Preechasilp and Wangkeeree SpringerPlus (2016)5:1345 DOI 10.1186/s40064-016-3001-z

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A characterization of nonemptiness and boundedness of the solution set for set-valued vector equilibrium problems via scalarization and stability results

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Pakkapon Preechasilp1^ and Rabian Wangkeeree2

'Correspondence: preechasilpp@gmail.com 1 Program in Mathematics, Faculty of Education, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand

Full list of author information is available at the end of the article

Abstract

In this paper, the existence theorems of solutions for generalized weak vector equilibrium problems are developed in real reflexive Banach spaces. Based on recession method and scalarization technique, we derive a characterization of nonemptiness and boundedness of solution set for generalized weak vector equilibrium problems. Moreover, Painleve-Kuratowski upper convergence of solution set is also discussed as an application, when both the objective mapping and the constraint set are perturbed by difference parameters.

Keywords: Equilibrium problem, Barrier cone, Pseudomonotone mappings, Stability analysis

Background

Let X be a real reflexive Banach space and Y be a real normed linear space. Suppose that, and C c Y is a nonempty, closed and convex pointed cone with int C = 0. Let K c X be a non-empty subset and a set-valued function F : K x K ^ 2Y\{0}, the following generalized weak vector equilibrium problem (GWVEP) is to find x e K such that

xX e K such that F(XX,y) n (-int C) = 0, Vy e K, (GWVEP)

and the dual problem for (GWVEP), is so called (DGWVEP), is to find x e K such that

xX e K such that F(y, XX) n (int C) = 0, Vy e K. (DGWVEP)

Both (GWVEP) and (DGWVEP) have been extensively studied by many authors (see Ansari and Flores-Bazan 2006; Ansari et al. 2001a, b, 2002; Flores-Bazan and Flores-Bazan 2003; Ansari et al. 2001; Sadeqi and Alizadeh 2011; Zhong et al. 2011). An important and interesting topic for (GWVEP) and (DGWVEP) is to study the nonemptiness and boundedness of the solution sets. As far as we known, the first paper which discussed this issues was Flores-Bazan and Flores-Bazan (2003) in the case where F is vector-valued. They studied the existence of solutions of (GWVEP) under the asymptotic analysis, where neither compactness of K nor any coercivity condition is assumed in

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reflexive Banach spaces. By using idea of recession method in Flores-Bazan and Flores-Bazan (2003), Ansari and Flores-Bazan (2006) gave some necessary and sufficient conditions for nonemptiness and boundedness of the solution set of (GWVEP). In 2011, Sadeqi and Alizadeh (2011) discussed and improved some results of Ansari and Flores-Bazan (2006). They gave the conditions under which the solution set of (GWVEP) is non-empty, convex and weakly compact subset in reflexive Banach spaces. After a thorough review of the literature and according to our knowledge, we found that the convexity assumed for second variable of F is an essential assumption (see also Chen et al. 2008; Flores-Bazan 2000; Fang and Huang 2007).

On the other hand, the stability analysis of the solution mappings to generalized vector equilibrium problem is an important topic in vector optimization theory. Recently, the lower semicontinuity, (Holder) continuity of the solution maps to (GWVEP) are discussed in Li and Li (2011), Gong (2008), Chen et al. (2009), Xu and Li (2013). Among those papers, we observe that the linear scalarization technique is one effective to deal with the lower semicontinuity and (Holder) continuity of solution mappings to (GWVEP). Based on the linear scalarization, the solution sets for (GWVEP) is the union of family of the solution set to scalarized equilibrium problems with respect to the linear map on dual cone. In natural, the union of family of solution sets to scalarized equilibrium problems is finer than the solution set to (GWVEP). In order to obtain the equality, convexity in second variable of F is assumed.

Motivated and Inspired by above works, the aim of this paper is to consider a (GWVEP) with a set-valued map on unbounded constraint set in reflexive Banach spaces. We first collect the characterization results of the nonemptiness and bounded-ness of the solution set of (GWVEP). By using the linear scalarization technique, we characterize the nonemptiness and boundedness of the solution set of (GWVEP) in terms of nonemptiness and boundedness of a family of scalar equilibrium problem with respect to linear maps in connected base for dual cone of C. Finally, we give the stability results for the solution maps to (GWVEP) in the sense of Painleve-Kuratowski upper convergence of solution set.

The paper is organized as follows. In "Preliminaries" section, we introduce some basic notations and preliminary results. In "Characterization of nonemptiness and bound-edness of the solution set" section, by using a scalarization technique, we establish the nonemptiness and boundedness of solution set for (GWVEP) in reflexive Banach spaces. In "Stability analysis" section, we give an application to the stability of the solution sets for (GWVEP).

Preliminaries

Throughout this paper, unless otherwise specified, we always assume that X is a real reflexive Banach space, Y is a real normed space with dual space Y* and C c Y is a nonempty, closed, convex and pointed cone with int C = 0. Let

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

be the dual cone of C. Clearly,

y e C ^(f, y) > 0, Vf e C*,

y e intC &(f,y) > 0, Vf e C*.

