A characterization of nonemptiness and boundedness of the solution set for setvalued vector equilibrium problems via scalarization and stability results
 Pakkapon Preechasilp^{1}Email authorView ORCID ID profile and
 Rabian Wangkeeree^{2}
Received: 30 March 2016
Accepted: 4 August 2016
Published: 12 August 2016
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, Painlevé–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
Background
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, (Hölder) 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 (Hölder) 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 setvalued map on unbounded constraint set in reflexive Banach spaces. We first collect the characterization results of the nonemptiness and boundedness 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 Painlevé–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 boundedness 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
Definition 1
 (i)upper Cconvex, if for any \( x,y\in K \)and for any \( t\in [0,1] \),$$\begin{aligned} tF(x) + (1t)F(y) \subseteq F(tx + (1t)y) + C; \end{aligned}$$
 (ii)lower Cconvex, if for any \(x,y\in K\) and for any \(t\in [0,1]\),$$\begin{aligned} F(tx + (1t)y) \subseteq tF(x) + (1t)F(y)  C; \end{aligned}$$
 (iii)
Cconvex, if F is both upper Cconvex and lower Cconvex.
Remark 1
If F is a upper Cconvex map on K, then for any \(x\in K , F(x) + C\) is convex set.
We first recall the wellknown concept of monotone mapping for a real setvalued mapping.
Definition 2
 (i)monotone on K, if for any \(x,y\in K\)$$\begin{aligned} z+ z' \le 0, \quad \forall z\in f(x,y),z'\in f(y,x); \end{aligned}$$
 (ii)pseudomonotone on K, if for any \(x,y\in K\)$$\begin{aligned} z \ge 0, \quad \forall z\in f(x,y) \Rightarrow z' \le 0, \ \forall z'\in f(y,x). \end{aligned}$$
It is wellknown that every monotone map is pseudomonotone map.
In the case where F is a vector setvalued, the concept of monotonicity can be also extended as follows.
Definition 3
 (i)Cmonotone if, for all \(x, y \in K \),$$\begin{aligned} F(x,y) + F(y,x) \subseteq  C; \end{aligned}$$
 (ii)Cpseudomonotone type I if, for all \(x, y \in K \),$$\begin{aligned} F(x,y)\cap (int \ C) = \varnothing \Rightarrow F(y,x) \cap (int \ C) = \varnothing ; \end{aligned}$$
 (iii)Cpseudomonotone type II if, for all \(x, y \in K \),$$\begin{aligned} F(x,y)\cap (int \ C) = \varnothing \Rightarrow F(y,x) \subseteq  C; \end{aligned}$$
 (iv)\(\xi \)monotone w.r.t. \(C^{*} \)if, for any \(\xi \in C^{*} \) and for any \( x, y \in K \),$$\begin{aligned} \xi (z) + \xi (z') \le 0, \quad \forall z\in F(x,y), \;\; \forall z'\in F(y,x); \end{aligned}$$
 (v)\( \xi \)pseudomonotone w.r.t. \(C^{*}\) if, for any \( \xi \in C^{*} \) and for any \( x, y \in K \),$$\begin{aligned} \xi (z) \ge 0,\quad \forall z\in F(x,y) \Rightarrow \xi (z') \le 0, \quad \forall z'\in F(y,x). \end{aligned}$$
Remark 2
 (1)
It is clear that Cmonotone mapping is Cpseudomonotone type I and type II and Cpseudomonotone type II implies Cpseudomonotone type I.
 (2)
Every Cmonotone mapping is \(\xi \)pseudomonotone w.r.t. \( C^{*} \).
 (3)
Every Cpseudomonotone type II mapping is \( \xi \)pseudomonotone w.r.t. \( C^{*} \), Indeed, for any \( \xi \in C^{*} \) and for any \( x,y\in K \) satisfying \( \xi (z) \ge 0 \) for all \( z\in F(x,y) \), we have \( z\notin int \ C \) and so \( F(x,y) \cap (int \ C) = \varnothing \). \( F(y,x) \subseteq  C \) implies that \( \xi (z') \le 0 \) for all \( z'\in F(y,x) \). But, Cpseudomonotone type I may not implies \( \xi \)pseudomonotone w.r.t. \( C^{*} \).
Example 1
Example 2
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 pathconnected 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 \( \varnothing \).
Definition 5
 (i)
upper semicontinuous(u.s.c.) on K iff, for every \( x \in K \) and every neighborhood N(F(x)) of F(x) , there exists a neighborhood N(x) of x such that \( F(N(x)) \subseteq N(F(x)) \);
 (ii)
lower semicontinuous(l.s.c.) on K iff, for every \( x \in K , u \in F(x) \) and every neighborhood N(u) of u, there exists a neighborhood N(x) of x such that \( F(x')\cap N(u) \ne \varnothing \) for every \( x' \in N(x) \).
