Math Sci (2016) 10:41-45 DOI 10.1007/s40096-016-0176-y

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ORIGINAL RESEARCH

Lower semicontinuity of solutions for order-perturbed parametric vector equilibrium problems

Shunyou Xia12 • Shuwen Xiang2 • Yanlong Yang2 • Deping Xu3

Received: 7 September 2014/Accepted: 7 April 2016/Published online: 22 April 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract The lower semicontinuity of the (weak) efficient solution mappings for parametric vector equilibrium problems under more weaker assumptions is established. Some examples are developed to illustrate our results are real generalization different from recent ones in the literature and to describe the essential conditions of the latest results in the references are not real essential.

Keywords Order-perturbed parametric vector equilibrium problems • Lower semicontinuity • Cone lower semicontinuity • Efficient solutions

Mathematics Subject Classification 49K40 • 90C31

& Shunyou Xia

xiashunyou@126.com

Shuwen Xiang shwxiang@vip.163.com

Yanlong Yang yylong1980@163.com

Deping Xu xdpsc@163.com

1 School of Mathematics and Computer Science, Guizhou Education University, Guiyang 550018, China

2 College of Computer Science and Technology, Guizhou University, Guiyang 550025, China

3 College of Information Management, Chengdu University of Technology, Chengdu 610059, China

Introduction

Several classes of problems, including the vector variational inequality problem, the vector complementarity problem, the vector optimization problem and the vector saddle point problem, have been unified as a model of the vector equilibrium problem, which has been intensively studied in the literature (see [1-16]). One of the important topics in optimization theory is the stability analysis of the solution mappings for vector equilibrium problems. Stability may be understood as some types of lower or upper semicontinuity. Recently, the semicontinuity, especially the lower semicontinuity, of the solution mappings for parametric vector equilibrium problems has been intensively studied in various directions (see [1, 2, 4, 6, 10, 15] and references therein).

Anh and Khanh first obtained the semicontinuity of the solution mappings of parametric multivalued vector quasi-equilibrium problems (see [1]), and then obtained verifiable sufficient conditions for solution sets of general quasi-variational inclusion problems to have these semicontinu-ity-related properties and discussed in detail a traffic network problem as a sample for employing the main results in practical situations (see [2]), and latter established sufficient conditions for lower and Hausdorff lower semicontinuity, upper semicontinuity, and continuity of solution mappings of parametric quasi-equilibrium problems in topological vector spaces (see [3]).

Gong and Yao, by virtue of a density result and a scalar-ization technique, first discussed the lower semicontinuity of the set of efficient solutions to parametric vector equilibrium problems with monotone bifunctions (see [4]), and studied the continuity of the solution mapping to parametric weak vector equilibrium problems (see [5]), recently, established the lower semicontinuity of solutions to the parametric

generalized strong vector equilibrium problem without the assumptions of monotonicity of the objective mapping and compactness of the constraint mapping (see [6]). Huang et al. used local existence results to establish the lower semicon-tinuity of solution mappings for parametric implicit vector equilibrium problems (see [7]). By using a new proof which is different from the ones of [4,5], Chen et al. established the lower semicontinuity and continuity of the solution mappings to a parametric generalized vector equilibrium problem (see [9]). Li and Fang investigated the lower semicontinuity of the solutions mapping to parametric generalized Ky Fan inequality under a weaker assumption than C-strict monotonicity (see [10]). Recently, by using an idea of [4], Li et al. established the continuity of solution mappings to a parametric generalized strong vector equilibrium problem for set-valued mappings under an assumption which is different from the C-strict monotonicity (see [11]). Kimura and Yao discussed the semicontinuity of solution mappings of parametric vector quasi-equilibrium problems (see [13]). Cheng and Zhu obtained a lower semicontinuity result of the solution mapping to weak vector variational inequalities in finite-dimensional spaces by using the scalarization method (see [14]).

Zhang et al. obtained the lower semicontinuity of solution mappings for parametric vector equilibrium problems under the Hoder-related assumptions [15]. Wangkeeree et al. extended the results in [15] to the case of set-valued mappings on parametric strong vector equilibrium problems (see [16]).