Since int C = 0, for any fixed e e int C, it proved in Huang et al. (2014) that the dual cone C * of C has a following weak * compact base C *0.

C*0 := {f e C* : (f, e) = 1},

where a subset D c C* is said to be a base of C* ^ 0 / D and C* c Ut>0tD. A vector x e K is called weak efficient solution to the (GWVEP) if

F(x,y) n (-int C) = 0, Vy e K, (1)

and weak efficient solution to the (DGWVEP) if

F (y, x) n (intC) = 0, Vy e K. (2)

Denote by S^ (K, F) and SD(K, F) the set of all weak efficient solution to the (GWVEP) and (DGWVEP), respectively.

Definition 1 (Zhong et al. 2011) Let K be a non-empty convex subset of X. For a given closed convex cone C of a real normed space Y, the set-valued map F : K ^ 2Y\{0} is said to be

(i) upper C-convex, if for any x, y e Kand for any t e [0,1] tF(x) + (1 - t)F(y) c F(tx + (1 - t)y) + C;

(ii) lower C-convex, if for any x, y e K and for any t e [0,1], F(tx + (1 - t)y) c tF(x) + (1 - t)F(y) - C;

(iii) C-convex, if F is both upper C-convex and lower C-convex.

Remark 1 If F is a upper C-convex map on K, then for any x e K, F(x) + C is convex set.

We first recall the well-known concept of monotone mapping for a real set-valued mapping.

Definition 2 A bifunction f : K x K ^ 2R\{0} is said to be

(i) monotone on K, if for any x, y e K

z + z! < 0, Vz e f (x,y),z! e f (y,x);

(ii) pseudomonotone on K, if for any x, y e K

z > 0, Vz ef (x,y) ^ z' < 0, Vz' ef (y,x).

It is well-known that every monotone map is pseudomonotone map. In the case where F is a vector set-valued, the concept of monotonicity can be also extended as follows.

Definition 3 Let C c Y be a nonempty, closed, convex and pointed cone with int C = 0. A set-valued map F : K x K ^ 2Y\{0} is said to be

(1) C-monotone if, for all x, y e K, F (x, y) + F (y, x) c — C;

(ii) C-pseudomonotone type I if, for all x, y e K,

F(x,y) n (-int C) = 0 ^ F(y, x) n (int C) = 0;

(iii) C-pseudomonotone type II if, for all x, y e K, F(x,y) n (-int C) = 0 ^ F(y,x) c -C;

(iv) f-monotone w.r.t. C*if, for any f e C* and for any x, y e K, £(z) + £(z') < 0, Vz e F(x,y), Vz' e F(y, x);

(v) f-pseudomonotone w.r.t. C* if, for any f e C* and for any x, y e K, Hz) > 0, Vz e F (x, y) ^ Hz') < 0, Vz' e F (y, x).

Remark 2 (1) It is clear that C-monotone mapping is C-pseudomonotone type I and type II and C-pseudomonotone type II implies C-pseudomonotone type I.

(2) Every C-monotone mapping is f-pseudomonotone w.r.t. C *.

(3) Every C-pseudomonotone type II mapping is f-pseudomonotone w.r.t. C*, Indeed, for any f e C* and for any x, y e K satisfying £(z) > 0 for all z e F(x, y), we have z / -int C and so F(x,y) n (-int C) = 0. F(y,x) c -C implies that £(z') < 0 for all z' e F(y, x). But, C-pseudomonotone type I may not implies f-pseudomonotone w.r.t. C *.

Example 1 Let X = R,K = [0,1], Y = R2, C = R+. Define F : K x K ^ 2Y\{0} by

{(x, — x) if x = y,

{(y — x)} x [0, (y — x)] ify — x > 0, {(y — x)} x [(y — x), 0] ify — x < 0.

Thus, clearly that F is f-pseudomonotone on K w.r.t. C* = C. Indeed, for any x, y e K and f e C* if f(F(x, y)) > 0, then y — x > 0. This implies that

F(y,x) = {(x - y)} x [(x - y), 0] c -R+ ^ < 0, Vz e F(y,x).

But C-pseudomonotone type II in the case when x = y.

Example 2 Let X = R, K = [0, Y = R2, C = R+ and C* = C. Define F : K x K ^ 2y\{0} by

F(x,y) = {0} x [0, |y - x|], Vx,y e K.

Thus, clearly that for any x, y e K, F(x,y) n (-intC) = 0 implies that F(y, x) n (int C) = 0. Hence, F is pseudomonotne on K type I, but not C-pseudomono-tone type II.

Moreover, for any f e C* and x, y e K, we then have £ (F(x,y)) = £ (F(y,x)) > 0.

Therefore, F is not f-pseudomonotone on K w.r.t. C * as shown in the following example.

