Existence and Duality of Generalized ε-Vector Equilibrium Problems

Abstract

We consider a generalized ε-vector equilibrium problem which contain vector equilibrium problems and vector variational inequalities as special cases. By using the KKM theorem, we obtain some existence theorems for the generalized ε-vector equilibrium problem. We also investigate the duality of this generalized ε-vector equilibrium problem and discuss the equivalence relation between solutions of primal and dual problems.

1. Introduction

It is known that vector equilibrium problems provide a unified model for several different problems appearing in the fields of vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems. Many important results for various kinds of vector equilibrium problems and their extensions have been extensively investigated, see [1–8] and the references therein. Ansari et al. [9] introduced an implicit vector variational problem which contain vector equilibrium problems and vector variational inequalities as special cases. They also established an equivalent relationship between the solution sets for an implicit variational problem and its dual problem. Li and Zhao [10] introduced a dual scheme for a mixed vector equilibrium problem by using the method of Fenchel conjugate functions and established a relationship between the solution sets of the primal and dual problems. Sun and Li [11] introduced a dual scheme for an extended Ky Fan inequality. By using the obtained duality assertions, they also obtained some Farkas-type results which characterize the optimal value of this extended Ky Fan inequality.

From the computational viewpoint, the algorithms proposed in the literature for solving nonlinear optimization problems, in general, can only obtain approximate solutions (𝜀-optimal solutions) of such problems. In this regard, many researchers have studied optimality conditions for 𝜀-solutions for scalar and vector optimization problems, see [12–16] and the references cited therein.

However, there are very little results for optimality conditions for 𝜀-solution (approximate solution) of vector equilibrium problems. Moreover, the study of approximate vector equilibrium problems is very important since many approximate optimization problems may be considered as their special cases, see [17–19] and the references cited therein. Sach et al. [20] introduced new versions of 𝜀-dual problems of a vector quasi-equilibrium problem with set-valued maps and obtained an 𝜀-duality result between approximate solutions of the primal and dual problems. X. B. Li and S. J. Li [21] considered parametric scalar and vector equilibrium problems and obtained sufficient conditions for Hausdorff continuity and Berge continuity of approximate solution mappings for parametric scalar and vector equilibrium problems. Sun and Li [22] considered a generalized multivalued 𝜀-vector variational inequality, formulated its dual problem, and proved duality results between the primal and dual problems.

To the best of our knowledge, there is no paper to deal with the generalized 𝜀-vector equilibrium problems. Motivated by the work reported in [3–5, 9, 22], in this paper, we first introduce a generalized 𝜀-vector equilibrium problem (GVEP)_𝜀 and establish an existence theorem for (GVEP)_𝜀 by using the known KKM Theorem. Then, we discuss duality results between (GVEP)_𝜀 and its dual problem. We also show that our results on existence contain known results in the literature as special cases.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions and introduce a generalized 𝜀-vector equilibrium problem (GVEP)_𝜀. In Section 3, by using the known KKM Theorem, we establish an existence theorem for the (GVEP)_𝜀. As a special case, we also derive some existence results for a generalized 𝜀-vector variational inequality introduced and studied by Sun and Li [22]. In Section 4, we give a dual 𝜀-vector variational inequality (DVEP)_𝜀 for (GVEP)_𝜀 and prove an equivalence relation between (GVEP)_𝜀 and (DVEP)_𝜀.

2. Mathematical Preliminaries

Throughout this paper, let 𝑋 and 𝑌 be two Banach spaces, and let ℒ(𝑋,𝑌) be the set of all linear continuous operators from 𝑋 to 𝑌. Let 𝐾⊆𝑌 be a convex cone with a nonempty interior int𝐾. Given 𝑥,𝑦∈𝑌, we define the following ordering relations:𝑦<𝑦_𝐾≤𝑥_𝐾⟺𝑦−𝑥∈−int𝐾,𝑦≮_𝐾𝑥⟺𝑦−𝑥∉−int𝐾,𝑥⟺𝑦−𝑥∈−𝐾,𝑦≰_𝐾𝑥⟺𝑦−𝑥∉−𝐾.(2.1) Given two sets 𝐴,𝐵⊆𝑌, we also consider the following set ordering relationships:𝐴𝐴<≮_𝐾𝐵⟺𝑦<_𝐾𝑥,∀𝑦∈𝐴,𝑥∈𝐵,_𝐾𝐴≤_𝐾𝐵⟺𝑦≮_𝐾𝑥,∀𝑦∈𝐴,𝑥∈𝐵,𝐴≰_𝐾𝐵⟺𝑦≤_𝐾𝑥,∀𝑦∈𝐴,𝑥∈𝐵,𝐵⟺𝑦≰_𝐾𝑥,∀𝑦∈𝐴,𝑥∈𝐵.(2.2)

