# Hölder Continuity of Solutions to Parametric Generalized Vector Quasiequilibrium Problems

Authors: Z. Y. Peng
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By using a linear scalarization method, we establish sufficient
conditions for the Hölder continuity of the solution mappings to a parametric generalized
vector quasiequilibrium problem with set-valued mappings. These results extend the
recent ones in the recent literature, (e.g., Li et al. (2009), Li et al. (2011)). Furthermore, two examples are given
to illustrate the obtained result.


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## Body

## 1. Introduction

The vector equilibrium problem has been attracting great interest because it provides a unified model for several important problems such as vector variational inequalities, vector complementarity problems, vector optimization problems, vector min-max inequality, and vector saddle point problems. Many different types of vector equilibrium problems have been intensively studied for the past years; see, for example, [1–3] and the references therein.It is important to derive results for parametric vector equilibrium problems concerning the properties of the solution mapping when the problems data vary. Among many desirable properties of vector equilibrium problems, the stability analysis of solutions is an essential topic in vector optimization theory and applications. In general, stability may be understood as lower (upper) semicontinuity, continuity, Lipschitz and Hölder continuity and so on. Recently, semicontinuity, especially lower semicontinuity, of solution mappings to parametric vector variational inequalities and parametric vector equilibrium problems has been intensively studied in the literature; see [4–12]. On the other hand, Hölder continuity of solutions to parametric vector equilibrium problems has also been discussed recently; see [13–22], although there are less works in the literature devoted to this property than to semicontinuity. There have been many papers devoted to discussing the local uniqueness and Hölder continuity of the solutions to parametric variational inequalities and parametric equilibrium problems; see [14–20] and the references therein. Yên [14] obtained Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Ait Mansour and Riahi [15] proved Hölder continuity of the unique solution for a parametric vector equilibrium problem under the concepts of strong monotonicity. Bianchi and Pini [16] introduced the concept of strong pseudomonotonicity and got the Hölder continuity of the unique solution of a parametric vector equilibrium problem. Bianchi and Pini [17] extend the results of [16] to vector equilibrium problems. Anh and Khanh [18] generalized the main results of [16] to the vector case and obtained Hölder continuity of the unique solutions for two classes of perturbed generalized vector equilibrium problems. Anh and Khanh [19] further discussed uniqueness and Hölder continuity of the solutions for perturbed generalized vector equilibrium problems, which improved remarkably the results in [16, 18]. Anh and Khanh [20] extended the results of [19] to the case of perturbed generalized vector quasiequilibrium problems and obtained Hölder continuity of the unique solutions.For general perturbed vector quasiequilibrium problems, it is well known that a solution mapping is, in general, a set-valued one, but not a single-valued one. Naturally, there is a need to study Hölder continuous properties of the set-valued solution mappings. Under the Hausdorff distance and the strong quasimonotonicity, Lee et al. [21] first showed that the set-valued solution mapping for a parametric vector variational inequality is Hölder continuous. Recently, by virtue of the strong quasimonotonicity, Ait Mansour and Aussel [22] discussed Hölder continuity of set-valued solution mappings for parametric generalized variational inequalities. Li et al. [23] introduced an assumption, which is weaker than the corresponding ones of [16, 18], and established the Hölder continuity of the set-valued solution mappings for two classes of parametric generalized vector quasiequilibrium problems in general metric spaces. Li et al. [24] extended the results of [23] to perturbed generalized vector quasiequilibrium problems. Later, S. J. Li and X. B. Li [25] use a scalarization technique to obtain the Hölder continuity of the set-valued solution mappings for a parametric vector equilibrium problem in general metric spaces.Motivated by the work reported in [21, 23, 25], this paper aims at establishing sufficient conditions for Hölder continuity of the solution sets for a class of parametric generalized vector quasiequilibrium problem (PGVQEP, in short) with set-valued mapping, by using a linear scalarization method. The main results in this paper are different from corresponding results in [23, 24] and overcome the drawback, which requires the knowledge of detailed values of the solution mapping in a neighborhood of the point under consideration. Our main results also extend and improve the corresponding ones in [25].The rest of the paper is organized as follows. In Section2, we introduce the (PGVQEP) and define the solution and ξ-solution to the (PGVQEP). Then, we recall some notions and definitions which are needed in the sequel. In Section 3, we discuss Hölder continuity of the solution mapping for the (PGVQEP) and compare our main results with the corresponding ones in the recent literature. We also give two examples to illustrate that our main results are applicable.