Definition 4 A topological space E is said to be connected iff, it is not the union of two disjoint nonempty open sets. Moreover, E is said to be path-connected iff, any two points of E can be joined by a path.

The following lemma, which gives an equivalent characterization of connected spaces, plays an important role in our proof.

Lemma 1 A topological space E is connected if and only if the only subsets of E which are both open and closed are E and 0.

Definition 5 Let F : K ^ 2Y be a set-valued mapping with nonempty values. F is said to be

(i) upper semicontinuous(u.s.c.) on K iff, for every x e K and every neighborhood N(F(x)) of F(x) , there exists a neighborhood N(x) of x such that F(N(x)) c N(F(x));

(ii) lower semicontinuous(l.s.c.) on K iff, for every x e K, u e F(x) and every neighborhood N(u) of u, there exists a neighborhood N(x) of x such that F(x') n N(u) = 0 for every x' e N (x).

Proposition 1 (Aubin and Ekeland 1984; Ferro 1989)

(i) F is l.s.c. at 2 if and only if for any sequence {2n} c A with 2n ^ 2 and any x e F(2), there exists xn e F(2n) such that xn ^ x.

(ii) F is weakly l.s.c. at 2 if and only if for any sequence {2n} c A with 2n ^ 2 and any x e F(2), there exists xn e F(2n) such that xn ^ x.

(iii) If F has compact values (i.e., F (2) is a compact set for each 2 e A), then F is u.s.c. at 2 if and only if for any sequence {2n} c A with 2n ^ 2 and for any xn e F (2n), there exists x e F(2) and a subsequence {xnk} of {xn} such that xnk ^ x.

We collect the following well-known KKM-Fan lemma.

Lemma 2 (Fan 1984) Let M be a nonempty, closed and convex subset of X and F : M ^ 2M\{0} be a set-valued map. Suppose that for any finite set {x1,..., xm} c M, one has

(i) conv{x1,. . . , Xm } C U j_ 1F(xi) (i.e., F is a KKM map on M);

(ii) F(x) is closed for every x e M; and

(iii) F(x) compact for some x e M.

Then HxsmF (x) = 0.

Now, we recall the fundamental tools used throughout this paper. This leads to the concepts of asymptotic cone and asymptotic function through its epigraph.

d e X : 3tk ^ Xk e X such that — Xk ^ d

where or "o — limn^TO xn = x" means convergence in the weak topology. In case K is convex subset, Kx can also be determined by the following formula

K<x = {d e X : xo + td e K, Vt > 0, Vxo e K}. The barrier cone of K is defined by

barrK =\ £ * e K * : sup(£ *, y) < >.

[ yeK J

Proposition 2 (Ansari and Flores-Bazan 2006, Proposition 2.1) The following holds:

(i) K1 ç K2 implies Kl ç K^;

(ii) (K + x)œ = Kœ, Vx e X;

(iii) let {K'},-e/ be any family of nonempty sets in X , then

(nie/KM CHieiK^. (3)

If, in addition, nieIK' = 0 and each set II' is closed and convex, then we obtain an equality in (3).

Lemma 3 (Adly et al. 2004) Let K be a nonempty, closed and convex subset of a real reflexive Banach space X with int(barr K) = 0. Then there is no sequence {xn} c K with \\xn\\ ^ to such that origin is a weak limit of n , i.e. n ^ 0.

Ilxn y llxn II

Lemma 4 (Fan and Zhong 2008) Let K be a nonempty, closed, convex subset of a real reflexive Banach space X with int (barr (K)) = 0. Then there exists no sequence {dn} c Kx with each \\dn\\ = 1 such thatdn ^ 0.

Lemma 5 (Fan and Zhong 2008) Let (M, d) be a metric space and ß0 e M be a given point. Let K : M ^ 2X be a set-valued mapping with nonempty valued and upper semicontinuous at /z0. Then there exists a neighborhood N(p0) of ß0 such that (K(p)<x>) C (K(^0))«>for all ß e N(ß0).

Characterization of nonemptiness and boundedness of the solution set

In this section, we shall prove the characterization of nonemptiness and boundedness of the solution set for (GWVEP) which states that under suitable conditions.

First of all, we recall the existing assumptions and results which can be found in Ansari and Flores-Bazan (2006), Zhong et al. (2011), Sadeqi and Alizadeh (2011).

Assumption 1 (Zhong et al. 2011; Ansari and Flores-Bazan 2006) The set-valued map F : K x K ^ 2y\{0} is such that:

(F0) F(x, x) = {0} for all x e K.

(F1) For any x,y e K, F(x,y) n (-int C) = 0 ^ F(y,x) c —C (C pseudomonotone type II).

(F2) For any x e K, F(x, ■) : K ^ 2Y\{0} is C-convex.

(F3) For any x,y e K, the set {z e [x,y] : F(z,y) n (-int C) = 0} is closed, where [x, y]

denotes the closed line segment joining x and y . (F4) For any x e K, F(x, •) is weakly lower semicontinuous. (F5) For any y e K, {x e K : F(y, x) n (int C) = 0} is convex.

