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Approximate weakly efficient solutions of setvalued vector equilibrium problems
Journal of Inequalities and Applications volume 2018, Article number: 181 (2018)
Abstract
In this paper, we introduce a new kind of approximate weakly efficient solutions to the setvalued vector equilibrium problems with constraints in locally convex Hausdorff topological vector spaces; then we discuss a relationship between the weakly efficient solutions and approximate weakly efficient solutions. Under the assumption of near conesubconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tuckertype and Lagrangetype optimality conditions for setvalued vector equilibrium problems, respectively.
1 Introduction
Vector optimization problems, vector variational inequality problems, vector complementarity problems, and vector saddle point problems are particular cases of vector equilibrium problems. As an extensive mathematical model, the vector equilibrium problem is a hot topic in the fields of operations research and nonlinear analysis (see [1–8]). Gong [2–4] obtained optimality conditions for vector equilibrium problems with constraints under the assumption of coneconvexity, and by using a nonlinear scalarization function and Ioffe subdifferentiability he derived optimality conditions for weakly efficient solutions, Henig solutions, super efficient solutions, and globally efficient solutions to nonconvex vector equilibrium problems. Long et al. [5] obtained optimality conditions for Henig efficient solutions to vector equilibrium problems with functional constrains under the assumption of near conesubconvexlikeness. Luu et al. [7, 8] established sufficient and necessary conditions for efficient solutions to vector equilibrium problems with equality and inequality constraints and obtained the Fritz John and Karush–Kuhn–Tucker necessary optimality conditions for locally efficient solutions to vector equilibrium problems with constraints and sufficient conditions under assumptions of appropriate convexities.
It is well known that models describe only simplified versions of real problems and numerical algorithms generate only approximate solutions. Hence it is interesting and meaningful to have a theoretical analysis of the notion of an approximate solution. For example, Loridan [9, 10] introduced the concept of ϵsolutions in general vector optimization problems.
As far as we know, there are few papers dealing with approximate weakly efficient solutions to the setvalued vector equilibrium problems. Li et al. [11] introduced a new kind of approximate solution set of a vector approximate equilibrium problem; it is uncertain if ϵ tends to zero, whether or not the approximate solution set equals to the original solution set? It is a natural question how to define approximate weakly efficient solutions to the setvalued vector equilibrium problems and under what condition the set of approximate weakly efficient solutions equals to the set of weakly efficient solutions? This has great theoretical significance and applicable value in the research of optimality conditions for approximate weakly efficient solutions to the setvalued vector equilibrium problems.
On the other hand, convexity plays an important role in the study of vector equilibrium problems. In 2001, Yang et al. [12] introduced a new convexity, named near conesubconvexlikeness, and proved that it is a generalization of coneconvexness and conesubconvexlikeness. In 2005, Sach (see [13]) introduced another new convexity called icconeconvexness, Xu et al. [14] proved that near conesubconvexlikeness is also a generalization of icconeconvexness. Up to now, near conesubconvexlikeness is considered to be the most generalized convexity.
Motivated by works in [3, 12, 15], in this paper, we introduce a new kind of approximate weakly efficient solutions to the setvalued vector equilibrium problems and reveal the relationship between weakly efficient solutions and approximate weakly efficient solutions. We establish KuhnTucker type and Lagrangetype optimality conditions for setvalued vector equilibrium problems under the assumption of the near conesubconvexlikeness.
The organization of the paper is as follows. Some preliminary facts are given in Sect. 2 for our later use. Section 3 is devoted to the relationship between weakly efficient solutions and approximate weakly efficient solutions. In Sect. 4, we establish Kuhn–Tuckertype sufficient and necessary optimality conditions for approximate weakly efficient solutions to the setvalued vector equilibrium problems. In Sect. 5, we establish Lagrangetype sufficient and necessary optimality conditions for approximate weakly efficient solutions to the setvalued vector equilibrium problems. At the end of the paper, we draw some conclusions.