## 2. Preliminaries

Throughout this paper, if not other specified,∥·∥ and d(·,·) denote the norm and metric in any metric space, respectively. Let B(0,δ) denote the closed ball with radius δ≥0 and center 0 in any metric linear spaces. Let X,Λ,M, Y be metric linear spaces. Let Y* be the topological dual space of Y. Let C⊂Y be a pointed, closed, and convex cone with intC≠∅, where intC denotes the interior of C. Let C*:={f∈Y*:f(y)≥0,forally∈C} be the dual cone of C. Since intC≠∅, the dual cone C* of C has a weak* compact base. Letting e∈intC be given, then Be*:={ξ∈C*:∥ξ∥=1} is a weak* compact base of C*.LetN(λ0)⊂Λ and N(μ0)⊂M be neighborhoods of considered points λ0 and μ0, respectively. Let K:X×Λ⇉X be a set-valued mapping, and let F:X×X×M⇉Y be a set-valued mapping. For each λ∈N(λ0) and μ∈N(μ0), consider the following parameterized generalized vector quasiequilibrium problem of finding x0∈K(x0,λ) such that(PGVQEP)F(x0,y,μ)⊂Y∖-int⁡C,∀y∈K(x0,λ).For eachλ∈N(λ0) and μ∈N(μ0), let(2.1)E(λ)∶={x∈X∣x∈K(x,λ)}.
Let S(λ,μ) be the solution set of (PGVQEP), that is,(2.2)S(λ,μ)∶={x∈E(λ)∣F(x,y,μ)⊂Y∖-int⁡C,∀y∈K(x,λ)}.For eachξ∈C*∖{0}, each λ∈N(λ0) and μ∈N(μ0), let Sξ(λ,μ) denote the set of ξ-solution set to (PGVQEP), that is,(2.3)Sξ(λ,μ)∶={x∈E(λ):inf⁡z∈F(x,y,μ)f(z)≥0,∀y∈K(x,λ)}.Special Case
(i)
WhenK(x,λ)=K(λ), that is, K does not depend on x, the (PGVQEP) reduces to the parametric generalized vector equilibrium problem (PGVEP) considered by Li et al. [23].(ii)
IfF:X×X×M→ℝ, the (PGVQEP) collapses to the quasiequilibrium problem (QEP) considered by Anh and Khanh [26].(iii)
IfK(x,λ)=K(λ) and F is a vector-valued mapping, that is, F:X×X×M→Y, the (PGVQEP) reduce to the parametric Ky Fan inequality (PKI) considered by S. J. Li and X. B. Li [25].Now we recall some basic definitions and their properties which are needed in this paper.Definition 2.1 (classical notion).
A set-valued mappingG:M⇉X is said to be ℓ·α-Hölder continuous at μ0 if there is a neighborhood U(μ0) of μ0 such that, for all μ1,μ1∈U(μ0),
(2.4)G(μ1)⊆G(μ2)+lB(0,dα(μ1,μ2)),
where ℓ≥0 and α>0.Definition 2.2.
A set-valued mappingG:X×Λ⇉Y is said to be (ℓ1·α1,ℓ2·α2)-Hölder continuous at (x0,λ0) if and only if there exists neighborhoods N(x0) of x0 and N(μ0) of μ0 such that, forallx1,x2∈N(x0),forallλ1,λ2∈N(λ0),
(2.5)G(x1,λ1)⊆G(x2,λ2)+(l1dα1(x1,x2)+l2dα2(λ1,λ2))B(0,1),
where ℓ1,ℓ2≥0 and α1,α2>0.Definition 2.3 (see [25]).
A set-valued mappingG:M⇉Y is said to be (ℓ·α)-Hölder continuous with respect to e∈intC at μ0 if and only if there exists neighborhoods N(μ0) of μ0 such that, for all μ1,μ2∈N(μ0),
(2.6)G(μ1)⊆G(μ2)+ldα(μ1,μ2)[-e,e],
where ℓ≥0, α>0 and [-e,e]={x:x∈e-C,x∈-e+C}.Definition 2.4.