Let 𝜑∶𝑋×𝑋→𝑌 and 𝑔∶𝑋→𝑌 be two vector-valued mappings satisfying 𝜑(𝑥,𝑥)=0, for any 𝑥∈𝑋. Consider the following generalized 𝜀-vector equilibrium problem (GVEP)_𝜀:Find𝑥₀∈𝑋suchthat𝜑𝑥₀,𝑥+𝑔(𝑥)−𝑔𝑥₀+𝜀‖‖𝑥−𝑥₀‖‖≮_𝐾0,forany𝑥∈𝑋,(GVEP)_𝜀 where 𝜀∈𝐾. We say that 𝑥₀ is an 𝜀-solution of (GVEP)_𝜀 if and only if𝜑𝑥₀,𝑥+𝑔(𝑥)−𝑔𝑥₀+𝜀‖‖𝑥−𝑥₀‖‖≮_𝐾0,forany𝑥∈𝑋.(2.3)

If 𝜀=0, then (GVEP)_𝜀 collapses to the following generalized vector equilibrium problem (GVEP), introduced and studied by Li and Zhao [10]:Find𝑥₀∈𝑋suchthat𝜑𝑥₀,𝑥+𝑔(𝑥)−𝑔𝑥₀≮_𝐾0,forany𝑥∈𝑋.(GVEP)

If 𝑔=0 and 𝜀=0, then (GVEP)_𝜀 becomes the following vector equilibrium problem (VEP)Find𝑥₀∈𝑋suchthat𝜑𝑥₀,𝑥≮_𝐾0,forany𝑥∈𝑋.(VEP)

Let 𝑇∶𝑋→ℒ(𝑋,𝑌) be a vector-valued mapping. If 𝜑(𝑥₀,𝑥)=⟨𝑇(𝑥₀),𝑥−𝑥₀⟩, for any 𝑥∈𝑋, (GVEP)_𝜀 collapses to the following generalized 𝜀-vector variational inequality (GVVI)_𝜀, introduced and studied by Sun and Li [22]:Find𝑥₀∈𝑋suchthat⟨𝑇𝑥₀,𝑥−𝑥₀⟩+𝑔(𝑥)−𝑔𝑥₀+𝜀‖‖𝑥−𝑥₀‖‖≮_𝐾0,forany𝑥∈𝑋,(GVVI)_𝜀 where 𝜀∈𝐾 and ⟨𝑇(𝑥₀),𝑥−𝑥₀⟩ is the evaluation of 𝑇(𝑥₀) at 𝑥−𝑥₀.

In this paper, we consider the generalized 𝜀-vector equilibrium problem (GVEP)_𝜀 and establish the existence theorem for solutions of (GVEP)_𝜀. Then, we give a dual 𝜀-vector equilibrium problem (DVEP)_𝜀 for (GVEP)_𝜀 and prove an equivalence relation between (GVEP)_𝜀 and (DVEP)_𝜀.

At first, let us recall some important definitions.

Definition 2.1 (see [23]). Let 𝑄 be a nonempty subset of 𝑌. A point 𝑦̂∈𝑄 is said to be a weak maximal point of 𝑄, if there is no 𝑦^∈𝑄 such that 𝑦̂<_𝐾𝑦^. The set of all maximal points of 𝑄 is called the weak maximum of 𝑄 and is denoted by WMax_𝐾𝑄. The weak minimum of 𝑄,WMin_𝐾𝑄, is defined analogously.