2 Preliminaries
Let X be a real topological vector space, and let Y and Z be real locally convex Hausdorff topological vector spaces with topological dual spaces \(Y^{*}\) and \(Z^{*}\), respectively. Let \(C\subset Y\) and \(D\subset Z\) be pointed closed convex cones with \(\operatorname{int}C\neq \emptyset \) and \(\operatorname{int}D\neq \emptyset \). The dual cones \(C^{*}\) of C and \(D^{*}\) of D are defined as \(C^{*}=\{\phi \in Y^{*}:\phi (c)\geq 0, \forall c\in C\}\) and \(D^{*}=\{\psi \in Z^{*}:\psi (d)\geq 0,\forall d\in D\}\), respectively. Let \(X_{0}\) be a nonempty convex subset in X, and let \(G:X_{0}\rightarrow 2^{Z}\) and \(\Phi:X_{0}\times X_{0}\rightarrow 2^{Y}\) be mappings.
We denote by \(L(Z,Y)\) the set of all continuous linear operators from Z to Y. A subset \(L^{+}(Z,Y)\) of \(L(Z,Y)\) is defined as \(L^{+}(Z,Y)=\{T\in L(Z,Y):T(D)\subset C\}\).
We denote the feasible set by
Consider the setvalued vector equilibrium problem with constraints (for short, ΦSVEPC): find \(x\in A\) such that
where \(P\cup \{0\}\) is a convex cone in Y.
Definition 2.1
A vector \(\bar{x}\in A\) satisfying
is called a weakly efficient solution to the ΦSVEPC. The set of all weakly efficient solutions to the ΦSVEPC is denoted by \(X_{W\min }(\Phi,A)\).
Let \(F:X_{0}\rightarrow 2^{Y}\) be a setvalued map. We consider the following setvalued optimization problem:
We assume that the feasible set \(A\subset X_{0}\) of (SOP) is nonempty.
Definition 2.2
A feasible solution x̄ of (SOP) is said to be a weakly efficient solution of (SOP) if there exists \(\bar{y}\in F(\bar{x})\) such that \((F(A)\bar{y})\cap (\operatorname{int}C)=\emptyset\). In this case, \((\bar{x},\bar{y})\) is said to be a weakly efficient pair to (SOP).
Definition 2.3
Let \(\epsilon \in C\). A feasible solution x̄ of (SOP) is said to be an ϵweakly efficient solution of (SOP) if there exists \(\bar{y}\in F(\bar{x})\) such that \((F(A)\bar{y}+\epsilon)\cap ( \operatorname{int}C)=\emptyset\). In this case, \((\bar{x},\bar{y})\) is said to be an ϵweakly efficient pair to (SOP).
Let \(\bar{T}\in L^{+}(Z,Y)\). Consider the following unconstrained setvalued optimization problem induced by (SOP):
where \(L(x,\bar{T})=F(x)+\bar{T}(G(x))\), \((x,\bar{T})\in X_{0} \times L^{+}(Z,Y)\).
Definition 2.4
A vector \(\bar{x}\in X_{0}\) is said to be a weakly efficient solution of \((\mathrm{USOP})_{\bar{T}}\) if there exists \(\bar{y} \in F(\bar{x})\) such that \((L(X_{0},\bar{T})\bar{y})\cap ( \operatorname{int}C)=\emptyset\), where \(L(X_{0},\bar{T})=\bigcup_{x\in X_{0}}L(x,\bar{T})\). In this case, \((\bar{x},\bar{y})\) is said to be a weakly efficient pair to \((\mathrm{USOP})_{\bar{T}}\).
Definition 2.5
Let \(\epsilon \in C\). A vector \(\bar{x}\in X_{0}\) is said to be an ϵweakly efficient solution of \((\mathrm{USOP})_{\bar{T}}\) if \(\exists \bar{y}\in F(\bar{x})\) such that \((L(X_{0},\bar{T})\bar{y}+ \epsilon)\cap (\operatorname{int}C)=\emptyset\), where \(L(X_{0},\bar{T})= \bigcup_{x\in X_{0}}L(x,\bar{T})\). In this case, \((\bar{x}, \bar{y})\) is said to be an ϵweakly efficient pair to \((\mathrm{USOP})_{\bar{T}}\).