Proposition 1
 (i)
F is l.s.c. at \(\bar{\lambda }\) if and only if for any sequence \(\{\lambda _n\} \subset \Lambda \) with \(\lambda _n\rightarrow \bar{\lambda }\) and any \(\bar{x}\in F(\bar{\lambda })\), there exists \(x_n \in F(\lambda _n)\) such that \(x_n \rightarrow \bar{x}\).
 (ii)
F is weakly l.s.c. at \(\bar{\lambda }\) if and only if for any sequence \(\{\lambda _n\} \subset \Lambda \) with \(\lambda _n\rightharpoonup \bar{\lambda }\) and any \(\bar{x}\in F(\bar{\lambda })\), there exists \(x_n \in F(\lambda _n)\) such that \(x_n \rightarrow \bar{x}\).
 (iii)
If F has compact values (i.e., \(F(\lambda )\) is a compact set for each \(\lambda \in \Lambda \)), then F is u.s.c. at \(\bar{\lambda }\) if and only if for any sequence \(\{\lambda _n\}\subset \Lambda \) with \(\lambda _n\rightarrow \bar{\lambda }\) and for any \(x_n\in F(\lambda _n)\), there exists \(\bar{x}\in F(\bar{\lambda })\) and a subsequence \(\{x_{n_k}\}\) of \(\{x_{n}\}\) such that \(x_{n_k}\rightarrow \bar{x}\).
We collect the following wellknown KKMFan lemma.
Lemma 2
 (i)
\( conv \{ x_1,\ldots ,x_m \} \subset \cup _{i=1}^m F(x_i ) \) (i.e., F is a KKM map on M);
 (ii)
F(x) is closed for every \( x \in M \); and
 (iii)
F(x) compact for some \(x \in M\).
Proposition 2
 (i)
\( K^1\subseteq K^2 \) implies \( K^1_\infty \subseteq K^2_\infty \);
 (ii)
\( \left( K+x\right) _\infty = K_\infty , \ \forall x\in X \);
 (iii)let \( \{K^i\}_{i\in I} \) be any family of nonempty sets in X , then$$\begin{aligned} \left( \cap _{i\in I} K^i \right) _\infty \subset \cap _{i\in I} K^i_\infty . \end{aligned}$$(3)
Lemma 3
(Adly et al. 2004) Let K be a nonempty, closed and convex subset of a real reflexive Banach space X with \( int(\text {barr } K) \ne \varnothing \). Then there is no sequence \( \{x_n\} \subset K \) with \( \Vert x_n \Vert \rightarrow \infty \) such that origin is a weak limit of \( \dfrac{x_n}{\Vert x_n\Vert } \), i.e. \( \dfrac{x_n}{\Vert x_n\Vert } \rightharpoonup 0 \).
Lemma 4
(Fan and Zhong 2008) Let K be a nonempty, closed, convex subset of a real reflexive Banach space X with \( int\,(barr\,(K))\ne \varnothing \). Then there exists no sequence \( \{d_n\}\subset K_\infty \) with each \( \Vert d_n\Vert =1 \) such that \( d_n \rightharpoonup 0 \).
Lemma 5
(Fan and Zhong 2008) Let (M, d) be a metric space and \( \mu _0\in M \) be a given point. Let \( K : M\rightarrow 2^X \) be a setvalued mapping with nonempty valued and upper semicontinuous at \( \mu _0 \). Then there exists a neighborhood \( N(\mu _0) \) of \( \mu _0 \) such that \( (K(\mu )_\infty ) \subset (K(\mu _0))_\infty \) for all \( \mu \in N(\mu _0) \).
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 FloresBazán (2006), Zhong et al. (2011), Sadeqi and Alizadeh (2011).
Assumption 1
 \( (F_0) \) :

\( F(x,x) =\{0\} \) for all \( x\in K \).
 \( (F_1) \) :

For any \( x,y\in K , F(x,y)\cap (int \ C) = \varnothing \Rightarrow F(y,x) \subseteq C\) (C pseudomonotone type II ).
 \( (F_2) \) :

For any \( x\in K , F(x,\cdot ):K\rightarrow 2^Y\backslash \{\varnothing \} \) is Cconvex.
 \( (F_3) \) :

For any \( x,y\in K \) , the set \( \{ z\in [x,y] : F(z,y)\cap (int \ C) = \varnothing \} \) is closed, where [x, y] denotes the closed line segment joining x and y .
 \( (F_4) \) :

For any \( x\in K , F(x,\cdot ) \) is weakly lower semicontinuous.
 \( (F_5) \) :

For any \( y\in K , \{x\in K : F(y,x) \cap (int \ C) = \varnothing \} \) is convex.
