A,for each x,y e A(x), F(x,y, ¡) + F(y,x, ¡) e -C.

(ii) F(■, ■, ■) is called C-strictly monotone (i.e., C-strongly monotone in [13]) on A(x) x A(x) x A,iff F is a C-monotone on A(x) x A(x) x A, and for any given ¡x e A, for eachx,y e A(x) with x = y, F(x,y, ¡) + F(y,x, ¡) e - int C.

(iii) F(x, ■, ¡) is called C-convex if, for each xi, x2 e A(x) and t e [0,1], tF(x, x1, ¡) + (1 - t)F(x, x2, ¡) e F(x, tx1 + (1 - t)x2, ¡) + C.

(iv) F(x, ■, ¡x) is called C-like-convex on A(x), iff for any x1, x2 e A(x) and any t e [0,1], there exists x3 e A(^) such that tF(x,x1, ¡) + (1 - t)F(x,x2, ¡) e F(x,x3, ¡) + C.

(v) A set D c Y is called a C-convex set, iff D + C is a convex set in Y.

Throughout this paper, we always assume Vw(¡) = 0 and VsO) = 0 for all ¡¡e A. This paper aims at investigating the semicontinuity of the strong solution mappings to (PGVEP).

Lemma 2.1 [14] Let F(x, A(^), ¡) be a C-convex set for each ¡i e A and x e A(^). If int C = 0, then VwO) = Ufe c*\{0} Vf (¡).

The notation B(X, S) denotes the open ball with center X e A and radius S >0. Let F: A ^ 2X be a set-valued mapping, and let there be given X e A.

Definition 2.5 [15]

(i) F is called lower semicontinuous (l.s.c., for short) at X, iff for any open set V satisfying V n F(X) = 0, there exists S >0, such that for every X e B(X, S),

V n F(X) = 0.

(ii) F is called upper semicontinuous (u.s.c., for short) atX, iff for any open set V satisfying F(X) c V, there exists S >0, such that for every X e B(X, S), F(X) c V.

We say F is l.s.c. (resp. u.s.c.) on A, iff it is l.s.c. (resp. u.s.c.) at each X e A. F is said to be continuous on A, iff it is both l.s.c. and u.s.c. on A.

Proposition 2.1 [15]

(i) F is l.s.c. at X if and only if for any sequence {Xn}c A with Xn ^ X and any x e F (X ), there exists xn e F(Xn), such that xn ^ x.

(ii) If F has compact values (i.e., F (X) is a compact set for each X e A), then F is u.s.c. at X if and only if for any sequences {Xn}c A with Xn ^ X and {xn} with xn e F (Xn), there exist xe F(X) and a subsequence {xnk} of {xn}, such that x nk ^ x.

The following lemma plays an important role in the proof of the lower semicontinuity of the solution map Vs(x).

Lemma 2.2 [16, Theorem 2, p.114] The union F = U ieI Ft of a family of l.s.c. set-valued mappings F ¡from a topological space X into a topological space Y is also an l.s.c. set-valued mappingfrom X into Y, where I is an index set.

Let Q : X ^ 2Y be a set-valued mapping between two topological spaces. The lower limit ofQ is defined as

Liminfx^xo Q(x) = {y e Y : Vxa ^ x0,3ya e Q(xa),ya ^ y}.

Proposition 2.2 [17]

(i) Liminfx^xo Q(x) is a closed set.

(ii) Q is l.s.c. at x0 e dom Q := {x|Q(x) = 0} if and only if Q(x0) c Liminfx^xo Q(x).

3 Lower semicontinuity of strong solution sets to (PGVEP)

In this section, we discuss the lower semicontinuity of the strong solutions for (PVEP). Firstly, we obtain two important lemmas relevant to the (PVEP) as follows.

Lemma 3.1 Letf e C*\ {0}. Suppose the following conditions are satisfied:

(i) A(-) is continuous with compact convex values on A.

(ii) For each ¡x e A, F(•, •, •) is continuous on B x B x ¡x.