LetF:X×X×Λ⇉Y be a set-valued mapping with nonempty values; F(x,·,μ) is called C-like convex on A(λ) if and only if for any x1,x2∈X and any t∈[0,1], there exists x3∈X such that
(2.7)tF(x,x1,λ)+(1-t)F(x,x2,λ)⊂F(x,x3,λ)+C.Remark 2.5.
If for eachμ∈N(μ0) and each x∈E(N(λ0)), F(x,·,μ) is C-like convex on E(N(λ0)), then F(x,E(N(λ0)),μ)+C is a convex set.


## 3. Main Results

In this section, we mainly discuss the Hölder continuity of the solution mappings to (PGVQEP).Lemma 3.1.
Suppose thatN(λ0),N(μ0) are the given neighborhoods of λ0, μ0, respectively.(a)
If for eachx,y∈E(N(λ0)), F(x,y,·) is m1·γ1-Hölder continuous with respect to e∈intC at μ0∈M, then for any ξ∈Be*, the function φξ(x,y,·)=infz∈F(x,y,·)ξ(z) is m1·γ1-Hölder continuous at μ0.(b)
If for eachx∈E(N(λ0)) and μ∈N(E(μ0)), F(x,·,μ) is m2·γ2-Hölder continuous with respect to e∈intC on E(N(λ0)), then for each ξ∈Be*, φξ(x,·,μ)=infz∈F(x,·,μ)ξ(z) is also m2·γ2-Hölder continuous on E(N(λ0)).Proof.
(a) By assumption, there exists a neighborhoodN(μ0) of μ0, such that for all μ1,μ2∈N(μ0),forallx,y∈E(N(λ0)):x≠y,
(3.1)F(x,y,μ1)⊂F(x,y,μ2)+m1dγ1(μ1,μ2)[-e,e].
So, for anyz1∈F(x,y,μ1), there exist z2∈F(x,y,μ2) and e0∈[-e,e] such that
(3.2)z1=z2+m1dγ1(μ1,μ2)e0.
Then, by the linearity ofξ, we have
(3.3)ξ(z1)-ξ(z2)=m1dγ1(μ1,μ2)ξ(e0).
It follows fromξ(e)=1,e0∈[-e,e], and the structure of [-e,e] that
(3.4)ξ(e0)≥-1.
Therefore, (3.3) and (3.4) together yield that
(3.5)-m1dγ1(μ1,μ2)≤ξ(z1)-ξ(z2).
Sincez1 is arbitrary and ξ(z2)≥infz∈F(x,y,μ2)ξ(z), we have
(3.6)-m1dγ1(μ1,μ2)≤inf⁡z∈F(x,y,μ1)ξ(z)-inf⁡z∈F(x,y,μ2)ξ(z).
Due to the symmetry betweenμ1 and μ2, the same estimate is also valid, that is,
(3.7)-m1dγ1(μ1,μ2)≤inf⁡z∈F(x,y,μ2)ξ(z)-inf⁡z∈F(x,y,μ1)ξ(z).
Thus, it follows (3.6) and (3.7) that
(3.8)|inf⁡z∈F(x,y,μ1)ξ(z)-inf⁡z∈F(x,y,μ2)ξ(z)|=|φξ(x,y,μ1)-φξ(x,y,μ2)|≤m1dγ1(μ1,μ2)
and the proof is completed.
(b) As the proof of(b) is similar to (a), we omit it. Then the proof is completed.Lemma 3.2.
If for eachμ∈N(μ0) and each x∈E(N(λ0)), F(x,·,μ) is C-like convex on E(N(λ0)), that is, F(x,E(N(λ0)),μ)+C is a convex set, then
(3.9)S(λ,μ)=∪ξ∈C*∖0⁡Sξ(λ,μ)=∪ξ∈Be*⁡Sξ(λ,μ).Proof.
In a similar way to the proof of Lemma  3.1 in [8], with suitable modifications, we can obtain the conclusion.Theorem 3.3.