Several definitions of generalized convexities have been introduced in the literature.
Definition 2.6
The map \(F:X_{0}\rightarrow 2^{Y}\) is said to be Cconvex on \(X_{0}\) if, for all \(x_{1},x_{2}\in X_{0}\) and \(\lambda \in [0,1]\), we have
Definition 2.7
([16])
The map \(F:X_{0}\rightarrow 2^{Y}\) is said to be Csubconvexlike on \(X_{0}\) iff there exists \(\theta \in \operatorname{int}C\) such that, for all \(x_{1},x_{2}\in X_{0}\), \(\lambda \in [0,1]\), \(y_{i} \in F(x_{i})\), \(i=1,2\), and \(\alpha >0\), there exists \(x_{3}\in X_{0}\) such that
Definition 2.8
([17])
The map \(F:X_{0}\rightarrow 2^{Y}\) is said to be generalized Csubconvexlike on \(X_{0}\) iff there exists \(\theta \in \operatorname{int}C\) such that, for all \(x_{1},x_{2}\in X_{0}\), \(\lambda \in [0,1]\), and \(\alpha >0\), there exist \(x_{3}\in X_{0}\) and \(\rho >0\) such that
Definition 2.9
([12])
The map \(F:X_{0}\rightarrow 2^{Y}\) is called nearly Csubconvexlike on \(X_{0}\) iff clcone\((F(X_{0})+C)\) is convex.
If \(\emptyset \neq S_{1}\subset Y\), \(\emptyset \neq S_{2}\subset Y\), \(\bar{y}\in Y\), and \(\psi \in Y^{*}\), then
and
3 Approximate weakly efficient solutions
Firstly, we introduce approximate weakly efficient solutions to the setvalued vector equilibrium problems with constraints.
Definition 3.1
Let \(\epsilon \in C\). A vector \(\bar{x}\in A\) satisfying
is called an ϵweakly efficient solution to the ΦSVEPC. The set of all ϵweakly efficient solutions to the ΦSVEPC is denoted by \(\epsilon \text{}X_{W\min }(\Phi,A)\).
Let \(\Upsilon:X_{0}\times X_{0}\rightarrow 2^{Y}\) be a mapping. Consider the following unconstrained setvalued vector equilibrium problem (for short, ϒUSVEP): find \(x\in X_{0}\) such that
where \(P\cup \{0\}\) is a convex cone in Y.
Definition 3.2
Let \(\epsilon \in C\). A vector \(\bar{x}\in X_{0}\) satisfying
is called an ϵweakly efficient solution to the ϒUSVEP. The set of all ϵweakly efficient solutions to the ϒUSVEP is denoted by \(\epsilon \text{}X_{W\min }( \Upsilon,X_{0})\).
Proposition 3.1
For any \(\epsilon \in C\), we have
Proof
If \(x\notin \epsilon \text{}X_{W\min }(\Phi,A)\), then there exists \(\bar{y}\in A\) such that
Thus there exists \(\bar{z}\in \Phi (x,\bar{y})\) such that
Since C is a convex cone, from \(\epsilon \in C\) and (3.1) we have
Hence
and thus \(x\notin X_{W\min }(\Phi,A)\). Then we obtain
□
Next, we show that in the proposition the relationship may be strict when \(\epsilon \in C\setminus {\{0\}}\).
Example 3.1
Let \(X=R^{1}\), \(A=[0,2]\), \(Y=R^{2}\), \(C=R_{+}^{2}\), and \(\epsilon =(x_{0},y_{0})\in C\setminus {\{0\}}\). Let \(\Phi:A\times A \longrightarrow 2^{Y}\) be defined by \(\Phi (x,y)=\{(p,q):q\geq p^{2}x \}\cap ([y,y]\times [0,+\infty))\), \(\forall x,y\in A\). It is obvious that \(X_{W\min }(\Phi,A)=\{0\}\); however, \(\epsilon \text{}X_{W \min }(\Phi,A)=[0,\delta ]\), where \(\delta =\min \{\max \{x_{0}^{2},y _{0}\},2\}\).
