Approximate weakly efficient solutions of set valued vector equilibrium problems

R E S E A R C H

Open Access

Approximate weakly efficient solutions of

set-valued vector equilibrium problems

Jian Chen

_{, Yihong Xu}

_{and Ke Zhang}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Nanchang University, Nanchang, China

Abstract

In this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued 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 cone-subconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued vector equilibrium problems, respectively.

MSC: 90C33; 90C46; 90C59

Keywords: Set-valued vector equilibrium problem; Approximate weakly efficient solution; Near cone-subconvexlikeness; Optimality condition

1 Introduction

Vector optimization problems, vector variational inequality problems, vector complemen-tarity problems, and vector saddle point problems are particular cases of vector equilib-rium problems. As an extensive mathematical model, the vector equilibequilib-rium 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 cone-convexity, and by using a nonlinear scalarization function and Ioffe subdifferentiability he derived optimality conditions for weakly efficient solu-tions, Henig solusolu-tions, super efficient solusolu-tions, and globally efficient solutions to non-convex 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 cone-subconvexlikeness. 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 neces-sary 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 nu-merical algorithms generate only approximate solutions. Hence it is interesting and mean-ingful to have a theoretical analysis of the notion of an approximate solution. For example, Loridan [9,10] introduced the concept of-solutions in general vector optimization prob-lems.

As far as we know, there are few papers dealing with approximate weakly efficient solu-tions to the set-valued vector equilibrium problems. Li et al. [11] introduced a new kind of approximate solution set of a vector approximate equilibrium problem; it is uncertain iftends to zero, whether or not the approximate solution set equals to the original so-lution set? It is a natural question how to define approximate weakly efficient soso-lutions to the set-valued vector equilibrium problems and under what condition the set of ap-proximate weakly efficient solutions equals to the set of weakly efficient solutions? This has great theoretical significance and applicable value in the research of optimality condi-tions for approximate weakly efficient solucondi-tions to the set-valued vector equilibrium prob-lems.

On the other hand, convexity plays an important role in the study of vector equilib-rium problems. In 2001, Yang et al. [12] introduced a new convexity, named near subconvexlikeness, and proved that it is a generalization of convexness and cone-subconvexlikeness. In 2005, Sach (see [13]) introduced another new convexity called ic-cone-convexness, Xu et al. [14] proved that near cone-subconvexlikeness is also a gener-alization of ic-coneconvexness. Up to now, near cone-subconvexlikeness 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 set-valued vector equilibrium problems and reveal the re-lationship between weakly efficient solutions and approximate weakly efficient solutions. We establish Kuhn-Tucker type and Lagrange-type optimality conditions for set-valued vector equilibrium problems under the assumption of the near cone-subconvexlikeness.

The organization of the paper is as follows. Some preliminary facts are given in Sect.2 for our later use. Section3is devoted to the relationship between weakly efficient solutions and approximate weakly efficient solutions. In Sect.4, we establish Kuhn–Tucker-type suf-ficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. In Sect.5, we establish Lagrange-type suf-ficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems. At the end of the paper, we draw some con-clusions.

2 Preliminaries

LetXbe a real topological vector space, and letYandZbe real locally convex Hausdorff topological vector spaces with topological dual spacesY∗andZ∗, respectively. LetC⊂Y andD⊂Zbe pointed closed convex cones withintC=∅andintD=∅. The dual conesC∗ ofCandD∗ofDare defined asC∗={φ∈Y∗:φ(c)≥0,∀c∈C}andD∗={ψ∈Z∗:ψ(d)≥ 0,∀d∈D}, respectively. LetX0be a nonempty convex subset inX, and letG:X0→2Zand

:X0×X0→2Y be mappings.

We denote byL(Z,Y) the set of all continuous linear operators fromZ toY. A subset L+_{(Z,Y) ofL(Z,Y) is defined asL}+_{(Z,Y) ={T∈L(Z,Y) :T(D)⊂C}.}

We denote the feasible set by

A=x∈X0:G(x)∩(–D)=∅

Consider the set-valued vector equilibrium problem with constraints (for short, -SVEPC): findx∈Asuch that

(x,y)∩(–P) =∅, ∀y∈A,

whereP∪ {0}is a convex cone inY.