However, all these results are with respect to the fixed order relationship in the object space, that is, the cone partial order is not perturbed by the parameters. Motivated by the idea of variational domination structure, the main results in [15] will be obtained under more weaker conditions in this paper.

The organization of this work is as follows. In Sect. 2, we introduce the efficient solutions to parametric vector equilibrium problems with the cone partial order being perturbed by the parameters and recall some basic notions. In Sect. 3, we discuss the lower semicontinuity of the (weak) efficient solution mappings for parametric vector equilibrium problems in the case of weakly conditions. Some examples are given to illustrate that the assumptions of the main results in our work or in [15] are only sufficient for some special problems and indicate also that our outcomes are real extension from the corresponding ones in [15].

Preliminaries

Let X and Z be two metric spaces, and let Y be a metric vector space and C be a pointed closed convex cone in Y with nonempty interior intC; the zero element in Y is denoted by h. Let A be a nonempty subset of X and F be a

vector-valued mapping from A x A into Y. A vector equilibrium problem, in short (VEP), is described as: Find x e A such that F(x,y) 2 — C\{h}, for all y e A A point x e A is said to be an efficient solution to (VEP)

F(x, y) e -C\{0}, for all y e A

When the subset A of X and the function F are perturbed by the parameter k e K, where K C Z, a parametric vector equilibrium problem, in short (PVEP), is a problem as following:

Find x e A(k) such that F(x, y, k) e —C\{h}, for all y

e A(k)

where A : K ! 2X\{/} is a set-valued mapping, and F:B x B x K C X x X x Z ? Y is a vector-valued mapping with A(K) = U keK A(k) C B.

A point x e A(k) is said to be an efficient solution to (PVEP) if

F(x, y, k) e — C\{h}, for all y e A(k)

In this work, we consider (PVEP) with cone C being also perturbed by parameter k e K, described as follows:

Find x e A(k) such that F(x, y, k)

e —C(k)\{0}, for all y e A(k)

where, for each k e K, C(k) is a pointed closed convex cone in Y with nonempty interior, that is to say, C:K ? 2Y is a cone-valued mapping. In this case, we call the (PVEP) as parametric vector order-perturbed equilibrium problem, in short (PVOPEP).

A point x e A(k) is called an efficient solution to (PVOPEP) if

F(x, y, k) e — C(k)\{h}, for all y e A(k)

The set of efficient solutions to (PVOPEP) is denoted by S(k), i.e.,

S(k) := {x e A(k)|F(x,y, k) e —C(k)\{0}, 8y e A(k)}

It is easy to see that S is a set-valued mapping S:K ? 2X. Throughout this work, we always assume that S(k) = /; for all k e K. Next, we recall some basic definitions and their properties which will be needed in the following.

BX(k, d) denotes the open ball with center k and radius d > 0 in a metric space X, dX( •, •) denotes the distance in X,and the distance from x to the set A C X is denoted by dx(x, A).

Definition 2.1 ([17]) A set-valued mapping S: K ? 2X is said to be

1. lower semicontinuous (l.s.c.) at k0 e K if for any open set V satisfying V \ S(k0) = /, there exists d > 0 such that for every k e BK(k0, d),V \ S(k) = /;

2. upper semicontinuous (u.s.c.) at k0 2 K if for any open set V satisfying S(k0) C V, there exists d > 0 such that for every k e BK(k0, d),S(k) C V;

3. l.s.c.(resp.,u.s.c.) on K if it is l.s.c.(resp.,u.s.c.) at each

k e K;

4. continuous on K if it is both l.s.c. and u.s.c. on K. Proposition 2.1 ([18, 19])

1. S:K ? 2X is l.s.c. at k0 e K if and only if for any sequence {kn} C K with kn ? k0 and any x0 e S(k0), there exists xn e S(kn) such that xn ? x0.

2. If S has compact values (i.e., S(k) is a compact set for each k e K), then S is u.s.c. at k0 e K if and only if for any sequence {kn} C K with kn ? k0 and for any xn e S(kn), there exist x0 e S(k0) and a subsequence {xnk} of {xn} such that xnk ! xo.