(iii) For each ¡x e A, x e A(x) \ Vf (¡), there exist y e Vf (¡) and r >0, such that

F (x, y, ¡x) + F (y, x, ¡x) + B( 0, dr (x, y)) c -C. Then Vf (•) is l.s.c. on A.

Proof Suppose to the contrary that there exists ¡x0 e A, such that Vf (•) is not l.s.c. at ¡x0. Then there exist a sequence {¡n} with ¡xn ^ ¡x0 and x0 e Vf (¡0), such that for any xn e Vf (¡n), xn ^ x0.

Since A(-) is l.s.c. at ¡0, there exists xn e A(xn), such that xn ^ x0. Obviously, xn e A(xn) \ Vf (¡n). By (iii), there exists yn e Vf (¡n) such that

F (xn, yn, ¡¡n) + F (yn, xn, ¡n)+B( 0, dr (xn, yn)) c -C. (3.1)

For yn eA(jxn), because A(-) is u.s.c. at ¡0 with compact values, there exist y0 eA(¡0) and a subsequence {ynk} of {yn}, such that y„k ^ y0. In particular, for (3.1), we have

F(xnk,ynk, ¡nk) + F(ynk,xnk, ¡nk)+B(0, dr(xnk,ynk)) c -C. (3.2)

Taking the limit as nk ^ we have

F(x0,y0, ¡0) + F(y0,x0, ¡i0)+B(0, dr(x0,y0)) c -C. (3.3)

Noting that xo e Vf (¡0) and yo e A(^0), we have

f (F(x0,y0, ¡0)) > 0. (3.4)

Moreover, since y„k e Vf (¡nk) and x„k e A(^nk), it follows from the continuity off and F that

f (F(y0,x0, ¡0)) > 0. (3.5)

By (3.4), (3.5), and the linearity off, we have

f (F(x0,y0, ¡0) + F(y0,x0, ¡0)) > 0. (3.6)

Assume that x0 = y0; by (3.3), we obtain F(x0,y0, ¡0) + F(y0,x0, ¡0) c - int C. Thus, we have

f (F(x0,y0, ¡0) + F(y0,x0, ¡0)) < 0,

which contradicts (3.6). Therefore x0 = y0, which leads to a contradiction. Hence, for each f e C* \ {0}, Sf (■) is l.s.c. on A. □

Lemma 3.2 Letf e C* \ {0}. Suppose the following conditions are satisfied:

(i) A is nonempty compact set.

(ii) F(■, ■) is continuous on B x B.

(iii) For each xeA \ Vf there exist y e Vf and r >0, such that

F(x,y)+F(y,x) +B(0, dr(x,y)) c -C. Let us define the set-valued mapping H: C* \{0}^ 2A by H f) = Vf, feC*\{0}; then we see that H(■) is l.s.c. on C* \ {0}.

Proof By using Lemma 3.1 and following a similar way to the proof to Lemma 3.1 of [18], we can obtain the conclusion. □

Now, we discuss the lower semicontinuity of strong solution mappings to (PGVEP).

Theorem 3.1 Letf e C*\ {0}. Suppose the following conditions are satisfied:

(i) A(-) is continuous with compact convex values on A.

(ii) For each ¡x e A and x e A(x), F(x, •, ¡x) is C-like-convex on A(x).

(iii) For each ¡x e A, F(•, •, •) is continuous on B x B x ¡x.

(iv) For each ¡x e A, x e A(x) \ Vf (x), there exist y e Vf (x) and r >0, such that

F (x, y, ¡x) + F (y, x, ¡x) + B( 0, dr (x, y)) c -C.

(v) F (A(x), A(x), ¡x) are bounded subsets of Y for each ¡x e A.

(vi) CB = 0 and int C = 0.

Then Vs(•) is lower semicontinuous on A.

Proof We prove the result in the following three steps. Step 1. We first show that

Since for any x e A(x), F(x, ■, x) is C-like-convex, then F(x, A(x), x) + C is a convex set. From Lemma 2.1, we have

Since Vf (x) = 0, for eachf e C*\ {0}. Then, by definition, we have

U Vf (x) c Vs(x) c Vw(x).