Definition 2.1 A vectorx¯∈Asatisfying

(x,¯ y)∩(–intC) =∅, ∀y∈A,

is called a weakly efficient solution to the-SVEPC. The set of all weakly efficient solutions to the-SVEPC is denoted byXWmin(,A).

LetF:X0→2Ybe a set-valued map. We consider the following set-valued optimization

problem:

(SOP) minF(x),

s.t.x∈A=x∈X0:G(x)∩(–D)=∅

.

We assume that the feasible setA⊂X0of (SOP) is nonempty.

Definition 2.2 A feasible solutionx¯of (SOP) is said to be a weakly efficient solution of (SOP) if there existsy¯∈F(x) such that (F(A) –¯ y)¯ ∩(–intC) =∅. In this case, (x,¯ y) is said¯ to be a weakly efficient pair to (SOP).

Definition 2.3 Let∈C. A feasible solutionx¯of (SOP) is said to be an-weakly efficient solution of (SOP) if there existsy¯∈F(x) such that (F(A) –¯ y¯+)∩(–intC) =∅. In this case, (x,¯ ¯y) is said to be an-weakly efficient pair to (SOP).

LetT¯ ∈L+_{(Z,Y). Consider the following unconstrained set-valued optimization}

prob-lem induced by (SOP):

(USOP)T¯ min

x∈X0 L(x,T),¯

whereL(x,T) =¯ F(x) +T(G(x)), (x,¯ T)¯ ∈X0×L+(Z,Y).

Definition 2.4 A vectorx¯∈X0is said to be a weakly efficient solution of (USOP)T¯ if there existsy¯∈F(x) such that (L(X¯ 0,T¯) –y)¯ ∩(–intC) =∅, whereL(X0,T¯) =x∈X0L(x,T¯). In this case, (x,¯ ¯y) is said to be a weakly efficient pair to (USOP)T¯.

Definition 2.5 Let ∈C. A vectorx¯ ∈X0 is said to be an-weakly efficient solution

of (USOP)T¯ if ∃¯y∈F(x) such that (L(X¯ 0,T) –¯ y¯+)∩(–intC) =∅, where L(X0,T¯) =

Definition 2.6 The mapF:X0→2Yis said to beC-convex onX0if, for allx1,x2∈X0and

λ∈[0, 1], we have

λF(x1) + (1 –λ)F(x2)⊂F

λx1+ (1 –λ)x2

+C.

Definition 2.7([16]) The mapF:X0→2Y is said to beC-subconvexlike onX0iff there

existsθ∈intCsuch that, for allx1,x2∈X0,λ∈[0, 1],yi∈F(xi),i= 1, 2, andα> 0, there existsx3∈X0such that

αθ+λy1+ (1 –λ)y2∈F(x3) +C.

Definition 2.8([17]) The mapF:X0→2Y is said to be generalizedC-subconvexlike on

X0iff there existsθ ∈intCsuch that, for allx1,x2∈X0,λ∈[0, 1], andα> 0, there exist

x3∈X0andρ> 0 such that

αθ+λF(x1) + (1 –λ)F(x2)⊂ρF(x3) +C.

Definition 2.9 ([12]) The mapF:X0→2Y is called nearly C-subconvexlike on X0 iff

clcone(F(X0) +C) is convex.

If∅ =S1⊂Y,∅ =S2⊂Y,y¯∈Y, andψ∈Y∗, then

ψ(S1)≥ψ(S2) stands forψ(s1)≥ψ(s2), ∀s1∈S1,s2∈S2,

and

ψ(S1)≥ψ(y) stands for¯ ψ(s1)≥ψ(y),¯ ∀s1∈S1.

3 Approximate weakly efficient solutions

Firstly, we introduce approximate weakly efficient solutions to the set-valued vector equi-librium problems with constraints.