Definition 2.2 (see [23]). Let 𝑔∶𝑋→𝑌 be a vector-valued mapping. 𝑔 is said to be 𝐾-convex if for any 𝑥₁,𝑥₂∈𝑋 and 𝛼∈[0,1], 𝑔𝛼𝑥₁+(1−𝛼)𝑥₂≤_𝐾𝛼𝑔𝑥₁+(1−𝛼)𝑔𝑥₂.(2.4) Furthermore, 𝑔 is said to be 𝐾-concave, if −𝑔 is 𝐾-convex.

Definition 2.3 (see [24]). A vector mapping 𝑔∶𝑋→𝑌 is said to be 𝜀-convex with 𝜀∈𝐾 if for any 𝑥₁,𝑥₂∈𝑋 and 𝛼∈[0,1], 𝑔𝛼𝑥₁+(1−𝛼)𝑥₂≤_𝐾𝛼𝑔𝑥₁+(1−𝛼)𝑔𝑥₂+𝜀𝛼(1−𝛼)‖‖𝑥₁−𝑥₂‖‖.(2.5)

Definition 2.4 (see [22]). Let 𝑔∶𝑋→𝑌 be an 𝜀-convex mapping with 𝜀∈𝐾.(i)A set-valued mapping 𝑔^∗𝑥_0,𝜀∶ℒ(𝑋,𝑌)⇉𝑌 defined by 𝑔^∗𝑥_0,𝜀(Λ)=WMax_𝐾⟨Λ,𝑥⟩−𝑔(𝑥)−𝜀‖‖𝑥−𝑥₀‖‖∶𝑥∈𝑋,foranyΛ∈ℒ(𝑋,𝑌),(2.6) is called the 𝜀-conjugate mapping of 𝑔 at 𝑥₀∈𝑋.(ii)A set-valued mapping 𝑔^𝑥∗∗_0,𝜀∶𝑋⇉𝑌 defined by 𝑔^𝑥∗∗_0,𝜀(𝑥)=WMax_𝐾⟨Λ,𝑥⟩−𝑔^∗𝑥_0,𝜀(Λ)∶Λ∈ℒ(𝑋,𝑌),forany𝑥∈𝑋,(2.7) is called the 𝜀-biconjugate mapping of 𝑔 at 𝑥₀∈𝑋.

Definition 2.5 (see [25]). Let 𝑔∶𝑋→𝑌 be a given mapping. A subdifferential of 𝑔 at 𝑥₀∈𝑋 is defined as 𝜕𝑔𝑥₀=Λ∈ℒ(𝑋,𝑌)∶𝑔(𝑥)−𝑔𝑥₀≮_𝐾⟨Λ,𝑥−𝑥₀⟩,∀𝑥∈𝑋.(2.8)

Definition 2.6 (see [26]). Let 𝑔∶𝑋→𝑌 be a given mapping with 𝜀∈𝐾. An 𝜀-subdifferential of 𝑔 at 𝑥₀∈𝑋 is defined as 𝜕_𝜀𝑔𝑥₀=Λ∈ℒ(𝑋,𝑌)∶𝑔(𝑥)−𝑔𝑥₀≮_𝐾⟨Λ,𝑥−𝑥₀⟩−𝜀‖‖𝑥−𝑥₀‖‖,∀𝑥∈𝑋.(2.9)

Remark 2.7. (i) It is easy to note that, when 𝜀=0,𝜕_𝜀𝑔(𝑥₀)=𝜕𝑔(𝑥₀) and 𝜕_𝜀𝑔(𝑥₀)=𝜕(𝑔+𝜀‖⋅−𝑥₀‖)(𝑥₀).
(ii) Note that Li and Guo [26, Section 3] have given some existence theorems of 𝜀-subdifferential for a vector-valued mapping under the condition that the cone 𝐾 is connected (i.e., 𝐾∪(−𝐾)=𝑌).

Next, we give KKMtheorem needed for the proof of the existence results.

Definition 2.8 (see [27]). A set-valued mapping 𝐺∶𝑋⇉𝑌 is called the KKM mapping if, for each finite subset {𝑥₁,𝑥₂,…,𝑥_𝑛} of 𝑋, co𝑥₁,𝑥₂,…,𝑥_𝑛⊂^𝑛_𝑖=1𝐺𝑥_𝑖,(2.10) where co{𝑥₁,𝑥₂,…,𝑥_𝑛} is the convex hull of {𝑥₁,𝑥₂,…,𝑥_𝑛}.