Proposition 3.2
For any \(\epsilon_{1},\epsilon_{2}\in C\), if \(\epsilon_{2} \epsilon_{1}\in C\), then
Proof
If \(x\notin \epsilon_{2}\text{}X_{W\min }(\Phi,A)\), then there exists \(y_{1}\in A\) such that
Thus there exists \(z_{1}\in \Phi (x,y_{1})\) such that
Since C is a convex cone, from \(\epsilon_{2}\epsilon_{1}\in C\) and (3.2) we have that
Hence
and thus \(x\notin \epsilon_{1}\text{}X_{W\min }(\Phi,A)\), so we obtain
□
In what follows, we discuss the relationship between the approximate weakly efficient solutions and weakly efficient solutions to the setvalued vector equilibrium problems with constraints.
Proposition 3.3
We have:
Proof
Firstly, we prove that
From Proposition 3.1 we can see that, for any \(\epsilon \in C \setminus {\{0\}}\), we have
Hence
Next, we prove that
Suppose \(\bar{x}\notin X_{W\min }(\Phi,A)\). Then there exists \(y_{0}\in A\) such that
and hence we can find \(z_{0}\in \Phi (\bar{x},y_{0})\) such that
Then there exists a neighborhood \(U_{0}\) of 0 in Y such that
Choosing \(\epsilon_{0}\in (C\cap U_{0})\setminus {\{0\}}\), we have
Since \(z_{0}\in \Phi (\bar{x},y_{0})\), we have
Hence \(\bar{x}\notin \epsilon_{0}\text{}X_{W\min }(\Phi,A)\), and therefore
Thus
From this we obtain
□
4 Kuhn–Tuckertype optimality conditions
In this section, under the assumption of near Csubconvexlikeness, we establish Kuhn–Tuckertype sufficient and necessary optimality conditions for approximate weakly efficient solutions to the setvalued vector equilibrium problems, which generalize the relevant results given by Gong [2] and Yang [17].
Definition 4.1
Let \(\bar{x}\in X_{0}\), and let \(\varphi:X_{0}\rightarrow 2^{Y\times Z}\) be an ordered pair mapping defined as \(\varphi (x)=(\Phi (\bar{x},x)+ \epsilon,G(x))\), \(\forall x\in X_{0}\).
By definition, φ is nearly \(C\times D\)subconvexlike on \(X_{0}\) if and only if \(\operatorname{cl}(\operatorname{cone}(\varphi (X_{0})+C\times D))\) is convex, where \(\varphi (X_{0})=\bigcup_{x\in X_{0}}\varphi (x)= \bigcup_{x\in X_{0}}(\Phi (\bar{x},x)+\epsilon,G(x))\).
Lemma 4.1
([15])
If \(y^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\), \(c_{0}\in \operatorname{int}C\), then \(y^{*}(c_{0})>0\).
Theorem 4.1
Suppose that φ is nearly \(C\times D\)subconvexlike on \(X_{0}\) and that there exists \(x_{0}\in X_{0}\) such that \(G(x_{0}) \cap (\operatorname{int}D)\neq \emptyset \). If x̄ is an ϵweakly efficient solution to the ΦSVEPC, then there exist \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and \(k^{*}\in D^{*}\) such that
Proof
Since x̄ is an ϵweakly efficient solution to the ΦSVEPC, we have
Next, we prove that
Suppose to the contrary that there exist \(\hat{\lambda }\geq 0\) and \(\hat{x}\in X_{0}\) such that
Thus,
and
From \(0\notin \operatorname{int}D\) we have \(\hat{\lambda }>0\). Since D is a convex cone, combining with (4.4), we have
It is clear that
Thus \(\hat{x}\in A\). Since \(\operatorname{int}C\cup \{0\}\) is a cone, by (4.3) we get
Since C is a convex cone, we have
which contradicts (4.1), and thus we obtain (4.2).