Definition 3.1 Let∈C. A vectorx¯∈Asatisfying

(x,¯ y) +∩(–intC) =∅, ∀y∈A,

is called an-weakly efficient solution to the-SVEPC. The set of all-weakly efficient solutions to the-SVEPC is denoted by-XWmin(,A).

Letϒ:X0×X0→2Y be a mapping. Consider the following unconstrained set-valued

vector equilibrium problem (for short,ϒ-USVEP): findx∈X0such that

ϒ(x,y)∩(–P) =∅, ∀y∈X0,

Definition 3.2 Let∈C. A vectorx¯∈X0satisfying

ϒ(x,¯ y) +∩(–intC) =∅, ∀y∈X0,

is called an-weakly efficient solution to theϒ-USVEP. The set of all-weakly efficient solutions to theϒ-USVEP is denoted by-XWmin(ϒ,X0).

Proposition 3.1 For any∈C,we have XWmin(,A)⊂-XWmin(,A).

Proof Ifx∈/-XWmin(,A), then there existsy¯∈Asuch that

(x,y) +¯ ∩(–intC)=∅. Thus there existsz¯∈(x,y) such that¯

¯

z+∈–intC. (3.1)

SinceCis a convex cone, from∈Cand (3.1) we have

¯

z∈–intC–⊂–intC–C⊂–intC. Hence

(x,y)¯ ∩(–intC)=∅,

and thusx∈/XWmin(,A). Then we obtain

XWmin(,A)⊂-XWmin(,A).

Next, we show that in the proposition the relationship may be strict when∈C\ {0}. Example3.1 LetX=R1_{,A= [0, 2],Y=R}2_,C=R2

+, and= (x0,y0)∈C\ {0}. Let:A×

A−→2Y_{be defined by(x,y) ={(p,q) :q≥p}2_{–x}∩([–y,y]×[0, +∞)),∀x,y∈A. It is}

obvi-ous thatXWmin(,A) ={0}; however,-XWmin(,A) = [0,δ], whereδ=min{max{x20,y0}, 2}.

Proposition 3.2 For any1,2∈C,if2–1∈C,then

1-XWmin(,A)⊂2-XWmin(,A).

Proof Ifx∈/2-XWmin(,A), then there existsy1∈Asuch that

(x,y1) +2

∩(–intC)=∅.

Thus there existsz1∈(x,y1) such that

SinceCis a convex cone, from2–1∈Cand (3.2) we have that

z1+1=z1+2– (2–1)∈–intC–C= –intC.

Hence

(x,y1) +1

∩(–intC)=∅,

and thusx∈/1-XWmin(,A), so we obtain

1-XWmin(,A)⊂2-XWmin(,A).

In what follows, we discuss the relationship between the approximate weakly efficient solutions and weakly efficient solutions to the set-valued vector equilibrium problems with constraints.

Proposition 3.3 We have:

∈C\{0}

-XWmin(,A) =XWmin(,A).

Proof Firstly, we prove that

XWmin(,A)⊂

∈C\{0}

-XWmin(,A).

From Proposition3.1we can see that, for any∈C\ {0}, we have

XWmin(,A)⊂-XWmin(,A).

Hence

XWmin(,A)⊂

∈C\{0}

-XWmin(,A).

Next, we prove that

∈C\{0}

-XWmin(,A)⊂XWmin(,A).

Supposex¯∈/XWmin(,A). Then there existsy0∈Asuch that

(x,¯ y0)∩(–intC)=∅, (3.3)

and hence we can findz0∈(x,¯ y0) such that

Then there exists a neighborhoodU0of 0 inYsuch that

z0+U0⊂–intC.

Choosing0∈(C∩U0)\ {0}, we have

z0+0∈–intC.

Sincez0∈(x,¯ y0), we have

(x,¯ y0) +0

∩(–intC)=∅.

Hencex¯∈/0-XWmin(,A), and therefore

¯

x∈/

∈C\{0}

-XWmin(,A).