Assume that for eachξ∈Be*, the ξ-solution set for (PGVQEP) exists in a neighborhood N(λ0)×N(μ0) of the considered point (λ0,μ0)∈Λ×M. Assume further that the following conditions hold.(i)
K(·,·) is (ℓ1·α1,ℓ2·α2)-Hölder continuous in E(N(λ0))×N(μ0).(ii)
For eachx,y∈E(N(λ0)), F(x,y,·) is m1·γ1-Hölder continuous with respect to e∈intC at μ0∈M.(iii)
For eachx∈E(N(λ0)) and μ∈N(E(μ0)), F(x,·,μ) is m2·γ2-Hölder continuous with respect to e∈intC on E(N(λ0)).(iv)
forallξ∈Be*,μ∈N(μ0),forallx,y∈E(N(λ0)):x≠y, there exists two constants h>0 and β>0 such that
(3.10)hdβ(x,y)≤d(inf⁡z∈F(x,y,μ)ξ(z),R+)+d(inf⁡z∈F(y,x,μ)ξ(z),R+).(v)
α1γ2=β and h>2m2ℓ1γ2.Then, for any ξ¯∈Be*, there exists open neighborhoods N(ξ¯) of ξ¯, Nξ¯(λ0) of λ0 and Nξ¯(μ0) of μ0, such that the ξ-solution set Sξ(·,·) on N(ξ¯)×Nξ¯(λ0)×Nξ¯(μ0) satisfies the following Hölder condition: forallξ∈N(ξ¯),forall(λ1,μ1),(λ2,μ2)∈Nξ¯(λ0)×Nξ¯(μ0),
(3.11)d(xξ(λ1,μ1),xξ(λ2,μ2))≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2)+(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2),
where xξ(λi,μi)∈Sξ(λi,μi),i=1,2.Proof.
Let(λ1,μ1),(λ2,μ2)∈Nξ¯(λ0)×Nξ¯(μ0) be arbitrarily given. For all ξ∈Be*, x,y∈X, and μ∈M, we set φξ(x,y,·)∶=infz∈F(x,y,·)ξ(z) for the sake of convenient statement in the sequel. We prove that (3.11) holds by the following three steps.Step 1.
We first show that, for allxξ(λ1,μ1)∈Sξ(λ1,μ1), for all xξ(λ1,μ2)∈Sξ(λ1,μ2),
(3.12)d(xξ(λ1,μ1),xξ(λ1,μ2))≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2).
Obviously, if xξ(λ1,μ1)=xξ(λ1,μ2), we have that (3.12) holds. So we suppose xξ(λ1,μ1)≠xξ(λ1,μ2). Since xξ(λ1,μ1)∈K(xξ(λ1,μ1),λ1),xξ(λ1,μ2)∈K(xξ(λ1,μ2),λ1), and by the Hölder continuity of K(·,λ1), there exists x1∈K(xξ(λ1,μ1),λ1) and x2∈K(xξ(λ1,μ2),λ1) such that
(3.13)d(xξ(λ1,μ1),x2)≤l1dα1(xξ(λ1,μ1),xξ(λ1,μ2)),d(xξ(λ1,μ2),x1)≤l1dα1(xξ(λ1,μ1),xξ(λ1,μ2)).
Sincexξ(λ1,μ1),xξ(λ1,μ2) are ξ-solutions to (PGVQEP) at parameters (λ1,μ1),(λ1,μ2), respectively, we obtain
(3.14)φξ(xξ(λ1,μ1),x1,μ1)≥0,φξ(xξ(λ1,μ2),x2,μ2)≥0.
By virtue of (iv), we get that
(3.15)hdβ(xξ(λ1,μ1),xξ(λ1,μ2))≤d(ϕξ(xξ(λ1,μ2),xξ(λ1,μ1),μ1),R+)+d(ϕξ(xξ(λ1,μ1),xξ(λ1,μ2),μ1),R+),
which together with (3.14) yields that
(3.16)hdβ(xξ(λ1,μ1),xξ(λ1,μ2))≤|ϕξ(xξ(λ1,μ2),xξ(λ1,μ1),μ1)-ϕξ(xξ(λ1,μ2),x2,μ2)|+|ϕξ(xξ(λ1,μ1),xξ(λ1,μ2),μ1)-ϕξ(xξ(λ1,μ1),x1,μ1)|≤|ϕξ(xξ(λ1,μ2),xξ(λ1,μ1),μ1)-ϕξ(xξ(λ1,μ2),xξ(λ1,μ1),μ2)|+|ϕξ(xξ(λ1,μ2),xξ(λ1,μ1),μ2)-ϕξ(xξ(λ1,μ2),x2,μ2)|+|ϕξ(xξ(λ1,μ1),xξ(λ1,μ2),μ1)-ϕξ(xξ(λ1,μ1),x1,μ1)|.