Since \(\operatorname{int}C\) and \(\operatorname{int}D\) are open sets, combining with (4.2), we have
Since φ is nearly \(C\times D\)subconvexlike on \(X_{0}\), by Definition 4.1 we can see that \(\operatorname{cl}(\operatorname{cone}(\varphi (X_{0})+C \times D))\) is convex. By the separation theorem for convex sets, there exists \((s^{*},k^{*})\in Y^{*}\times Z^{*}\setminus \{(0_{Y^{*}},0_{Z ^{*}})\}\) such that
Since \(\operatorname{cl}(\operatorname{cone}(\varphi (X_{0})+C\times D))\) is a cone, and there is a lower bound of \((s^{*},k^{*})\) on \(\operatorname{cl}(\operatorname{cone}(\varphi (X_{0})+C\times D))\), we have
Hence
Since \((0_{Y},0_{Z})\in C\times D\), we have
Thus
It is clear that
From (4.6) we get
Next, we prove that
Suppose to the contrary that there exists \(c_{0}\in C\) such that
When \(\delta_{1}\) is large enough, there exist \(x_{1}\in X_{0},y_{1} \in \Phi (\bar{x},x_{1}),z_{1}\in G(x_{1}),\delta_{2}^{\prime }\geq 0\), and \(d_{1}\in D\) such that
which contradicts (4.8). Hence we obtain
Similarly, we get
Thus
Then we need to prove that
Suppose to the contrary that
Since \((s^{*},k^{*})\neq (0_{Y^{*}},0_{Z^{*}})\), we have
From \(k^{*}\in D^{*}\setminus \{0_{Z^{*}}\}\), \(s^{*}=0_{Y^{*}}\), and (4.7) we can see that
On the other hand, there exists \(x_{0}\in X_{0}\) such that \(G(x_{0}) \cap (\operatorname{int}D)\neq \emptyset \), and thus there exists \(p\in G(x_{0})\cap (\operatorname{int}D)\), so that by Lemma 4.1 we obtain
which contradicts (4.9). Hence
The proof is complete. □
Corollary 4.1
Suppose that \(\bar{x}\in A\), \(0\in \Phi (\bar{x},\bar{x})\), φ is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D) \neq \emptyset \). If x̄ is a weakly efficient solution to the ΦSVEPC, then there exist \(s^{*}\in C^{*}\setminus \{0_{Y^{*}} \}\) and \(k^{*}\in D^{*}\) such that \(\min k^{*}(G(\bar{x}))=0\) and
Proof
In the proof of Theorem 4.1, letting \(\epsilon =0\), we see that
From \(\bar{x}\in A\) we have
Thus there exists \(q\in G(\bar{x})\) such that \(q\in D\), and since \(k^{*}\in D^{*}\), we have
Letting \(x=\bar{x}\) in (4.11), by \(0\in \Phi (\bar{x},\bar{x})\) we have
From \(q\in G(\bar{x})\) we have \(k^{*}(q)\geq 0\). Combining with (4.12), we obtain
Thus
Combining with (4.13), we get
From \(0\in \Phi (\bar{x},\bar{x})\) and (4.14) it follows that
Combining with (4.11), we obtain (4.10). The proof is complete. □
Theorem 4.2
Assume that

(i)
\(\bar{x}\in A\) and

(ii)
there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that
$$ s^{*}(y)+s^{*}(\epsilon)+k^{*}(z)\geq 0, \quad \forall x\in X_{0},y\in \Phi (\bar{x},x),z\in G(x). $$
Then x̄ is an ϵweakly efficient solution to the ΦSVEPC.