Thus

∈C\{0}

-XWmin(,A)⊂XWmin(,A).

From this we obtain

∈C\{0}

-XWmin(,A) =XWmin(,A).

4 Kuhn–Tucker-type optimality conditions

In this section, under the assumption of nearC-subconvexlikeness, we establish Kuhn– Tucker-type sufficient and necessary optimality conditions for approximate weakly effi-cient solutions to the set-valued vector equilibrium problems, which generalize the rele-vant results given by Gong [2] and Yang [17].

Definition 4.1 Letx¯∈X0, and letϕ:X0→2Y×Zbe an ordered pair mapping defined as

ϕ(x) = ((x,¯ x) +,G(x)),∀x∈X0.

By definition,ϕis nearlyC×D-subconvexlike onX0if and only ifcl(cone(ϕ(X0) +C×D))

is convex, whereϕ(X0) =

x∈X0ϕ(x) =

x∈X0((x,¯ x) +,G(x)).

Lemma 4.1([15]) If y∗∈C∗\ {0Y∗},c0∈intC,then y∗(c0) > 0.

Theorem 4.1 Suppose thatϕis nearly C×D-subconvexlike on X0 and that there exists

x0∈X0such that G(x0)∩(–intD)=∅.Ifx is an¯ -weakly efficient solution to the-SVEPC,

then there exist s∗∈C∗\ {0Y∗}and k∗∈D∗such that

Proof Sincex¯is an-weakly efficient solution to the-SVEPC, we have

(x,¯ x) +∩(–intC) =∅, ∀x∈A. (4.1)

Next, we prove that

coneϕ(X0) +C×D

∩(–intC)×(–intD)=∅. (4.2)

Suppose to the contrary that there existλˆ≥0 andxˆ∈X0such that

_ˆ

λ(x,¯ x) +ˆ C+,G(x) +ˆ D∩(–intC)×(–intD)=∅. Thus,

_ˆ

λ(x,¯ x) +ˆ C+∩(–intC)=∅ (4.3)

and

_ˆ

λG(x) +ˆ D∩(–intD)=∅. (4.4)

From 0 /∈–intDwe haveλˆ> 0. SinceDis a convex cone, combining with (4.4), we have

G(x)ˆ ∩(–intD)=∅. It is clear that

G(x)ˆ ∩(–D)=∅.

Thusxˆ∈A. SinceintC∪ {0}is a cone, by (4.3) we get

(x,¯ x) +ˆ C+∩(–intC)=∅.

SinceCis a convex cone, we have

(x,¯ x) +ˆ ∩(–intC)=∅,

which contradicts (4.1), and thus we obtain (4.2).

Since –intCand –intDare open sets, combining with (4.2), we have

clconeϕ(X0) +C×D

∩(–intC)×(–intD)=∅. (4.5)

Since ϕ is nearly C × D-subconvexlike on X0, by Definition 4.1 we can see that cl(cone(ϕ(X0) +C×D)) is convex. By the separation theorem for convex sets, there exists

(s∗,k∗)∈Y∗×Z∗\ {(0Y∗, 0Z∗)}such that

s∗,k∗clconeϕ(X0) +C×D

Since cl(cone(ϕ(X0) +C × D)) is a cone, and there is a lower bound of (s∗,k∗) on cl(cone(ϕ(X0) +C×D)), we have

s∗,k∗clconeϕ(X0) +C×D

≥0.

Hence

s∗,k∗ϕ(X0) +C×D

≥0. (4.6)

Since (0Y, 0Z)∈C×D, we have

s∗,k∗ϕ(X0)

≥0.

Thus

s∗(x,¯ x) ++k∗G(x)≥0, ∀x∈X0.

It is clear that

s∗(y) +s∗() +k∗(z)≥0, ∀x∈X0,y∈(x,¯ x),z∈G(x). (4.7)

From (4.6) we get

s∗(y++δ1c) +k∗(z+δ2d)

≥0, ∀x∈X0,y∈(x,¯ x),z∈G(x),δ1,δ2≥0,c∈C,d∈D. (4.8)

Next, we prove that

s∗(c)≥0, ∀c∈C.