Theorem 2.9 (see [27, KKM Theorem]). Let 𝐺∶𝑋⇉𝑌 be a 𝐾𝐾𝑀 mapping. If for each 𝑥∈𝑋, 𝐺(𝑥) is closed and is compact for at least one 𝑥∈𝑋, then, ⋂_𝑥∈𝑋𝐺(𝑥)≠∅.

3. Existence Theorems for (GVEP)_𝜀

In this section, we prove some existence results for the generalized 𝜀-vector equilibrium problem (GVEP)_𝜀 by using the known KKMtheorem. As a special case, we derive some existence results for the generalized 𝜀-vector variational inequality (GVVI)_𝜀.

Theorem 3.1. Suppose that the following conditions are satisfied:(i)𝜑∶𝑋×𝑋→𝑌 and 𝑔∶𝑋→𝑌 are two continuous mappings;(ii)for any 𝑦∈𝑋,𝐵_𝑦∶={𝑥∈𝑋∶𝜑(𝑥,𝑦)+𝑔(𝑦)−𝑔(𝑥)+𝜀‖𝑥−𝑦‖<_𝐾0} is convex;(iii)there exists a nonempty compact subset 𝐶 and 𝑥′∈𝐶 such that for any 𝑦∈𝑋⧵𝐶, one has 𝜑(𝑥′,𝑦)+𝑔(𝑦)−𝑔(𝑥′)+𝜀‖𝑥′−𝑦‖<_𝐾0.(3.1)Then, the generalized 𝜀-vector equilibrium problem (GVEP)_𝜀 is solvable.

Proof. Define a set-valued mapping 𝐺∶𝑋⇉𝑋 by: for any 𝑥∈𝑋, 𝐺(𝑥)∶=𝑦∈𝑋∶𝜑(𝑥,𝑦)+𝑔(𝑦)−𝑔(𝑥)+𝜀‖𝑥−𝑦‖≮_𝐾0.(3.2)
We first prove that 𝐺 is a KKMmapping. In fact, suppose that 𝐺(𝑥) is not a KKMmapping. Then, there exists a finite subset {𝑥^1,𝑥^2,…,𝑥^𝑛} of 𝑋 such that co𝑥^1,𝑥^2,…,𝑥^𝑛⊄^𝑛_𝑖=1𝐺𝑥^𝑖.(3.3) Let 𝑦′∈co{𝑥^1,𝑥^2,…,𝑥^𝑛}. Then, 𝑦′=∑^𝑛𝑖=1𝛼_𝑖𝑥^𝑖 for some 𝛼_𝑖∈[0,1],𝑖=1,2,…,𝑛 with ∑^𝑛𝑖=1𝛼_𝑖=1 and 𝑦^∉^𝑛_𝑖=1𝐺𝑥^𝑖.(3.4) So, for any 𝑖∈{1,2,…,𝑛}, we have 𝜑𝑥^𝑖,𝑦^+𝑔𝑦^−𝑔𝑥^𝑖+𝜀‖‖𝑥^𝑖−𝑦^‖‖<_𝐾0.(3.5) Hence, 𝑥^1,𝑥^2,…,𝑥^𝑛⊂𝐵_𝑦^′.(3.6) Since 𝐵_𝑦′ is convex, we have co𝑥^1,𝑥^2,…,𝑥^𝑛⊂𝐵_𝑦^′.(3.7) By 𝑦′∈co{𝑥^1,𝑥^2,…,𝑥^𝑛}, we have 𝑦^∈𝐵_𝑦′.(3.8) Thus, we have 𝜑𝑦^,𝑦^+𝑔𝑦^−𝑔𝑦^+𝜀‖𝑦′−𝑦′‖<_𝐾0,(3.9) which is a contradiction. Therefore, 𝐺 is a KKMmapping.
Next, we prove that, for any 𝑥∈𝑋,𝐺(𝑥) is closed. Indeed, let any sequence {𝑦_𝑛} with 𝑦_𝑛∈𝐺(𝑥) and 𝑦_𝑛→𝑦₀. Then, 𝜑𝑥,𝑦_𝑛+𝑔𝑦_𝑛−𝑔(𝑥)+𝜀‖‖𝑥−𝑦_𝑛‖‖≮_𝐾0.(3.10) Taking limit in (3.10) with 𝜑 and 𝑔 being two continuous mappings, we have 𝜑𝑥,𝑦₀+𝑔𝑦₀−𝑔(𝑥)+𝜀‖‖𝑥−𝑦₀‖‖≮_𝐾0.(3.11) Thus, 𝑦₀∈𝐺(𝑥) and 𝐺(𝑥) is closed.
Assumption (iii)implies that 𝐺(𝑥′)⊂𝐶. Hence, 𝐺(𝑥′) is compact. Therefore, the assumptions of Theorem 2.9 hold. By Theorem 2.9, we have ^𝑥∈𝑋𝐺(𝑥)≠∅.(3.12) Hence, there exist 𝑥₀∈𝑋 such that 𝜑𝑥,𝑥₀+𝑔𝑥₀−𝑔(𝑥)+𝜀‖‖𝑥−𝑥₀‖‖≮_𝐾0,(3.13) for any 𝑥∈𝑋. This completes the proof.