Proof
Suppose to the contrary that x̄ is not an ϵweakly efficient solution to the ΦSVEPC. Then we can find \(\hat{x} \in A\) such that
Thus there exists \(\hat{y}\in \Phi (\bar{x},\hat{x})\) such that
By \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and Lemma 4.1 we have
Choosing \(\hat{z}\in G(\hat{x})\cap (D)\), since \(k^{*}\in D^{*}\), we have \(k^{*}(\hat{z})\leq 0\). Combining with (4.15), we obtain
which contradicts (ii), and hence x̄ is an ϵweakly efficient solution to the ΦSVEPC. □
Corollary 4.2
Assume that

(i)
\(\bar{x}\in A\) and

(ii)
there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that
$$ s^{*}(y)+k^{*}(z)\geq 0, \quad \forall x\in X_{0},y\in \Phi (\bar{x},x),z \in G(x). $$
Then x̄ is a weakly efficient solution to the ΦSVEPC.
Proof
Letting \(\epsilon =0\) in Theorem 4.2, we get the conclusion. □
Corollary 4.3
Suppose that \(\bar{x}\in A\), \(0\in \Phi (\bar{x},\bar{x})\), φ is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D) \neq \emptyset \). Then x̄ is a weakly efficient solution to the ΦSVEPC if and only if there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that \(\min k^{*}(G(\bar{x}))=0\) and
Proof
This follows directly from Corollaries 4.1 and 4.2. □
Remark 4.1
Corollary 4.3 extends Theorem 3.1 of Gong [2] in the following aspects:

(i)
The vectorvalued function is extended to a setvalued function;

(ii)
The coneconvexity of φ is extended to near conesubconvexlikeness.
Corollary 4.4
Suppose that \(\bar{x}\in X_{0}\), \(\bar{y}\in F(\bar{x})\), \((F\bar{y},G)\) is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D) \neq \emptyset \). If \((\bar{x},\bar{y})\) is a weakly efficient pair to the (SOP), then there exist \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and \(k^{*}\in D^{*}\) such that \(\min k^{*}(G(\bar{x}))=0\) and
Proof
Letting \(\Phi (y,x)=F(x)\bar{y}\), it is clear that \(\Phi (y,x)\) depends only upon the second variable. Since \(\bar{y}\in F(\bar{x})\), we have \(0\in F(\bar{x})\bar{y}\), and hence \(0\in \Phi (\bar{x},\bar{x})\) and
Since \((\bar{x},\bar{y})\) is a weakly efficient pair to the (SOP), we can see that x̄ is a weakly efficient solution to the ΦSVEPC. By Corollary 4.1 we get the conclusion. □
Remark 4.2
From Remarks 3.1 and 3.3 in [18] we can see that if \((F\bar{y},G)\) is generalized \(C\times D\)subconvexlike on \(X_{0}\), then \((F\bar{y},G)\) is nearly \(C\times D\)subconvexlike on \(X_{0}\). Thus, Corollary 4.4 generalizes Theorem 4.2 in [17].
From Theorems 4.1 and 4.2 we obtain the following result.
Corollary 4.5
Suppose that \(\bar{x}\in A\), φ is nearly \(C\times D\)subconvexlike on \(X_{0}\) and that there exists \(x_{0} \in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D)\neq \emptyset \). Then x̄ is an ϵweakly efficient solution to the ΦSVEPC if and only if there exist \(s^{*}\in C^{*}\setminus \{0_{Y ^{*}}\}\) and \(k^{*}\in D^{*}\) such that
5 Lagrangetype optimality conditions
In this section, we present Lagrangetype sufficient and necessary optimality conditions for approximate weakly efficient solutions to the setvalued vector equilibrium problems, which generalize the relevant results given by Rong [15].
Theorem 5.1
Suppose that \(\bar{x}\in A\), \(0\in \Phi (\bar{x},\bar{x})\), φ is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D) \neq \emptyset \). If x̄ is an ϵweakly efficient solution to the ΦSVEPC, then there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \(\bar{T}(G(\bar{x})\cap (D))\subset ((\operatorname{int}C\cap \{0\})\setminus (\epsilon +\operatorname{int}C))\), and x̄ is an ϵweakly efficient solution to the ΨUSVEP, where \(\Psi:X_{0}\times X_{0}\rightarrow 2^{Y}\) is defined by
Proof
From the proof of Theorem 4.1 we see that there exist \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and \(k^{*}\in D^{*}\) satisfying (4.7).