Under Assumption 1, It is proved in Zhong et al. (2011) that, S^ (K, F) is nonempty if K is bounded subset of X . In the case where K is unbounded, it is needed to determine the behavior of F along some particular directions. We introduce the following cones.

Ri := {d e Kx : F(y,y + td) n (int C) = 0, Vy e K, t > 0}. (4)

The following lemma illustrates that the solution set SW (K, F) and S{W (K, F) are coincide no matter what K is bounded or not.

Lemma 6 (Sadeqi and Alizadeh 2011, Lemma 3.4) Let K be a nonempty, closed and convex subset of X and F : K x K ^ 2Y\{0} be a set valued map satisfying (F0) — (F3). Then

SW (K, F) = SD (K, F).

Theorem 1 (Sadeqi and Alizadeh 2011, Theorem 3.5) Let K be a nonempty, closed and convex subset of X and F : K x K ^ 2Y\{0} be a set valued map satisfying (F0) — (F5). If the set the solution set SWW (K, F) is nonempty, then

(Spw(K, F))x = (SW(K, F))x = Ri.

The following theorem is due to the result in Zhong et al. (2011), Ansari and Flores-Bazan (2006), Sadeqi and Alizadeh (2011).

Theorem 2 Let Kbe a nonempty closed convex subset of Xand F : K x K ^ 2Y \{0} be a set valued mapping satisfying assumptions (F0) — (F5). Suppose that int(barr(K)) = 0. Then the following statements are equivalent.

(i) the solution set of (K, F) is nonempty and bounded;

(ii) the solution set of SW (K, F) is nonempty and bounded;

(iii) Ri = {0};

(iv) there exists a bounded set B c K such that for every x e K\B, there exists some y e B such that F(y, x) n (int C) = 0.

Proof (i) ^ (ii) and (ii) ^ (iii) are obtained by Theorems 1 and 2, respectively.

(iii) ^ (iv) Suppose not, if (iv) does not hold, then there exists a sequence {xn} c K such that for each n, \\xn || > n and

F(y, xn) n (int C) = 0,

for every y e K with ||y|| < n. For fixed y e K and t > 0, without loss of generality, we may take a subsequence {xnk} of {xn} such that

e (0,1) and a — lim —= a — lim ..""".. = d0 e

\\Xnk — y\\ k^+<x\\X"k — yII k^+x\\X"k II (5)

Thanks to Lemma 3, one has d0 = 0. The lower C-convexity of F(x, ■) implies p(y>y + P^—Pi ^ (l - ^T"^V(y,y) + xnk) - C.

Wxnk - y\\J V llxnk - y\\J \\Xnk - y\\

It follows from F(y, y) = {0} and F(y, xnk) n (int C) = 0 that t (Xnk - y)

ny'y +«xnf-¥jn {MC) = 0

t (xn, — y)

Since y +---^ y + td0 and F is weakly lower semicontinuous at second argu-

l|Xnk — y\\

ment, we have that F(y, y + td0) n (int C) = 0, and so d0 e R1. This is a contradiction. Hence (iv) holds. (iv) ^ (ii) Let G : K ^ 2K be a set-valued mapping defined by

G(y) :={x e K : F(y, x) n (int C) = 0}, Vy e K. (6)

We first prove that G(y) is a closed subset of K . Indeed, for any xn e G(y) with Xn ^ Xo, we have F(y, xn) n (int C) = 0. It follows from the weakly lower semicontinuity of F(x, ■) that F(y, x0) n (int C) = 0. This shows that x0 e G(y) and so G(y) is closed.

Next, we will show that G is a KKM mapping. Suppose to the contrary that there exist a1, a2,..., an e (0,1) with a1 + a2 + •••+ an = 1, y1, y2,..., yn e K and y = aiyi + aiyi +-----h anyn e co{yi, y2,..., yn} such that y </ Uie{i,2,...,„} G(y,-). Then

F(yi,y) n (intC) = 0, i = 1,2,...,n. Using (F-l) yields

F (y, yi) n (-intC) = 0, i = 1,2,..., n. (7)

The upper C-convexity of F implies

aiF(y, yi) + a2F(y,y2) +-----h anF(y,yn) c F(y, y) + C = 0 + C c C.

This is a contradiction with (7). Therefore, G is KKM mapping.

We may assume that B is a bounded closed convex set (otherwise, consider the closed convex hull of B instead of B ). Let {y1,...,ym} be finite number of points in K and let M := co(B U {y1;y2,...,ym}). Then the reflexivity of the space X yields that M is weakly compact convex. We consider the set-valued mapping G' which defined by G'(y) := G(y) n M for all y e M. Then each G'(y) is a weakly compact convex subset of M and G' is a KKM mapping. We claim that

0 = HyeMG'(y) c B. (8)

By Lemma 2, the intersection in (8) is nonempty. Moreover, if there exists some x0 e nyeMG'(y) but x0 / B, then by (iv), we have F(y,x0) n (int C) = 0 for some y e B. Thus, x0 / G(y) and so x0 / G'(y), which is a contradiction to the choice of x0.