Now, we give an example to illustrate Theorem 3.1.

Example 3.2. Let 𝑋=𝑅,𝑌=𝑅²,𝐶=[−1,1],𝐾=𝑅²⁺, and 𝜀=(0,1). Suppose that 𝜑∶𝑅×𝑅→𝑅² and 𝑔∶𝑅→𝑅² are defined by 𝜑(𝑥,𝑦)=𝑔−(2𝑥𝑥)²=+𝑦−²2,𝑥−₂2𝑥²+𝑦²−||𝑥−𝑦||,(3.14),−2𝑥²,(3.15) respectively. Then, assumption (i) in Theorem 3.1 is clearly satisfied. It can be checked that, for any 𝑦∈𝑅, 𝐵_𝑦=𝑥+∈0𝑋,||𝑥∶−−𝑦||2𝑥<²+_𝑅2+𝑦²,−2𝑥²+𝑦²−||𝑥−𝑦||+−2𝑦²,−2𝑦²−−2𝑥²,−2𝑥²=𝑥∈𝑋∶−𝑦₂0,−𝑦²<_𝑅²⁺0.(3.16) Obviously, 𝐵_𝑦 is a convex set. Thus, assumption (ii) of Theorem 3.1 holds.
Obviously, [−1,1] is a nonempty compact set. Let 𝑥′=1∈[−1,1]. For any 𝑦∈𝑅⧵[−1,1], we have 𝜑𝑥^,𝑦+𝑔(𝑦)−𝑔𝑥^+𝜀‖𝑥′−𝑦‖==−−−(−𝑦2₂2+,−𝑦²2,)−+2+0,𝑦||1²−−||𝑦1||−𝑦||+−2𝑦²,−2𝑦²<_𝑅2+,−𝑦²0.(3.17) Thus, assumption (iii) of Theorem 3.1 holds. Obviously, 𝑥₀=0 is a solution of (GVEP)_𝜀.

Remark 3.3. When the function 𝑔 is 𝐾-concave and 𝜑(⋅,𝑦) is 𝐾-convex, we have that condition (ii) of Theorem 3.1 holds, that is, 𝐵_𝑦 is convex.
Indeed, let 𝑥₁,𝑥₂∈𝐵_𝑦 and 𝛼∈[0,1]. Then, we have 𝜑𝜑𝑥𝑥₁,𝑦+𝑔(𝑦)−𝑔𝑥₁+𝜀‖‖𝑥₁−𝑦‖‖∈−int𝐾,₂,𝑦+𝑔(𝑦)−𝑔𝑥₂+𝜀‖‖𝑥₂−𝑦‖‖∈−int𝐾.(3.18) Since 𝑔 is 𝐾-concave and 𝜑(⋅,𝑦) is 𝐾-convex, we have 𝜑𝑥₁∈+𝛼(1𝜑−𝑥𝛼₁)𝑥₂,𝑦+𝑔(𝑦)−𝑔𝛼𝑥₁+(1−𝛼)𝑥₂+𝜀‖‖𝛼𝑥₁+(1−𝛼)𝑥₂−𝑦‖‖+(1−𝛼),𝑦𝜑+𝑥𝑔₂(𝑦)−𝑔𝑥₁+𝜀‖‖𝑥₁−𝑦‖‖⊆=−−iinntt𝐾𝐾.−int𝐾,−𝑦𝐾+−𝑔𝐾(𝑦−)−𝐾𝑔𝑥₂+𝜀‖‖𝑥₂−𝑦‖‖−𝐾−𝐾−𝐾(3.19) So, 𝐵_𝑦 is convex and we have the following corollary.