Since \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\), we can find \(c_{0} \in \operatorname{int}C\) such that \(s^{*}(c_{0})=1\). Define the operator \(\bar{T}:Z\rightarrow Y\) by
Thus \(\bar{T}(D)=k^{*}(D)c_{0}\subset C\). It is evident that
Letting \(x=\bar{x}\) in (4.7), since \(0\in \Phi (\bar{x},\bar{x})\), we have
Noticing that \(z\in D\), we obtain
Thus
Next, we prove
Suppose to the contrary that there exists \(\tilde{z}\in G(\bar{x}) \cap (D)\) such that
Thus \(\bar{T}(\tilde{z})\epsilon \in \operatorname{int}C\). By the definition of T̄ we have
Combining with (5.1), we have
On the other hand, from \(s^{*}\in C^{*}\setminus \{0_{Y^{*}}\}\) and Lemma 4.1 it follows that
which, together with (5.5), gives
which contradicts (5.4). Thus we obtain (5.3).
The combination of (5.2) and (5.3) leads to
Finally, we prove that x̄ is an ϵweakly efficient solution to the ΨUSVEP.
In fact, by the definition of T̄ and (4.7) we obtain
Since \(s^{*}(\operatorname{int}C)<0\), we have
Consequently,
It is evident that x̄ is an ϵweakly efficient solution to the ΨUSVEP. □
Theorem 5.2
Assume that

(i)
\(\bar{x}\in A\) and

(ii)
there exists \(\bar{T}\in L^{+}(Z,Y)\) such that x̄ is an ϵweakly efficient solution to the ΨUSVEP, where \(\Psi:X_{0}\times X_{0}\rightarrow 2^{Y}\) is defined by
$$ \Psi (y,x)=\Phi (y,x)+\bar{T}\bigl(G(x)\bigr). $$
Then x̄ is an ϵweakly efficient solution to the ΦSVEPC.
Proof
Since x̄ is an ϵweakly efficient solution to the ΨUSVEP, we have
Since C is a convex cone, we obtain
On the other hand, for any \(x\in A\), we have \(G(x)\cap (D)\neq \emptyset \). Thus there exists \(z_{x}\in G(x)\) such that \(z_{x}\in D\). It follows from \(\bar{T}\in L^{+}(Z,Y)\) that \(\bar{T}(z_{x})\in C\), thus \(C\bar{T}(z_{x})\subset C+C\subset C\), and hence \(C\subset \bar{T}(z_{x})+C\); since \(z_{x}\in G(x)\), it is evident that \(C\subset \bar{T}(G(x))+C\).
Thus
It follows from (5.6) that
Since \(0\in C\), it is evident that
Hence x̄ is an ϵweakly efficient solution to the ΦSVEPC. □
From Theorems 5.1 and 5.2 we obtain the following result.
Corollary 5.1
Suppose that \(\bar{x}\in A\), \(0\in \Phi (\bar{x},\bar{x})\), φ is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D) \neq \emptyset \). Then x̄ is an ϵweakly efficient solution to the ΦSVEPC if and only if there exists \(\bar{T} \in L^{+}(Z,Y)\) such that x̄ is an ϵweakly efficient solution to the ΨUSVEP, where \(\Psi:X_{0}\times X_{0}\rightarrow 2^{Y}\) is defined by
Corollary 5.2
Suppose that \((F\bar{y},G)\) is nearly \(C\times D\)subconvexlike on \(X_{0}\) and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap ( \operatorname{int}D)\neq \emptyset \). If \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the (SOP), then there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \(\bar{T}(G(\bar{x})\cap (D)) \subset ((\operatorname{int}C\cap \{0\})\setminus (\epsilon +\operatorname{int}C))\), and \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the \((\mathrm{USOP})_{\bar{T}}\).