Suppose to the contrary that there existsc0∈Csuch that

s∗(c0) < 0.

Whenδ1is large enough, there existx1∈X0,y1∈(x,¯ x1),z1∈G(x1),δ2≥0, andd1∈D

such that

δ1s∗(c0) < –s∗(y1+) –k∗

z1+δ2d1

,

which contradicts (4.8). Hence we obtain

s∗(c)≥0, ∀c∈C. Similarly, we get

Thus

s∗∈C∗, k∗∈D∗. Then we need to prove that

s∗= 0Y∗.

Suppose to the contrary that

s∗= 0Y∗.

Since (s∗,k∗)= (0Y∗, 0Z∗), we have k∗= 0Z∗.

Fromk∗∈D∗\ {0Z∗},s∗= 0Y∗, and (4.7) we can see that

k∗G(x)≥0, ∀x∈X0. (4.9)

On the other hand, there existsx0∈X0such thatG(x0)∩(–intD)=∅, and thus there exists

p∈G(x0)∩(–intD), so that by Lemma4.1we obtain

k∗(p) < 0,

which contradicts (4.9). Hence

s∗∈C∗\ {0Y∗}.

The proof is complete.

Corollary 4.1 Suppose thatx¯∈A, 0∈(x,¯ x),¯ ϕis nearly C×D-subconvexlike on X0,and

there exists x0∈X0such that G(x0)∩(–intD)=∅.Ifx is a weakly efficient solution to the¯

-SVEPC,then there exist s∗∈C∗\ {0Y∗}and k∗∈D∗such thatmink∗(G(x)) = 0¯ and

mins∗(y) +k∗(z) :x∈X0,y∈(x,¯ x),z∈G(x)

= 0. (4.10)

Proof In the proof of Theorem4.1, letting= 0, we see that

s∗(y) +k∗(z)≥0, ∀x∈X0,y∈(x,¯ x),z∈G(x). (4.11)

Fromx¯∈Awe have G(x)¯ ∩(–D)=∅.

Thus there existsq∈G(x) such that¯ q∈–D, and sincek∗∈D∗, we have

Lettingx=x¯in (4.11), by 0∈(x,¯ x) we have¯

k∗(z)≥0, ∀z∈G(x).¯ (4.13)

Fromq∈G(x) we have¯ k∗(q)≥0. Combining with (4.12), we obtain k∗(q) = 0.

Thus

0∈k∗G(x)¯ . (4.14)

Combining with (4.13), we get

mink∗G(x)¯ = 0.

From 0∈(x,¯ x) and (¯ 4.14) it follows that 0∈s∗(x,¯ x)¯ +k∗G(x)¯ .

Combining with (4.11), we obtain (4.10). The proof is complete.

Theorem 4.2 Assume that

(i) x¯∈Aand

(ii) there exists∗∈C∗\ {0Y∗}andk∗∈D∗such that

s∗(y) +s∗() +k∗(z)≥0, ∀x∈X0,y∈(x,¯ x),z∈G(x).

Thenx is an¯ -weakly efficient solution to the-SVEPC.

Proof Suppose to the contrary thatx¯is not an-weakly efficient solution to the-SVEPC. Then we can findxˆ∈Asuch that

(x,¯ x) +ˆ ∩(–intC)=∅. Thus there existsyˆ∈(x,¯ x) such thatˆ

ˆ

y+∈–intC.

Bys∗∈C∗\ {0Y∗}and Lemma4.1we have

s∗(y) +ˆ s∗() < 0. (4.15)

Choosingˆz∈G(x)ˆ ∩(–D), sincek∗∈D∗, we havek∗(z)ˆ ≤0. Combining with (4.15), we obtain

s∗(y) +ˆ s∗() +k∗(ˆz) < 0,

Corollary 4.2 Assume that

(i) x¯∈Aand

(ii) there exists∗∈C∗\ {0Y∗}andk∗∈D∗such that s∗(y) +k∗(z)≥0, ∀x∈X0,y∈(x,¯ x),z∈G(x).