Corollary 3.4. Suppose that the following conditions are satisfied:(i)𝜑(⋅,𝑦)∶𝑋→𝑌 is a continuous 𝐾-convex mapping and 𝑔∶𝑋→𝑌 is a continuous 𝐾-concave mapping;(ii)there exists a nonempty compact subset 𝐶 and 𝑥′∈𝐶 such that for any 𝑦∈𝑋⧵𝐶, one has 𝜑𝑥^,𝑦+𝑔(𝑦)−𝑔𝑥^+𝜀‖‖𝑥^−𝑦‖‖<_𝐾0.(3.20)Then, the generalized 𝜀-vector equilibrium problem (GVEP)_𝜀 is solvable.

By Theorem 3.1 and Corollary 3.4, we can derive the following existence results for the generalized 𝜀-vector variational inequality (GVVI)_𝜀.

Theorem 3.5. Suppose that the following conditions are satisfied:(i)𝑇∶𝑋→ℒ(𝑋,𝑌) and 𝑔∶𝑋→𝑌 are two continuous mappings;(ii)for any 𝑦∈𝑋, 𝐵_𝑦∶={𝑥∈𝑋∶⟨𝑇(𝑦),𝑥−𝑦⟩+𝑔(𝑥)−𝑔(𝑦)+𝜀‖𝑥−𝑦‖<_𝐾0} is convex;(iii)there exists a nonempty compact subset 𝐶 and 𝑥′∈𝐶 such that for any 𝑦∈𝑋⧵𝐶, one has 𝑇(𝑦),𝑥^−𝑦+𝑔𝑥^−𝑔(𝑦)+𝜀‖𝑥′−𝑦‖<_𝐾0.(3.21)Then, the generalized 𝜀-vector variational inequality (GVVI)_𝜀 is solvable.

Remark 3.6. If the function 𝑔 is 𝐾-convex, we can prove that condition (ii) of Theorem 3.5 holds by the similar method of Remark 3.3.

Corollary 3.7. Suppose that the following conditions are satisfied:(i)𝑇∶𝑋⇉ℒ(𝑋,𝑌) is a continuous mapping, and 𝑔∶𝑋→𝑌 is a continuous 𝐾-convex mapping;(ii)there exists a nonempty compact subset 𝐶 and 𝑥′∈𝐶 such that for any 𝑦∈𝑋⧵𝐶, one has 𝑇(𝑦),𝑥^−𝑦+𝑔𝑥^−𝑔(𝑦)+𝜀‖𝑥′−𝑦‖<_𝐾0.(3.22)Then, the generalized 𝜀-vector variational inequality (GVVI)_𝜀 is solvable.

Remark 3.8. (i) In case when 𝑇 is a set-valued mapping, similar results of Theorem 3.5 and Corollary 3.7 have been studied by Sun and Li [22].
(ii) Theorem 3.5 and Corollary 3.7 can be viewed as an extension of Theorem 1 in Yang [3].

4. Dual Results for (GVEP)_𝜀

In this section, we first introduce the dual generalized 𝜀-vector equilibrium problems for (GVEP)_𝜀. Then, we establish an equivalence between primal and dual problems.

We first recall that 𝑔 is said to be externally stable at 𝑥∈𝑋 if 𝑔(𝑥)∈𝑔^𝑥∗∗_0,𝜀(𝑥). The external stability was introduced in [25] when the vector conjugate function is defined via the set of efficient points.

The following result plays an important role in our study.