The following lemma illustrates that the solution set \( S^P_W(K,F) \) and \( S^D_W(K,F) \) are coincide no matter what K is bounded or not.
Lemma 6
Theorem 1
The following theorem is due to the result in Zhong et al. (2011), Ansari and FloresBazán (2006), Sadeqi and Alizadeh (2011).
Theorem 2
 (i)
the solution set of \( S^P_W(K,F) \) is nonempty and bounded;
 (ii)
the solution set of \( S^D_W(K,F) \) is nonempty and bounded;
 (iii)
\( R_1=\{0\} \);
 (iv)
there exists a bounded set \( B \subset K \) such that for every \( x \in K \backslash B \), there exists some \( y \in B \) such that \( F(y,x) \cap (int \ C) \ne \varnothing \).
Proof
(i) \( \Leftrightarrow \) (ii) and (ii) \( \Rightarrow \) (iii) are obtained by Theorems 1 and 2, respectively.
Let \( z \in \cap _{y\in M} G(y) \). Then, by (8) we get \( z \in B \), and so \( z\in \cap _{i=1}^m (G(y_i)\cap B) \). This shows that the collection \( \{G(y) \cap B : y \in K \} \) has finite intersection property. For each \( y \in K \), it follows from the weak compactness of \( G(y) \cap B \) that \( \cap _{y\in K} (G(y)\cap B) \) is nonempty, which coincides with the solution set of \( S^D_W(F,K) \). The proof is complete. \(\square \)
The following example show that Theorem 2 is applicable.
Example 3
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 \( \xi \in C^* \). We recall the concept of \( \xi \)efficient solution set for (GWVEP) as follows.
The following lemma characterizes relation between \( S^P_W(K,F) \) and \( S^P_{\xi }(K,F) \).
Lemma 7
Proof
The following corollary give the sufficient conditions for nonemptiness and boundedness of solution set for (GWVEP) in the case of real setvalued mappings.
It follows from Theorem 2, we can derive the following corollary in the case where \( F:K\times K \rightarrow 2^\mathbb {R}\backslash \{\varnothing \} \).
Corollary 1
Let K be a nonempty closed convex subset of X and \( F : K \times K \rightarrow 2^\mathbb R\backslash \{\varnothing \} \) be a setvalued mapping satisfying assumptions (F _{0})–(F _{4}).
 (i)
the solution set of \( S^P_W(K,F) \) is nonempty and bounded;
 (ii)
the solution set of \( S^D_W(K,F) \) is nonempty and bounded;
 (iii)
\( R=\{ d\in K_\infty : \sup _{z\in F(y,y+td)} z \le 0, \ \forall y\in K, t>0 \}=\{0\} \);
 (iv)
there exists a bounded set \( B \subset K \) such that for every \( x \in K \backslash B \), there exists \( y \in B \) such that \( z>0 \) for some \( z\in F(y,x) \).
Proof
We see that F satisfies the assumption \( (F_0) \)\( (F_4) \) in Theorem 2. It is easy to verify, by \( (F_2) \), that \( (F_5) \) is satisfied. \(\square \)
By virtue of Lemma 7, one sees that the solution set for (GWVEP) can be represented by union of real setvalued \( \xi (F) \) mappings. This means that the nonemptiness of \( S^P_\xi (K,F) \) guarantees the existence of solution for (GWVEP). We next establish the existence theorem for \( \xi \)weak efficient solution to the (GWVEP).
Lemma 8
\( R_1 = \{0\} \Rightarrow \cup _{\xi \in C^{*0}}R^{\xi }_{1} = \{0\}\).
Proof
The following example shows that the inverse implication of Lemma 8 may not be true.
The following example has been changed format.
Example 4
From the Corollary 1, we can obtain the following characterization corollary for \( \xi \)efficient solution \( S^P_{\xi }(K,F) \) and \( S^D_{\xi }(K,F) \).
Corollary 2
 (i)
the solution set of \( S^P_{\xi }(K,F) \) is nonempty and bounded;
 (ii)
the solution set of \( S^D_{\xi }(K,F) \) is nonempty and bounded;
 (iii)
\( R^{\xi }_1=\{0\} \);
 (iv)
there exists a bounded set \( B \subset K \) such that for every \( x \in K \backslash B \), there exists \( y \in B \) such that \( \xi (z)>0 \) for some \( z\in F(y,x) \).
Proof
For any fixed \( \xi \in C^*\backslash \{0\} \), we define \( \xi (F):K\times K\rightarrow 2^{\mathbb {R}}\backslash \{\varnothing \} \) as in (9). It is not hard to check that \( \xi (F) \) satisfies conditions \( (F_0) \)\( (F_4) \) in Corollary 1. \(\square \)
We now characterize the nonemptiness and boundedness of \( S^P_W(K,F) \) in term of nonemptiness and boundedness of the solution set \( S^P_\xi (K,F) \) for any \( \xi \in C^{*0} \).