Thenx is a weakly efficient solution to the¯ -SVEPC.

Proof Letting= 0 in Theorem4.2, we get the conclusion.

Corollary 4.3 Suppose thatx¯∈A, 0∈(x,¯ x),¯ ϕis nearly C×D-subconvexlike on X0,and

there exists x0∈X0such that G(x0)∩(–intD)=∅.Thenx is a weakly efficient solution to the¯

-SVEPC if and only if there exist s∗∈C∗\ {0Y∗}and k∗∈D∗such thatmink∗(G(x)) = 0¯ and

mins∗(y) +k∗(z) :x∈X0,y∈(x,¯ x),z∈G(x)

= 0.

Proof This follows directly from Corollaries4.1and4.2.

Remark4.1 Corollary4.3extends Theorem 3.1 of Gong [2] in the following aspects:

(i) The vector-valued function is extended to a set-valued function; (ii) The cone-convexity ofϕis extended to near cone-subconvexlikeness.

Corollary 4.4 Suppose thatx¯∈X0,y¯∈F(x), (F¯ –¯y,G)is nearly C×D-subconvexlike on

X0,and there exists x0∈X0such that G(x0)∩(–intD)=∅.If(x,¯ y)¯ is a weakly efficient pair

to the(SOP),then there exist s∗∈C∗\ {0Y∗}and k∗∈D∗such thatmink∗(G(x)) = 0¯ and s∗(y) =¯ mins∗F(x)+k∗G(x):x∈X0

.

Proof Letting(y,x) =F(x) –y, it is clear that¯ (y,x) depends only upon the second vari-able. Sincey¯∈F(x), we have 0¯ ∈F(x) –¯ y, and hence 0¯ ∈(x,¯ x) and¯

F(x) –y,¯ G(x)=(x,¯ x),G(x).

Since (x,¯ y) is a weakly efficient pair to the (SOP), we can see that¯ x¯ is a weakly efficient solution to the-SVEPC. By Corollary4.1we get the conclusion.

Remark4.2 From Remarks 3.1 and 3.3 in [18] we can see that if (F–y,¯ G) is generalized C×D-subconvexlike onX0, then (F–y,¯ G) is nearlyC×D-subconvexlike onX0. Thus,

Corollary4.4generalizes Theorem 4.2 in [17].

From Theorems4.1and4.2we obtain the following result.

Corollary 4.5 Suppose thatx¯∈A,ϕis nearly C×D-subconvexlike on X0and that there

exists x0∈X0such that G(x0)∩(–intD)=∅.Thenx is an¯ -weakly efficient solution to the

5 Lagrange-type optimality conditions

In this section, we present Lagrange-type sufficient and necessary optimality conditions for approximate weakly efficient solutions to the set-valued vector equilibrium problems, which generalize the relevant results given by Rong [15].

Theorem 5.1 Suppose thatx¯∈A, 0∈(x,¯ x),¯ ϕis nearly C×D-subconvexlike on X0,and

there exists x0∈X0such that G(x0)∩(–intD)=∅.Ifx is an¯ -weakly efficient solution to

the-SVEPC,then there existsT¯ ∈L+_{(Z,Y)such that–T(G(}¯ _{x)¯ ∩(–D))⊂((intC∩ {0})\}

(+intC)),andx is an¯ -weakly efficient solution to the-USVEP,where:X0×X0→2Y

is defined by

(y,x) =(y,x) +T¯G(x).

Proof From the proof of Theorem4.1we see that there exists∗∈C∗\ {0Y∗}andk∗∈D∗ satisfying (4.7).

Sinces∗∈C∗\ {0Y∗}, we can findc0∈intC such thats∗(c0) = 1. Define the operator

¯

T:Z→Y by

¯

T(z) =k∗(z)c0, ∀z∈Z.

ThusT¯(D) =k∗(D)c0⊂C. It is evident that

¯

T∈L+(Z,Y).