Lemma 4.1. Let 𝑔∶𝑋→𝑌 and 𝑥₀∈𝑋. Then, Λ∈𝜕_𝜀𝑔𝑥₀⟺⟨Λ,𝑥₀⟩−𝑔𝑥₀∈𝑔^∗𝑥_0,𝜀(Λ).(4.1)

Proof. It follows from the definitions of the 𝜀-conjugate function and the 𝜀-subdifferentials that Λ∈𝜕_𝜀𝑔𝑥₀(4.2) if and only if 𝑔(𝑥)−𝑔𝑥₀≮_𝐾⟨Λ,𝑥−𝑥₀⟩−𝜀‖‖𝑥−𝑥₀‖‖,forany𝑥∈𝑋,(4.3) equivalently, ⟨Λ,𝑥⟩−𝑔(𝑥)−𝜀‖‖𝑥−𝑥₀‖‖≯_𝐾⟨Λ,𝑥₀⟩−𝑔𝑥₀,forany𝑥∈𝑋,(4.4) which means that ⟨Λ,𝑥₀⟩−𝑔𝑥₀∈𝑔^∗𝑥_0,𝜀(Λ),(4.5) and the proof is completed.

It is shown in [28] that, if 𝜑(𝑥₀,⋅) is convex, then 𝜕𝜑_𝑥0(𝑥)≠∅, where 𝜕𝜑_𝑥0(𝑥) denotes the subdifferential of 𝜑 with respect to its second component. Now, we define the dual generalized 𝜀-vector equilibrium problem (DVEP)_𝜀 of (GVEP)_𝜀 as follows:Find𝑥₀∈𝑋,−Γ₀∈𝜕𝜑_𝑥0𝑥₀,and𝑦₀∈𝑔^∗𝑥_0,𝜀Γ₀suchthat𝑦₀−⟨Γ₀,𝑥₀⟩f>o̸r_𝐾a𝑔ny^∗𝑥_0,𝜀Γ(Γ∈)ℒ−⟨(Γ𝑋,,𝑥𝑌(₀D⟩)V.,EP)_𝜀 Moreover, (𝑥₀,Γ₀) is called a solution of (DVEP)_𝜀.

Remark 4.2. In [8], Sach et al. gave some examples to prove that the dual problems of [3, 5] are not suitable for the duality property of vector variational inequalities, and hence, all possible applications of them cannot be seen to be justified. This fact shows that, when dealing with duality in vector variational inequality problems which are generalizations of those considered in [3, 5], we must use dual problems different from those of [3, 5]. So, in this section we consider this problem (DVEP)_𝜀 called the dual problem of (GVEP)_𝜀.

In the following, we will discuss the relationships between the solutions of (GVEP)_𝜀 and (DVEP)_𝜀.

Theorem 4.3. Suppose that 𝑔 is externally stable at 𝑥₀∈𝑋. If 𝑥₀ is a solution of (GVEP)_𝜀 and 𝜕_𝜀(𝜑_𝑥0+𝑔)(𝑥₀)=𝜕𝜑_𝑥0(𝑥₀)+𝜕_𝜀𝑔(𝑥₀), then, there exists Γ₀∈ℒ(𝑋,𝑌) such that (𝑥₀,Γ₀) is a solution of (DVEP)_𝜀.

Proof. If 𝑥₀ is a solution of (GVEP)_𝜀, then, 𝜑𝑥₀,𝑥+𝑔(𝑥)−𝑔𝑥₀+𝜀‖‖𝑥−𝑥₀‖‖≮_𝐾0,forany𝑥∈𝑋.(4.6) So, it is easy to see that 𝜑𝑥₀,𝑥+𝑔(𝑥)−𝜑𝑥₀,𝑥₀+𝑔𝑥₀≮_𝐾⟨0,𝑥−𝑥₀⟩−𝜀‖‖𝑥−𝑥₀‖‖,forany𝑥∈𝑋,(4.7) which means that 0∈𝜕_𝜀𝜑_𝑥0+𝑔𝑥₀.(4.8) Hence, 0∈𝜕𝜑_𝑥0𝑥₀+𝜕_𝜀𝑔𝑥₀,(4.9) or, equivalently, there exists Γ₀∈ℒ(𝑋,𝑌) such that Γ₀∈−𝜕𝜑_𝑥0𝑥₀∩𝜕_𝜀𝑔𝑥₀.(4.10) Then, from (4.1), we get ⟨Γ₀,𝑥₀⟩−𝑔𝑥₀∈𝑔^∗𝑥_0,𝜀Γ₀.(4.11) Since 𝑔 is externally stable at 𝑥₀∈𝑋, we have 𝑔𝑥₀∈𝑔^𝑥∗∗_0,𝜀𝑥₀=WMax_𝐾⟨Γ,𝑥₀⟩−𝑔^∗𝑥_0,𝜀(Γ)∶Γ∈ℒ(𝑋,𝑌).(4.12) Thus, 𝑔𝑥₀≮_𝐾⟨Γ,𝑥₀⟩−𝑔^∗𝑥_0,𝜀(Γ),foranyΓ∈ℒ(𝑋,𝑌).(4.13) By (4.11), there exists 𝑦₀∈𝑔^∗𝑥_0,𝜀(Γ₀) such that 𝑦₀=⟨Γ₀,𝑥₀⟩−𝑔𝑥₀.(4.14) By (4.13) and (4.14), we get 𝑦₀−⟨Γ₀,𝑥₀⟩≯_𝐾𝑔^∗𝑥_0,𝜀(Γ)−⟨Γ,𝑥₀⟩,foranyΓ∈ℒ(𝑋,𝑌),(4.15) and the proof is completed.