Definition 2.2 A vector-valued mapping f: X ? Y is called cone lower semicontinuity (c.l.s.c.) at x0 e X if for each open set V of f(x0), there exists a neighborhood U of x0, such thatf(x) C V + C, for all x e U, where C C Y is a cone.

Definition 2.3 Let K be a topology space, Y be a topology vector space, and C:K ? 2Y be a cone-valued mapping, if for every k e K, C(k) is a closed convex pointed cone in Y. The closed unit ball with center h in Y is denoted by By(h). We call the cone-valued mapping C is an upper semicontinuous cone-valued mapping, if for each k e K and each open set Uin Y with U . C(k) \ By(h), there exists an open set V of k such that U . C(k) \ BY(h) for every k e V.

The main results

In this section, we present the lower semicontinuity of the solution mapping to (PVOPEP).

Theorem 3.1. Suppose that the following conditions are satisfied:

© A(-) is continuous with compact values on K.

® F(-, -, •) is c.l.s.c. on B x B x K.

® C(-) is an upper semicontinuous cone-valued mapping on K.

© If A(k)\S(k) ^ / for each k e K, then for each k e K,for each x e A(k)\S(k), there exist y e S(k) and a positive function M:K ? (0, + oo) which is upper semicontinuous on K, such that

dX(x,y)<M(k)- dY(F(x,y, k), Y\ — intC(k)) Then, S( ) is l.s.c. on K.

Proof Suppose to the contrary that there exists k0 e K such that S(-)is not l.s.c. at k0. Then, there exists a sequence {kn} C K with kn ? k0 and x0 e S(k0) with some open set Vi of x0, such that for any xn e S(kn), xn e Vi.

From x0 e S(k0), we have x0 e A(k0) and F(xo,y, k0) e —C(k0)\{0}, 8y e A(k0)

Since A(-) is l.s.c. at k0, there exists a sequence {xn} C -A(kn) such that xn e V1. Then, for the above open set V1, there exists a positive integer N; such that xn e V1, for all n > N. Obviously, we have xn e A(kn)\S(kn) for all n > N. For the sake of convenience, we consider n as still from one to infinity. By @, for each xn e A(kn)\S(kn), there exist yn e S(kn) and a positive function M:K ? (0, + ?) which is upper semicontinuous on K, such that

dX(xn,yn)<M(kn)- dY(F(xn,yn, kn), Y\ — intC(kn)) (2)

Since yn e A(kn), it follows from the upper semicontinuity and compactness of A(-) at k0 that there exist y0 e A(k0) and a subsequence {yni} of {yn} such that yni ! y0. In particular, from (2), we have

dX (xni, yni )< M(kni )• dY (F(xni, yni, kni), Y\ — intC(k^))

Since the distance function d(-, •) is continuous, F(-, •, •) is c.l.s.c., M(-) is u.s.c., and C(^) is an upper semicontinuous cone-valued mapping, then let i ? + ? on both sides of (3), we have

dX(xo,yo)<M(ko)^ dY(F(xo,yo, ko), Y\ — intC(ko)) (4)

If x0 = y0, by (4), we can obtain M(ko)- dY (F(xo, yo, ko), Y\ —intC(ko)) > dX (xo, yo) > 0

From M(k0) > 0, we have

dY(F(xo,yo, ko), Y\ — intC(ko)) > 0

Thus, we have F(xo, yo, ko) e —intC(ko))

which contradicts (1) as we see by taking y = y0. Therefore x0 = y0. This is impossible by the contradiction assumption. Thus, the proof is complete. □

Remark 3.1 The following examples indicate that the assumption @ in Theorem 3.1 in our work (or the assumption (iii) of Theorem 3.1 in [15]) cannot be applied to the case when A(k)\S(k) = /, for some k e K. So the assumption @ of Theorem 3.1 in our paper (or the assumption (iii) of Theorem 3.1 in [15])is not essential.

Example 3.1 Let X = Z = R,K1 = [±, K2 = [2, 3],

A(k) = B = [0, 1],

Y = R2

C(k) = R?