Lettingx=x¯in (4.7), since 0∈(x,¯ x), we have¯

s∗() +k∗(z)≥0, ∀z∈G(x)¯ ∩(–D). (5.1)

Noticing thatz∈–D, we obtain –T¯(z) = –k∗(z)c0∈intC∪ {0}.

Thus

–T¯G(x)¯ ∩(–D)⊂intC∪ {0}. (5.2)

Next, we prove

–T¯G(x)¯ ∩(–D)∩(+intC) =∅. (5.3)

Suppose to the contrary that there existsz˜∈G(x)¯ ∩(–D) such that

–T¯(˜z)∈+intC. (5.4)

Thus –T¯(˜z) –∈intC. By the definition ofT¯ we have s∗–T¯(z) –˜ =s∗–k∗(z)c˜ 0–

Combining with (5.1), we have

s∗–T¯(z) –˜ ≤0. (5.5)

On the other hand, froms∗∈C∗\ {0Y∗}and Lemma4.1it follows that

s∗(intC) > 0,

which, together with (5.5), gives

–T¯(˜z) –∈/intC,

which contradicts (5.4). Thus we obtain (5.3). The combination of (5.2) and (5.3) leads to

–T¯G(x)¯ ∩(–D)⊂intC∩ {0}\(+intC).

Finally, we prove thatx¯is an-weakly efficient solution to the-USVEP. In fact, by the definition ofT¯ and (4.7) we obtain

s∗(x,¯ x) +T¯G(x)+=s∗(x,¯ x)+s∗() +k∗G(x)≥0, ∀x∈X0.

Sinces∗(–intC) < 0, we have

(x,¯ x) +T¯G(x)+∩(–intC) =∅, ∀x∈X0.

Consequently,

(x,¯ x) +∩(–intC) =∅, ∀x∈X0.

It is evident thatx¯is an-weakly efficient solution to the-USVEP.

Theorem 5.2 Assume that

(i) x¯∈Aand

(ii) there existsT¯ ∈L+(Z,Y)such thatx¯is an-weakly efficient solution to the

-USVEP,where:X0×X0→2Y is defined by

(y,x) =(y,x) +T¯G(x).

Thenx is an¯ -weakly efficient solution to the-SVEPC.

Proof Sincex¯is an-weakly efficient solution to the-USVEP, we have

x∈X0

SinceCis a convex cone, we obtain

x∈X0

(x,¯ x) +C+∩(–intC) =∅. (5.6)

On the other hand, for anyx∈A, we haveG(x)∩(–D)=∅. Thus there existszx∈G(x) such thatzx∈–D. It follows fromT¯ ∈L+(Z,Y) thatT¯(zx)∈–C, thusC–T(z¯ x)⊂C+C⊂C, and henceC⊂ ¯T(zx) +C; sincezx∈G(x), it is evident thatC⊂ ¯T(G(x)) +C.

Thus

x∈A

(x,¯ x) +C+⊂ x∈A

(x,¯ x) +T¯G(x)+C+

⊂

x∈X0

(x,¯ x) +T¯G(x)+C+

= x∈X0

(x,¯ x) +C+.

It follows from (5.6) that

x∈A

(x,¯ x) +C+∩(–intC) =∅.

Since 0∈C, it is evident that

x∈A

(x,¯ x) +∩(–intC) =∅.

Hencex¯is an-weakly efficient solution to the-SVEPC.

From Theorems5.1and5.2we obtain the following result.

Corollary 5.1 Suppose thatx¯∈A, 0∈(x,¯ x),¯ ϕis nearly C×D-subconvexlike on X0,and

there exists x0∈X0such that G(x0)∩(–intD)=∅.Thenx is an¯ -weakly efficient solution

to the-SVEPC if and only if there existsT¯ ∈L+_{(Z,Y)such thatx is an¯ -weakly efficient}

solution to the-USVEP,where:X0×X0→2Y is defined by

(y,x) =(y,x) +T¯G(x).