Let z e nyeMG(y). Then, by (8) we get z e B, and so z e nrm=1(G(yi) n B). This shows that the collection {G(y) n B : y e K} has finite intersection property. For each y e K, it follows from the weak compactness of G(y) n B that nyeK (G(y) n B) is nonempty, which coincides with the solution set of S{W (F, K). The proof is complete. □

The following example show that Theorem 2 is applicable.

Example 3 Let X = R, K = [0, Y = R2, C = R+, e = (1,1) e intC. A set-valued map F : K x K ^ 2r2\{0} is defined by

F(x,y) = {y — x} x [(y — x), (1 + x)(y — x)], Vx,y e K.

We have that Kx = [0, and C*0 := {(xi, X2) e R2, xi + X2 = 1, xi > 0 and X2 > 0}. It is easily seen that F is satisfied conditions (F0)-(F4). To verify (F5) holds, we fixed y e [0, and consider the following set,

{x e K : F (y, x) n intC = 0} = {x e [0, : x - y < 0 or (1 + y)(x - y) < 0}

= {x e [0, : x < y} = [0, y] is convex set.

Obviously,

Ri = {d e : F(y, y + td) n (int C) = 0, Vy e K, t > 0} = {d e [0, : td < 0, Vt > 0 and Vy e [0, +^)} = {0}.

Hence, Theorem 2 concludes that (F, K) is nonempty and bounded. It follows from direct calculating that SW (F, K) = {0}.

In what follow, we shall discuss the relationship between the nonemptiness and boundedness of the solution set for (GWVEP) and the solution set for (GWVEP) which F is composed by f e C*. We recall the concept of f-efficient solution set for (GWVEP) as follows.

For any fixed f e Cthe real set-valued map f (F) : K x K ^ 2R\{0} is defined by %(F)(x,y) :={f(z) : z e F(x,y)}, Vx,y e K. (9)

A vector x e K is called f-weak efficient solution to the (GWVEP) if inf f(z) > 0, Vy e K,

zeF (x,y)

and f-weak efficient solution to the (DGWVEP) if sup f(z) < 0, Vy e K.

zeF (y,x)

Denote by (K, F) and Sf(K, F) the set of all f-weak efficient solution to the (GWVEP) and (DGWVEP), respectively. The following lemma characterizes relation between S^ (K, F) and Sp (K, F).

Lemma 7 Suppose that int C = 0 and for any x e K, F(x, K) + C is a convex set. Then,

SW (K, F) = Ut ec *\{0}Sf (K, F) = eC to SP (K, F).

Proof (2) Let x0 e UfeC»o Sp (K, F). Then there exists f0 e C such that

f0 (z) > 0 for all y e K for all z e F(x0, y). (10)

We claim that x0 e SWW (K, F). If not, then there exists y0 e K such that

fo (zo) < 0 for some zo e F(xo, yo).

This is a contradiction with (10). Hence, we have desired. (c) Let xo e S^ (K, F). Then,

F(xo, y) n (-int C) = 0 for all y e K.

This implies that

F(xo,K) n (-/nt C) = 0.

Since C is a pointed convex cone, we have

(F(xo,K) + C) n (-int C) = 0.

Using the separation theorem for convex sets, there exists some f' e Y*\{0} such that

inf{f '(F(xo,y) + c : y e K, c e C)} > sup{f '(-c) : c e C}. (11)

From (11), we get f' e C*\{0} and so

f '(z) > 0 for all z e F(x0, y) for all y e K.

By our hypothesis, we have Cis a weekly compact base for C* and we can choose

e e int C with f '(e) > 0. Setting f'' = we then have that f'' e Cand

f '(e)

f ''(z) > 0 for all z e F(xo, y) for all y e K. Hence, x0 e Sp„ (K, F) c Uf eC»oSp(K, F). This completes the proof. □

The following corollary give the sufficient conditions for nonemptiness and bounded-ness of solution set for (GWVEP) in the case of real set-valued mappings.

It follows from Theorem 2, we can derive the following corollary in the case where F : K x K ^ 2R\{0}.

Corollary 1 Let AT be a nonempty closed convex subset of X and F : K x K ^ 2R \{ 0} be a set-valued mapping satisfying assumptions (F0) — (F4). Suppose that int (barr(K)) = 0. Then the following statements are equivalent.

(i) the solution set of SW (K, F) is nonempty and bounded;

(ii) the solution set of SW (K, F) is nonempty and bounded;

(iii) R ={d e Km : supzef (y>y+td) z < 0, Vy e K, t > 0} = {0};

(iv) there exists a bounded set B c K such that for every x e K\B, there exists y e B such that z > 0 for some z e F(y, x).