F(x, y, k) = (k(y — x), k(y — x)). It follows from a direct computation that

S1(k) = {0}, 8k e K1 and S2(k) = {0}, 8k e K2

For each k e K U K2, for every x e A(k)\S(k) = (0,1], taking y = 0 e S(k); we have

dX (x, y) = x and dY (F(x, y, k), Y\ —intC(k)) = kx Obviously,

dx(x, y)> dY(F(x, y, k), Y\ — intC(k)), 8k e K1

dx(x, y)< dY(F(x, y, k), Y\ — intC(k)), 8k e K2

From above two inequalities, it easy to see that the assumption (iii) in Theorem 3.1. in [15] is violated when k e Kj. However, Si(0 and S2(-) are all continuous on K1 and K2, respectively. If we take M: Kj ? (0, + 00), defined by M(k) =4,8k e K1 = [1, §], then

dX(x,y)<M(k) dY(F(x,y, k), Y\ — intC(k)),8k e K1

Thus, the example 3.1 satisfies all the assumptions of Theorem 3.1. in our work, and then, it is obtained that S( ) is l.s.c. on K.Consequently, Theorem 3.1. in our work is real extension from Theorem 3.1. in [15]. □

Example 3.2 Let X = Y = R, C(k) = R?, K = (0, 1], A(k) = B = [k2, 1 ? k] and F(x, y, k) = k(y — x). It follows from a direct computation that S(k) = {k2}, Vk e K. And it is easy to check that S( ) is continuous on K. For any k e (0, 1], for each x e A(k)\S(k) = (k2,1 + k], taking the unique element y = k2 e S(k), we have

x — k2 = dX(x,y)> dY(F(x,y, k), Y\ — intC(k)) = k(x — k2)

Obviously, the assumption (iii) of Theorem 3.1. in [15] is violated, but if taking M: K ? (0, ? 00) as following:

M(k) = k, 8k e K where a is a constant number and a > 1.

Thus, the example 3.2 satisfies all the assumptions of Theorem 3.1. in our work, and it follows from Theorem 3.1. in our work that S( ) is l.s.c. on K. But the Theorem 3.1. in [15] is invalid.

Example 3.3 Let X = Y = Z = R, C(k) = R?, K = [0, 1], A(k) = B = [1 — k2, 1 ? k2] and F(x, y, k) = x(1 ? k — y).

It follows from a direct computation that S(k) = [1 — k2, 1 ? k2] = A(k). For any k e [0, 1], A(k)\S(k) = /. Obviously, we cannot take any x e A(k)\S(k). So the condition (iii) of Theorem 3.1 in [15] (or the assumption @ in Theorem 3.1 in our work) cannot be applied. But it is easy to check that S( ) is continuous on K.

Our approach can also be applied to study the lower semicontinuity of the weak solution mappings. A point x e A(k) is called a weak efficient solution to (PVOPEP) if

F(x, y, k) 2 —intC(k), 8y e A(k)

The set of weak efficient solutions to (PVOPEP) is denoted by SW(k), i.e.,

Sw(k) :={x e A(k)|F(x,y, k) 2 —intC(k),8y e A(k)g

We can also obtain the following theorem on the lower semicontinuity of the weak efficient solution map to (PVOPEP) with a trivial adaptation of the proof.

Theorem 3.2 Suppose that the following conditions are satisfied:

® A(-) is continuous with compact values on K. ® F(-, -, •) is c.l.s.c. on B x B x K. ® C(-) is an upper semicontinuous cone-valued mapping on K.

@ If A(k)\SW(k) = for each k e K, then for each k e K,for each x e A(k)\SW (k), there exist y e SW(-k) and a positive function M:K ? (0, + ?) which is upper semicontinuous on K, such that

dX(x,y)<M(k)- dY(F(x,y, k), Y\ — intC(k)) Then, SW(-) is l.s.c. on K.

Acknowledgement This work is supported by NSFC(70661001; 11161008), DPFME P.R.China(20115201110002), NSFGP([2014] 2005; [2015]7298).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creative commons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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