Corollary 5.2 Suppose that(F–y,¯ G)is nearly C×D-subconvexlike on X0and there exists

x0∈X0such that G(x0)∩(–intD)= ∅.If(x,¯ y)¯ is an-weakly efficient pair to the(SOP),then

there existsT¯ ∈L+_{(Z,Y)such that–T}¯_{(G(x)¯ ∩(–D))⊂((intC∩ {0})\(+intC)),and(x,¯ ¯y)}

is an-weakly efficient pair to the(USOP)T¯.

Proof Letting(y,x) =F(x) –y, since¯ y¯∈F(x), it is evident that 0¯ ∈(x,¯ x) and (F(x) –¯

¯

y,G(x)) = ((x,¯ x),G(x)).

Thus by Theorem5.1there existsT¯ ∈L+_{(Z,Y) such that –T}¯_{(G(x)¯ ∩(–D))⊂((intC∩}

{0})\(+intC)) andx¯is an-weakly efficient solution to the-USVEP, that is,

x∈X0

(x,¯ x) +T¯G(x)+∩(–intC) =∅.

Consequently,

x∈X0

F(x) –y¯+T¯G(x)+∩(–intC) =∅,

which is equivalent to

x∈X0

L(x,T) +¯ –y¯

∩(–intC) =∅.

Thus

L(X0,T¯) +–y¯

∩(–intC) =∅.

Hence, by Definition2.5, (x,¯ y) is an¯ -weakly efficient pair to the (USOP)T¯.

Remark5.1 If (F,G) isC×D-subconvexlike, then (F–y,¯ G) is nearlyC×D-subconvexlike. Thus Corollary5.2generalizes Theorem 3.1 in [15].

Corollary 5.3 Assume that

(i) x¯∈A,y¯∈F(x)¯ ,and

(ii) there existsT¯ ∈L+(Z,Y)such that(x,¯ y)¯ is an-weakly efficient pair to the(USOP)T¯.

Then(x,¯ ¯y)is an-weakly efficient pair to the(SOP).

Proof Letting(y,x) =F(x) –y, since¯ y¯∈F(x), it is evident that 0¯ ∈(x,¯ x) and (F(x) –¯

¯

y,G(x)) = ((x,¯ x),G(x)).

Since (x,¯ y) is an¯ -weakly efficient pair to the (USOP)T¯, we see thatx¯ is an-weakly

efficient solution to the-USVEP. Combining this with Theorem5.2, we conclude that

¯

xis an -weakly efficient solution to the-SVEPC; it is clear that (x,¯ y) is an¯ -weakly

efficient pair to the (SOP).

Remark5.2 Comparing with Theorem 3.2 in [15], this corollary is not required for the convexity of (F,G).

Corollary 5.4 Suppose thatx¯∈A,y¯∈F(x), (F¯ –¯y,G)is nearly C×D-subconvexlike on X0,

and there exists x0∈X0such that G(x0)∩(–intD)=∅.Then(x,¯ y)¯ is an-weakly efficient

pair to the(SOP)if and only if there existsT¯ ∈L+(Z,Y)such that(x,¯ y)¯ is an -weakly efficient pair to the(USOP)T¯.

6 Conclusions

In this paper, we discuss some relationships between approximate weakly efficient solu-tions and weakly efficient solusolu-tions of set-valued vector equilibrium problems. We con-clude that∈C\{0}-XWmin(,A) =XWmin(,A), and hence it is really “approximate”. The optimality conditions for set-valued vector equilibrium problems are established, and the results we obtained generalize those of Gong[2], Yang[17], and Rong[15]. As an exten-sive mathematical model, further research on approximate weakly efficient solutions of set-valued vector equilibrium problems seems to be of interest and value.

Funding

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).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally, and they all read and approved the final manuscript.

Authors’ information

Yihong Xu (1969-), Professor, Doctor, the major field of interest is in the area of set-valued optimization. Jian Chen, email: [email protected]. Ke Zhang, email:[email protected].

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 5 May 2018 Accepted: 22 June 2018 References

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