Theorem 4.4. If (𝑥₀,Γ₀) is a solution of (DVEP)_𝜀 and 𝜕_𝜀(𝜑_𝑥0+𝑔)(𝑥₀)=𝜕𝜑_𝑥0(𝑥₀)+𝜕_𝜀𝑔(𝑥₀), then, 𝑥₀ is a solution of (GVEP)_𝜀.

Proof. This is obtained by inverting the reasoning in the proof of Theorem 4.3 step by step.

Remark 4.5. Although the equality of the approximate subdifferentials used in Theorems 4.3 and 4.4 is difficult to be verified, Li and Guo [26, Section 5] have given some sufficient conditions for the validity of the equality under the condition that the cone 𝐾 is connected (i.e., 𝐾∪(−𝐾)=𝑌).

Now, we give an example to illustrate Theorem 4.4.

Example 4.6. Let 𝑋=𝑅,𝑌=𝑅²,𝐾=𝑅²⁺, and 𝜀=(1,1). Suppose that 𝜑∶𝑅×𝑅→𝑅² and 𝑔∶𝑅→𝑅² are defined by 𝑔𝜑(𝑥)𝑥=₀,𝑥𝑥−=|𝑥|||𝑥,₀𝑥||₂,||𝑥₀||,(4.16)−|𝑥|,(4.17) respectively. Then, for 𝑥₀=0, we have 𝜕_𝜀𝜑_𝑥0+𝜕𝑔𝜑_𝑥0𝑥₀=𝜕_𝜀𝑔𝑥₀={1,0},𝑥₀={0,0}.(4.18) Obviously, 𝜕_𝜀𝜑_𝑥0+𝑔𝑥₀=𝜕𝜑_𝑥0𝑥₀+𝜕_𝜀𝑔𝑥₀.(4.19) Moreover, for any Γ=(Γ₁,Γ₂)∈𝑅², 𝑔^∗𝑥_0,𝜀(Γ)=⎧⎪⎪⎨⎪⎪⎩𝑦¹,𝑦₂∈𝑅²∶𝑦₂≤Γ²²4,𝑦¹𝑦=0,ifΓ₁=1,₁,𝑦₂∈𝑅²∶𝑦₂=−1Γ₁−1²𝑦²¹+Γ²Γ₁−1𝑦¹,𝑦₁≥Γ²Γ₁−12,ifΓ¹≠1.(4.20) Then, for Γ₀=(0,0)∈𝜕𝜑_𝑥0(𝑥₀), we obtain that 𝑔𝑦^∗𝑥_0,𝜀Γ₀=𝑦₁,𝑦₂∈𝑅²∶𝑦₂=−𝑦²¹,𝑦₁≥0₀,=⟨Γ₀,𝑥₀⟩−𝑔𝑥₀=(0,0)∈𝑔^∗𝑥_0,𝜀Γ₀.(4.21) Obviously, (𝑥₀,Γ₀) is a solution of (DVEP)_𝜀 and 𝑥₀ is a solution of (GVEP)_2𝜀.

Acknowledgments

The authors thank the two anonymous referees for their valuable comments and suggestions, which helped improve this paper. This research was supported by the National Natural Science Foundation of China (70972056; 71172084) and the Fundamental Research Funds for the Central Universities (CDJSK11003).