Then, from Lemma 3.1, (3.13), we have
(3.17)hdβ(xξ(λ1,μ1),xξ(λ1,μ2))≤m1dγ1(μ1,μ2)+m2dγ2(xξ(λ1,μ2),x1)+m2dγ2(xξ(λ1,μ1),x2)≤m1dγ1(μ1,μ2)+2m2lγ2dα1γ2(xξ(λ1,μ1),xξ(λ1,μ2)).
The assumption (v) yields that
(3.18)d(xξ(λ1,μ1),xξ(λ1,μ2))≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2).
Hence, we have that (3.12) holds.Step 2.
Now we show that, for allxξ(λ1,μ2)∈Sξ(λ1,μ2), for all xξ(λ2,μ2)∈Sξ(λ2,μ2),
(3.19)d(xξ(λ1,μ2),xξ(λ2,μ2))≤(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2).
Obviously, we only need to prove that (3.19) holds when xξ(λ1,μ2)≠xξ(λ2,μ2). By virtue of assumption (i), there exists x1′∈K(xξ(λ2,μ2),λ1) and x2′∈K(xξ(λ1,μ2),λ2) such that
(3.20)d(xξ(λ2,μ2),x1′)≤l2dα2(λ1,λ2),d(xξ(λ1,μ2),x2′)≤l2dα2(λ1,λ2).
By the Hölder continuity ofK(·,·), there exists x1′′∈K(xξ(λ1,μ2),λ1) and x2′′∈K(xξ(λ2,μ2),λ2) such that
(3.21)d(x1′,x1′′)≤l1dα1(xξ(λ1,μ2),xξ(λ2,μ2)),d(x2′,x2′′)≤l1dα1(xξ(λ1,μ2),xξ(λ2,μ2)).
From the definition of ξ-solution for (PGVQEP), we have
(3.22)ϕξ(xξ(λ1,μ2),x1′′,μ2)≥0,ϕξ(xξ(λ2,μ2),x2′′,μ2)≥0.
From assumptions (ii)–(iv), (3.22), and Lemma 3.1, we have
(3.23)hdβ(xξ(λ1,μ2),xξ(λ2,μ2))≤d(ϕξ(xξ(λ1,μ2),xξ(λ2,μ2),μ2),R+)+d(ϕξ(xξ(λ2,μ2),xξ(λ1,μ2),μ2),R+)≤d(ϕξ(xξ(λ1,μ2),x(λ2,μ2),μ2)-ϕξ(xξ(λ1,μ2),x1′′,μ2))+d(ϕξ(xξ(λ2,μ2),xξ(λ1,μ2),μ2)-ϕξ(xξ(λ2,μ2),x2′′,μ2))≤d(ϕξ(xξ(λ1,μ2),x(λ2,μ2),μ2)-ϕξ(xξ(λ1,μ2),x1′,μ2))+d(ϕξ(xξ(λ1,μ2),x1′,μ2)-ϕξ(xξ(λ1,μ2),x1′′,μ2))+d(ϕξ(xξ(λ2,μ2),xξ(λ1,μ2),μ2)-ϕξ(xξ(λ2,μ2),x2′,μ2))+d(ϕξ(xξ(λ2,μ2),x2′,μ2)-ϕξ(xξ(λ2,μ2),x2′′,μ2))≤m2dγ2(x(λ2,μ2),x1′)+m2dγ2(x1′,x1′′)+m2dγ2(x(λ1,μ2),x2′)+m2dγ2(x2′,x2′′).
By virtue of (3.20)–(3.21) and (3.23), we can get
(3.24)hdβ(xξ(λ1,μ2),xξ(λ2,μ2))≤m2l2γ2dα2γ2(λ1,λ2)+m2l1γ2dα1γ2(xξ(λ1,μ2),xξ(λ2,μ2))+m2l2γ2dα2γ2(λ1,λ2)+m2l1γ2dα1γ2(xξ(λ1,μ2),xξ(λ2,μ2)).