Theorem 3
Let X be a reflexive Banach space and K be a closed convex subset of X with \( int (barrK) \ne \varnothing \). Let Y be a normed space and \( C^{*0} \) is a compact base of \( C^{*} \). Suppose that \( F : K \times K \rightarrow 2^Y\backslash \{\varnothing \} \) is a setvalued mapping satisfying assumptions F _{0} (F _{2})–(F _{4}) and (v) in Definition 3.
Then \( S_W^P(K,F) \) is nonempty and bounded if and only if for any \( \xi \in C^{*0} , S^P_{\xi }(K,F) \) is nonempty and bounded.
Proof
Theorem 4
 (i)
\( S^P_W(K,F) \) is nonempty and bounded;
 (ii)
For every \( \xi \in C^{*0} , S^P_{\xi }(K,F) \) is nonempty and bounded;
 (iii)
\( \cup _{\xi \in C^{*0} } R^{\xi }_{1} = \{0\} \).
Remark 3
 (i)
Condition \( (F_1) \) is relaxed to the condition \( (F^{\xi }_1) \).
 (ii)
Recession cone \( R_1=\{0\} \) is relaxed to the condition \( \cup _{\xi \in C^{*0}}R_1^\xi =\{0\} \).
 (iii)
Condition \( (F_5) \) is omitted.
The following example show that Theorem 4 is applicable.
Example 5
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 \( (\Lambda ,d_{\Lambda }) \) and \( (M, d_M) \) be two metric spaces. Let \( K(\lambda ) \) be perturbed by a parameter \( \lambda \), which varies over \( (\Lambda , d_{\Lambda }) \), that is, \( K : \Lambda \rightarrow 2^X \) is a setvalued mapping with nonempty, closed, and convex values. Let F be perturbed by a parameter \( \mu \), which varies over \( (M,d_{M}) \), that is, \( F : K \times K \times M \rightarrow 2^Y\backslash \{ \varnothing \} \) is a parametric setvalued mapping.
We say that the sequence \( \{A_n\} \) is of upper convergence in the sense of Painle\(\acute{\mathrm{v}}\)e–Kuratowski (P.K. convergence) Durea (2007) to A if \( \limsup _{n\rightarrow +\infty } A_n \subseteq A \).
The following theorem shows that under suitable situation, there exists a neighborhood \( N(\lambda _0)\times N(\mu _0) \) of \( (\lambda _0,\mu _0) \) such that \( S^P_W(\lambda ,\mu ) \) P.K. convergence to \( S^P_W(\lambda _0,\mu _0) \) in the neighborhood \( N(\lambda _0)\times N(\mu _0) \).
Theorem 5
 (I)
\( K(\cdot ) \) is continuous on \( \Lambda \) and \( int (\ barr \ K(\lambda _0)) \ne \varnothing \), for all \( \lambda \in \Lambda \) and has nonempty closed convex valued.
 (II)
For any \( \lambda \in \Lambda \) and \( x\in K(\lambda ) , F(x,x,\mu ) = \{0\}\).
 (III)
For any \( \lambda \in \Lambda \) and \( \mu \in M , F(\cdot ,\cdot ,\mu ) \) is \( \xi \)pseudomonotone on \( K(\lambda ) \) w.r.t. \( C^{*0} \).
 (IV)
For any \( \mu \in M \) and \( x\in K(\mu ) , F(x,\cdot ,\mu ) \) is Cconvex.
 (V)
For any \( \lambda \in \Lambda \) and \( \mu \in M , F(\cdot ,\cdot ,\cdot ) \) is continuous on \( K(\lambda )\times K(\lambda ) \times M \).
 (i)
there exists a neighborhood \( N(\lambda _0)\times N(\mu _0) \) such that \( S_W^P(\lambda ,\mu ) \) has a nonempty and bounded for all \( (\lambda ,\mu )\in N(\lambda _0)\times N(\mu _0) \).
 (ii)
\( \limsup _{(\lambda ,\mu )\rightarrow (\lambda _0,\mu _0)} S^P_W(\lambda ,\mu ) \subseteq S^P_W(\lambda _0,\mu _0) \).
Proof
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 \( S^P_W(K,F) \) in term of nonemptiness and boundedness of the solution set \( S^P_\xi (K,F) \) for any \( \xi \in C^{*0} \). As application, we discuss the stability result for the solution set to (PGWVEP) in the sense of Painlevé–Kuratowski upper convergence of set.
Declarations
Authors' contributions
Both authors read and approved the final manuscript.
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.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.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.
Authors’ Affiliations
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