Proof
Letting \(\Phi (y,x)=F(x)\bar{y}\), since \(\bar{y}\in F(\bar{x})\), it is evident that \(0\in \Phi (\bar{x},\bar{x})\) and \((F(x)\bar{y},G(x))=( \Phi (\bar{x},x),G(x))\).
Since \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the (SOP), we see that x̄ is an ϵweakly efficient solution to the ΦSVEPC.
Thus by Theorem 5.1 there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \(\bar{T}(G(\bar{x})\cap (D))\subset ((\operatorname{int}C\cap \{0\}) \setminus (\epsilon +\operatorname{int}C))\) and x̄ is an ϵweakly efficient solution to the ΨUSVEP, that is,
Consequently,
which is equivalent to
Thus
Hence, by Definition 2.5, \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the \((\mathrm{USOP})_{\bar{T}}\). □
Remark 5.1
If \((F,G)\) is \(C\times D\)subconvexlike, then \((F\bar{y},G)\) is nearly \(C\times D\)subconvexlike. Thus Corollary 5.2 generalizes Theorem 3.1 in [15].
Corollary 5.3
Assume that

(i)
\(\bar{x}\in A, \bar{y}\in F(\bar{x})\), and

(ii)
there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the \((\mathrm{USOP}) _{\bar{T}}\).
Then \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the (SOP).
Proof
Letting \(\Phi (y,x)=F(x)\bar{y}\), since \(\bar{y}\in F(\bar{x})\), it is evident that \(0\in \Phi (\bar{x},\bar{x})\) and \((F(x)\bar{y},G(x))=( \Phi (\bar{x},x),G(x))\).
Since \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the \((\mathrm{USOP})_{\bar{T}}\), we see that x̄ is an ϵweakly efficient solution to the ΨUSVEP. Combining this with Theorem 5.2, we conclude that x̄ is an ϵweakly efficient solution to the ΦSVEPC; it is clear that \((\bar{x}, \bar{y})\) is an ϵweakly efficient pair to the (SOP). □
Remark 5.2
Comparing with Theorem 3.2 in [15], this corollary is not required for the convexity of \((F, G)\).
Corollary 5.4
Suppose that \(\bar{x}\in A\), \(\bar{y}\in F(\bar{x})\), \((F\bar{y},G)\) is nearly \(C\times D\)subconvexlike on \(X_{0}\), and there exists \(x_{0}\in X_{0}\) such that \(G(x_{0})\cap (\operatorname{int}D)\neq \emptyset \). Then \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the (SOP) if and only if there exists \(\bar{T}\in L^{+}(Z,Y)\) such that \((\bar{x},\bar{y})\) is an ϵweakly efficient pair to the \((\mathrm{USOP}) _{\bar{T}}\).
Proof
6 Conclusions
In this paper, we discuss some relationships between approximate weakly efficient solutions and weakly efficient solutions of setvalued vector equilibrium problems. We conclude that \(\bigcap_{\epsilon \in C\setminus {\{0\}}}\epsilon \text{}X_{W \min }(\Phi,A)=X_{W\min }(\Phi,A)\), and hence it is really “approximate”. The optimality conditions for setvalued vector equilibrium problems are established, and the results we obtained generalize those of Gong[2], Yang[17], and Rong[15]. As an extensive mathematical model, further research on approximate weakly efficient solutions of setvalued vector equilibrium problems seems to be of interest and value.
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Authors’ information
Yihong Xu (1969), Professor, Doctor, the major field of interest is in the area of setvalued optimization. Jian Chen, email: 1169898604@qq.com. Ke Zhang, email: 2768482283@qq.com.
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This research was supported by the National Natural Science Foundation of China Grant (11461044), the Natural Science Foundation of Jiangxi Province (20151BAB 201027), and the Educational Commission of Jiangxi Province (GJJ12010).
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Chen, J., Xu, Y. & Zhang, K. Approximate weakly efficient solutions of setvalued vector equilibrium problems. J Inequal Appl 2018, 181 (2018). https://doi.org/10.1186/s1366001817730
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DOI: https://doi.org/10.1186/s1366001817730