Proof We see that F satisfies the assumption (F0)-(F4) in Theorem 2. It is easy to verify, by (F2), that (F5) is satisfied. □

By virtue of Lemma 7, one sees that the solution set for (GWVEP) can be represented by union of real set-valued £(F) mappings. This means that the nonemptiness of Sp (K, F) guarantees the existence of solution for (GWVEP). We next establish the existence theorem for f-weak efficient solution to the (GWVEP). By the idea of linear scalarization technique, for any f e Cwe first introduce the set

R := < d e Kx : sup H(z) < 0, Vy e K, t > 0 L

[ zeF (y,y+td) J

The following lemma shows that the condition of UfeC»oR = {0} is weaker than

Ri = {0}.

Lemma 8 Ri = {0} ^ Uf eC*oRf = {0}.

Proof Assume that Rf = {0}. Let d0 e UfeC»0Rf. Then there exists f0 e C*0 and d0 e Kx such that for every y e K and t > 0

£o (z) < 0 for all z e F(y, y + tdo). (12)

We claim that for any z e F(y,y + tdo), z / int C. If not, there exists z0 e F(y,y + td0) such that z e int C and so

Ho (zo) > 0, (13)

which leads to contradiction with (12). Hence, for every y e K and t > 0

F(y, y + tdo) n (int C) = 0.

By our hypothesis, d0 = 0.

The following example shows that the inverse implication of Lemma 8 may not be true. The following example has been changed format.

Example 4 Let X = R, K = [0, Y = R2, C = R+, e = (1,1) e int C. Define F : K x K ^ 2y\{0} by

Fy) _ I {0}x [0,1 -ly - x|], if 0 < ly - x| < 1, (x,y) _ \ [|y - x| - 1,0] X {0}, if |y - x| > 1.

Then Kx = [0, and C*0 = {(xi, xi) e R+ : xi + x2 = 1}. We see that for any y e R+, d e R and t > 0,

F+ td) r I {0} x [0,1], if 0 <\td\< 1,

(y,y +td) ^[0, +TO) x {0}, if \td\ > 1,

which implies that F (y, y + td) n intC = 0 for all y e R+, d e R and t > 0. Hence, R1 = [0, +to). But, for any f e C*0, we have for any y, d e R+ and t > 0

£(z) > 0, for all x e F(y,y + td),

which implies that d must be 0 , and so R = {0} for all f e C .

From the Corollary 1, we can obtain the following characterization corollary for f-effi-

cient solution Sp (K, F) and S^K, F).

Corollary 2 Let f e Cbe any given. Let K be a nonempty closed convex subset of X and F : K x K ^ 2Y\{0} be a set-valued mapping satisfying assumptions (F0), (F2)-(F4) and (v) in Definition 3. Suppose that int (barr (K)) = 0. Then the following statements are equivalent:

(i) the solution set of Sp (K, F) is nonempty and bounded;

(ii) the solution set of SD (K, F) is nonempty and bounded;

(iii) R = {0};

(iv) there exists a bounded set B c K such that for every x e K\B, there exists y e B such that f(z) > 0 for some z e F(y, x).

Proof For any fixed f e C*\{0}, we define f(F) : K x K ^ 2R\{0} as in (9). It is not hard to check that f (F) satisfies conditions (F0)-(F4) in Corollary 1. □

We now characterize the nonemptiness and boundedness of SW (K, F) in term of non-emptiness and boundedness of the solution set Sp(K, F) for any f e C

Theorem 3 Let X be a reflexive Banach space and K be a closed convex subset of X with int(barrK) = 0. Let Y be a normed space and C*0 is a compact base of C*. Suppose that F : K x K ^ 2y\{0} is a set-valued mapping satisfying assumptions (F0), (F2)- (F4) and (v) in Definition 3.

Then SW (K, F) is nonempty and bounded if and only if for any f e C *0, SfP (K, F) is nonempty and bounded.

Proof Suppose that for any f e CSp(K, F) is nonempty and bounded. Then by Corollary 2, R = {0}. We claim that SW (K, F) is nonempty and bounded. The nonemptiness of SW(K, F) is obvious, because of Sp (K, F) c SW (K, F). We only need to show that SW (K, F) is bounded. If not, there exists a sequence xn e S^ (K, F) such that l|x«|| ^ Since xn e S^(K, F), we then have

F(xn,y) n (-int C) = 0, for all y e K.

Thus, for every zn e F(xn,y), zn / —int C. Then there exists fn e Csuch that

fn(zn) > 0, for all z e F(xn,y), for all y e K.

By the f-pseudomonotonicity of F , we have

fn(z'n) < 0, for all z e F(y,x„), for all y e K (14)

Since C is compact, without loss of generality, we can assume that fn —>- f0 6 CFor any fixed y e K and t > 0, without loss of generality, we may take a subsequence {xnk} of {xn} such that

e (0,1) and w — lim --- = w — lim --- = d0 e K&.