Vw (x)= U Vf (x).

f eC*\{0}

By (3.7) and (3.8), we get

U Vf (x) c Vs (x) c y Vf (x).

(3.10)

f eC*\{0}

To show that

we first prove

Let us define the set-valued mapping H : C* \ {0}

H (f ) = Vf, f e C*\{0}.

By Lemma 3.2, we know that H(•) is lower semicontinuous on C* \ {0}.

Let x0 e |Jfe C*\{0} Vf. Then there exists f0 e C* \ {0} such that x0 e Vf = Hfo). Since C« = 0, letg e C« and set

fn =fo + (1/n)g.

Then fn e C«. We show that fn} converges tof0 with respect to the topology j(Y*, Y).

For any neighborhood U of 0 with respect to j (Y*, Y), there exist bounded subsets B; c Y (i = 1,2,..., m) and e >0 such that

f){f e Y*: supf(y)| < A c U.

1 JeB; >

Since B; is bounded andg e Y*, |g(B;)| is bounded for i = 1,...,m. Thus, there exists N such that

sup|(1/n)g(y)| < e, i = 1,...,m,n > N.

Hence (1/n)g e U, that is,fn -f0 e U. This means that {fn} converges tof0 with respect to j (Y *, Y).

Since H(f ) is l.s.c. atf0, for sequence {fn} c C* \ {0}, fn ^f0 and x0 e H(f0), there exists xn e H(fn) = Vfn c U/ e ct Vf, such that xn ^ x0. This means that

X0 ec^ Vf\.

VeC« '

By the arbitrariness of x0 e IJf e C*\{0} Vf, we have

U vf c d( u f.

f e C*\{0} f e C« 7

Then we can obtain the result that for each ¡x e A (3.11) is valid, and the validity of (3.7) follows readily from (3.10) and (3.11).

Step 2. For each ¡x e A, let S(¡) := Ufe C« Vf (¡). By a similar argument to the proof of Lemma 3.1, we find for eachf e C«, that Vf (■) is l.s.c. on A. It follows from Lemma 2.2 that S(■) is l.s.c. on A.

Step 3. Now we show that Vs(-) is lower semicontinuous on A. From Step 1, we have

S(x) c Vs(¡) c c/(5(x)), V^eA.

Let ¡x0 e A be any fixed point. Because of the lower semicontinuity of S(■) at ¡x0 and the closedness of the lower limit of S (see Proposition 2.2), together with the above inclusion relation, we have

Vs(x0) c cl(S(¡0)) c liminfS(xa) c liminf Vs(xa).

Hence, Vs(■) is l.s.c. at ¡0. By the arbitrariness of ¡x0, Vs(-) is l.s.c. on A. This completes the proof. □

Remark 3.2 In Theorem 3.1, by using the density technique, we obtain a sufficient condition for the lower semicontinuity of strong solutions to (PGVEP). Our approach is different from the corresponding ones in [7] (see Theorem 2.1 of [7]). Furthermore, the condition of C-strong (strict) monotonicity is not required, and the C-convexity of F is generalized to the C-like-convexity. Thus, Theorem 3.1 improves and extends the corresponding results in the literature [5, 7, 8,10,11]. The following example is given to illustrate the case.

Example 3.1 LetX = R, Y = R2, C = R+, A = [1,2], A(x) = [-1,1], and F(x,y, ¡x) = (1-2¡i2 + y, 2xx). For any/ e C* \ {0}, it follows from a direct computation that V/ (¡) = [0,1]. It is clear that A(-) is a continuous set-valued mapping with nonempty compact convex values and for each ¡x e A, F(■, ■, ■) is continuous on A(jx) x A(x) x ¡x. Therefore, conditions (i)-(ii) of Theorem 3.1 are satisfied. For any given ¡i e A and for any given x e A(x), we have, for any y1,y2 e A(x), t e [0,1],

F(x, tyi + (1 - t)y2, ¡x)

= (1 - 2x2 + ty1 + (1 - t)y2,2¡xx) = t( 1 - 2¡x2 + y1,2/xx) + (1 -1)(1 - 2x2 + y2,2/xx) e tF(x,y1, ¡) + (1 - t)F(x,y2, ¡x) - C.