\\Xnk — y II k^ + ^\\Xnk — y\\ k^+<x \\Xnk I

By Lemma 3, d0 = 0. Upper C-convexity of F implies

1--1-V(y,y) +---F(y,xm) C F(y,y + - tXnk -y) ) + C

llx»k - y\\) y y llxnk - y\\ y k V y llxnk - yll '

It follows from F (y, y) = {0} and (14) that for any fn

Since F is weakly lower semicontinuous at second variable and fn ^ f0, we have

fo (F(y, y + tdo)) < 0.

This implies that 0 = d0 e r\, which is a contradiction.

Conversely, we assume that S{W (K, F) is nonempty and bounded. We claim that Sp (K, F) is nonempty and bounded for all f e C*0. We consider the set A c C*0 as follows.

A :={f e C*0 : Sp(K, F) is nonempty and bounded }.

Clearly, A is nonempty. Firstly, we claim that A is open subset in C*0. If not, there exists f0 e A and a sequence fn e C*0 with fn ^ f0 such that / A. This implies that Rfn = {0} . Then there exists dn e Rf1 such that \\dn\\ = 1. Since C*0 is compact and \\dn\\ = 1, without loss of generality, we may assume that dn ^ d0 e K»\{0}. Since dn e Rfn, we have

fn(z') < 0 for all z' e F(y,y + tdn) for all y e K.

Since F is weakly lower semicontinuous at second variable and fn ^ f0, we have

f0 (z') < 0 for all z' e F(y, y + td0) for all y e K.

Thus 0 = d0 e Rf0. This implies that Spo (K, F) is not nonempty and bounded, which leads to a contradiction with f0 e A. Hence A is an open subset of C*0.

Finally, we claim that A is a closed subset of C*0. Let fn e A with fn ^ f0. In view of fn e A, we have Sfn (K, F) is nonempty and bounded. Let xn e Spn (K, F). Whereas Spn (K, F) c SW(K, F) and SW(K, F) is bounded, {xn} is also. We may assume that xn ^ x0 e K. Since xn e Sfn (K, F), then we have

fn(z) > 0 for all z e F(xn,y) for all y e K.

By f-pseudomonotonicity of F , we get

fn(z') < 0, for all z' e F(y,xn), for all y e K.

Since F is weakly lower semicontinuous at the second variable, letting n ^ to fo(z') < 0, for all z' e F(y,x0), for all y e K.

Hence, X0 e Sf0 (K, F). Thanks to Corollary 2, we get that X0 e Sp0 (K, F). The bounded-ness of SW (K, F) implies Sp0 (K, F) is also. This means that f0 e A and so A is closed. Since the base C*0 of C* is connected, we have A must be C*0. □

Theorem 4 Let X be a reflexive Banach space and K be a closed convex subset of X with int(barrK) = 0. Let Y be a normed space and C*0 is a compact base of C*. Suppose that F : K x K ^ 2y\{0} is a set-valued mapping satisfying assumptions (F0), (F2)-(F4) and (v) in Definition 3. Then the following statements are equivalent.

(i) SW (K, F) is nonempty and bounded;

(ii) For every f e CSp (K, F) is nonempty and bounded;

(iii) Uf ec = {0}.

Remark 3 Theorem 4 generalize Theorem 2, in the following three cases:

(i) Condition (F1) is relaxed to the condition (f|).

(ii) Recession cone Ri = {0} is relaxed to the condition UfeC»oRf = {0}.

(iii) Condition (F5) is omitted.

The following example show that Theorem 4 is applicable.

Example 5 Let X = R, K = [0, Y = R2, C = R+, e = (1,1) e intC. A set-valued map F : K x K ^ 2r2\{0} is defined by

F(x,y) = {(y - x)} x [(y - x), (e(y-x - 1) + (y - x)]

Then, clearly (F0), (F2) — (F4) and (v) in Definition 3 are satisfied. For any f e Cwe consider

R = <j d e : sup %(z) < 0, Vy e K, t > 0

ze(F (y,y+td))

= {d e [0, : ?(z) < 0, Vz e {td} x [td, (etd - 1) + td] and Vy e K, t > o} = {0}. It follows from Theorem 4 that, SW (K, F) is nonempty and bounded.

Stability analysis

In this section, we shall establish the stability theorem of solution set for (GWVEP) when the mapping F and the domain set K are perturbed by different parameters.

We first recall some important notions . Let (A, and (M, dM) be two metric spaces. Let K(!) be perturbed by a parameter I, which varies over (A, dA), that is, K : A ^ 2X is a set-valued mapping with nonempty, closed, and convex values. Let F be perturbed by a parameter which varies over (M, dM), that is, F : K x K x M ^ 2Y\{0} is a parametric set-valued mapping.

Consider the parametric generalized weak vector equilibrium problems, denoted by (PGWVEP), which consists in finding x e K(!) such that

F(x, y, n (-int C) = 0 Vy e K(!). (PGWVEP)

Denote by S{W(X, p) the set of all weak efficient solution to the (PGWVEP). Let {A n} be a sequence of nonempty subset of X . We define

limsupAn := {x e X : 3{nk},xnk e Ank such thatxnk ^ x}.

We say that the sequence {An} is of upper convergence in the sense of Painleve-Kura-towski (P.K. convergence) Durea (2007) to A if limsupn^+TO An c A.

The following theorem shows that under suitable situation, there exists a neighborhood N(X0) x N(p0) of (X0, p0) such that SW(X, p) P.K. convergence to SW(X0, p0) in the neighborhood N(X0) x N(p0).

Theorem 5 Let X be a real reflexive Banach space and K be a closed convex subset of X with int (barrK) = 0. Let Y be a normed space and C*0 is a compact base of C*. Suppose that F satisfies the following conditions:

(I) K(■) is continuous on A and int(barr K(X0)) = 0, for all X e A and has nonempty closed convex valued.

(II) For any X e A and x e K(X), F(x, x, p) = {0}.

(III) For any X e A and p e M, F(■, ■, p) is f-pseudomonotone on K(X) w.r.t. C*0.

(IV) For any p e M and x e K(p), F(x, ■, p) is C-convex.

(V) For any X e A and p e M, F(■, ■, ■) is continuous on K(X) x K(X) x M.

If SW (X0, p0) is nonempty and bounded, then the following statements hold.

(i) there exists a neighborhood N(X0) x N(p0) such that S{W(X, p) has a nonempty and bounded for all (X, p) e N(X0) x N(p0).

(ii) lim suP(X,p)^(i0,p0) sw(X> p) ^ (X0> po).

Proof (i) We claim that there exists a neighborhood N(X0) x N(p0) of (X0, p0) such that for any (X, p) e N(X0) x N(p0) and f e C*0

(X, p) := i d e K(X)TO : sup f(z) < 0, Vy e K, t > 0 I = {0}.

[ ze(F (y,y+td,p)) J

If not, there exists (Xn, pn) e A x M with (Xn, pn) ^ (X0, p0) and f' e C*0 such that R' (X„, Pn) = {0}.

Since K is lower semicontinuous at X0, for any y e K(X0), we have yn e K(Xn) such that yn ^ y. Together with pn ^ p0, we have (yn, pn) ^ (y, p0). Thus, we can select a sequence {dn} such that

dn e K(X„)TO and sup %'(z) < 0, Vy e K(Xn), t > 0.

zeF (yn,yn +tdn,^n) y '

with || dn || = 1 for all n = 1,2,.... Since {dn} is a bounded sequence in a reflexive Banach space X we can assume that dn ^ d0. It follows from Lemma 4 that d0 = 0. We claim that d0 e K(X0)TO. Since Kis upper semicontinuous at X0 and dn e K(Xn)TO, by Lemma 5,

we have that dn e K(!0)TO, for all sufficiently large n . By the closure of K(!0)TO, we have d0 e K(10W Notice that the continuity assumption of F , taking the limit in (15), we have

sup % '(z) < 0,

zeF (y,y+tdo,^o)

which implies that 0 = d0 e R- (Ao, /x0). This is a contradiction with S^ (A0, = 0, so we have the claim. (ii) We want to show that for any (A, ^ (A0, ^0),

lim sup SW (A, ç SW (Ao, ^o).

Let x e limsup(A,^)^(A0,M0) SW(A, Then there exits a sequence xnk e S^W(Ank, ) such that

Xnk ^ x as k ^ to. Since A" is upper semicontinuous at A0, for sufficiently large n we get that

K (An) ç K (Ao ) + - B, n

where B is a closed unit ball. By virtue of xnk e K(A„k), we get that

d(xnk, K(Ao)) < — ^ 0. k nk

It follows from K (A0 ) is closed and

that xx e K (Ao).

Since K is lower semicontinuous at A0, for any y e K(A0) there exists ynk e K(A„k ) with ynk ^ y. By our hypothesis, we get

F(Xnk,ynk, ^nk) n (-intC) = 0.

Continuity of F implies

F (XX, y, ^0) n (-intC) = 0.

Since the latest inequality holds for all y e K(A0). Hence, x e SW (Ao, ^0). D

Conclusions

In this paper, some characterizations of nonemptiness and boundedness of solution sets for generalized weak vector equilibrium problems are established in a reflexive Banach space. By using the linear scalarization method, we give a sufficient and necessary condition for the nonemptiness and boundedness of SW(K, F) in term of nonemptiness and boundedness of the solution set Sp(K, F) for any f e CAs application, we discuss the stability result for the solution set to (PGWVEP) in the sense of Painlevé-Kuratowski upper convergence of set.

Authors' contributions

Both authors read and approved the final manuscript.

Author details

1 Program in Mathematics, Faculty of Education, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand.

2 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

Acknowledgements

The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments in the original version of this paper. The first author was supported by Thailand Research Fund (TRG5880058).

Competing interests

Both authors declare that they have no competing interests.

Received: 30 March 2016 Accepted: 4 August 2016 Published online: 12 August 2016

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