Thus, for any given ¡i e A and for any given x e A(x), F(x, ■, ¡x) is R+-like-convex on A(x), that is, condition (iii) of Theorem 3.1 is satisfied. For any x e A(x) \ V/(¡), there exists y = 0 e V/(¡), such that

F (x, y, ¡x) + F(y, x, ¡i) + B( 0, dr (x, y))

= (2 - + x, 2¡xx) + B(0,dr(x, 0)) c -C.

Thus, the condition (iv) of Theorem 3.1 is satisfied. It is clear that F(A(^), A(^), ¡) are bounded subsets of Y for each ¡i e A, (R+)B = 0, and intR+ = 0. Consequently, by Theorem 3.1, Vs(-) is lower semicontinuous on A.

However, the condition of C-strong monotonicity does not hold. Indeed, for any x e A(n) \ V/(¡) = [-1,0), there exists y = -x e V/(¡) = [0,1], such that

F(x, y, ¡)+F(y, x, ¡) = (-4^2 + 2,0) e - int C.

Obviously, F(■, ■, ¡x) is not C-strongly monotone on AO) x A(^). Then Theorem 2.1 in [7] and Theorem 3.2 in [11] are not applicable, and the corresponding results in references [8, Theorem 4.2], [10, Theorem 3.1] are also not applicable.

Finally, we give an example to illustrate that the assumption (iv) of Theorem 3.1 is essential.

Example 3.2 Let X = R, Y = R2, C = R+. Let A = [4,5] and

A(x)=[x -4,1], x e A.

It is clear that A is a continuous set-valued mapping from A to X with nonempty compact convex values. Define the mapping F: A(x) x A(x) x A ^ R2 by

Obviously, we know that conditions (ii), (iii), (v), and (vi) are satisfied. We have, for anyf e C* \ {0}, 0 e Vf (¡), and

Vs(x) = Vf(¡) = {0,1}, ifx = 4 and Vs(¡) = {1}, if ¡e (4,5].

In fact, there exists x =2 e A(x) \ Vf (¡), for y = 0 e Vf (¡), we have

F (x, y, ¡x) + F (y, x, ¡x) + B( 0, dr (x, y))

Using a similar method, there exists x = 2 e A(x) \ Vf (x), for y = 1 e Vf (x), we have F(x,y, x) +F(y, x, x) +B(0, dr(x,y)) ^ -C. Thus, condition (iv) of Theorem 3.1 is not satisfied.

Now, we show that Vs(x) is not lower semicontinuous at x0 = 4. There exists 0 e Vs(4) and there exists a neighborhood of 0, for any neighborhood U(4) of 4, there exists

4 < x <5 such that x e U(4) and

By virtue of Definition 2.5, we know that Vs(x) is not lower semicontinuous at x0 = 4.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors contributed equally to the writing of this paper. Allauthors read and approved the finalmanuscript. Author details

'College of Science, Chongqing JiaoTong University, Chongqing, 400074, P.R. China. 2 Statistics School, Southwestern University of Finance and Economics, Chengdu, 611130, China.

Acknowledgements

This work was supported by the NaturalScience Foundation of China (Nos. 11301571,11271389, 71201126), the science foundation of Ministry of Education of China (No. 12XJC910001), the NaturalScience Foundation Project of ChongQing (No. 2012jjA00016) and the Education Committee Research Foundation of Chongqing (KJ130428).

Vx,y e A(x), x e A.

Received: 15 February 2014 Accepted: 5 August 2014 Published: 26 Aug 2014

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Cite this article as: Peng et al.: A note on solution lower semicontinuity for parametric generalized vector equilibrium problems. Journal of Inequalities and Applications 2014, 2014:325

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