Therefore, it follows from (v) that
(3.25)d(xξ(λ1,μ2),xξ(λ2,μ2))≤(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2)
and the conclusion (3.19) holds.Step 3.
Finally, by the arbitrariness ofxξ(λ1,μ1)∈Sξ(λ1,μ1),xξ(λ1,μ2)∈Sξ(λ1,μ2),xξ(λ2,μ2)∈Sξ(λ2,μ2), (3.12) and (3.19), we can easily get that
(3.26)d(xξ(λ1,μ1),xξ(λ2,μ2))≤d(xξ(λ1,μ1),xξ(λ1,μ2))+d(xξ(λ1,μ2),xξ(λ2,μ2))≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2)+(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2)
and the conclusion (3.11) holds. This completes the proof.Remark 3.4.
Theorem3.3 generalizes Lemma 3.3 in S. J. Li and X. B. Li [25] from vector-valued version to set valued version. Moreover, the assumption (H4) of Lemma  3.3 in [25] is removed.Now, we give an example to illustrate that Theorem3.3 is applicable under the case that the mapping F is set valued.Example 3.5.
LetX=Y=R,Λ=M=[0,1],C=ℝ+ and e=3/2∈intC. Let K:X×M⇉Y be defined by
(3.27)K(x,λ)=[λ2+x16,1]
and F:X×X×M⇉Y a set-valued mapping defined by
(3.28)F(x,y,λ)=[(1+λ)(x+12)(y-x),28-2x3/2].
Then, E(λ)=[λ2/15,1]. Consider that λ0=0.5 and N(λ0)=Λ. Direct computation shows that E(Λ)=E(N(λ0))=[0,1].It can be checked thatK(·,·) is ((1/16)·1,(3/2)·1)-Hölder continuous in E(N(λ0))×N(μ0); for all x,y∈E(N(λ0)), F(x,y,·) is 62.1-Hölder continuous with respect to e=3/2∈intC at λ0∈M; for each x∈E(N(λ0)) and λ∈N(E(λ0)), F(x,·,λ) is 3.1-Hölder continuous with respect to e∈intC on E(N(λ0)). Here ℓ1=1/16,α1=1,ℓ2=3/2,α2=1,m1=62,γ1=1,m2=3,γ2=1. Take β=1 and h=1/2, for any ξ∈Be* and for all x,y∈E(N(λ0)):x≠y, we have
(3.29)hdβ(x,y)≤d(inf⁡z∈F(x,y,μ)ξ(z),R+)+d(inf⁡z∈F(y,x,μ)ξ(z),R+)
and also have α1γ2=β and h>2m2ℓ1γ2=3/8. Hence, all assumptions of Theorem 3.3 hold, and thus it is valid.Theorem 3.6.
Assume that for eachξ∈Be*, the ξ-solution set for (PGVQEP) exists in a neighborhood N(λ0)×N(μ0) of the considered point (λ0,μ0)∈Λ×M. Assume further that the following conditions hold:(i)
K(·,·) is (ℓ1·α1,ℓ2·α2)-Hölder continuous in E(N(λ0))×N(μ0);(ii)
for eachx,y∈E(N(λ0)), F(x,y,·) is m1·γ1-Hölder continuous with respect to e∈intC at μ0∈M;(iii)
for eachx∈E(N(λ0)) and μ∈N(E(μ0)), F(x,·,μ) is m2·γ2-Hölder continuous with respect to e∈intC on E(N(λ0));(iv)
forallξ∈Be*,μ∈N(μ0),forallx,y∈E(N(λ0))(x≠y), there exist two constants h>0 and β>0 such that
(3.30)hdβ(x,y)≤d(inf⁡z∈F(x,y,μ)ξ(z),R+)+d(inf⁡z∈F(y,x,μ)ξ(z),R+);(v)
forallx∈E(N(λ0)),forallμ∈N(μ0), F(x,·,μ) is C-like convex on E(N(λ0));(vi)
α1γ2=β and h>2m2ℓ1γ2.Then there exist neighborhoods Ñ(λ0) of λ0 and Ñ(μ0) of μ0, such that the solution set S(·,·) on Ñ(λ0)×Ñ(μ0) is nonempty and satisfies the following Hölder continuous condition, for all (λ1,μ1),(λ2,μ2)∈Ñ(λ0)×Ñ(μ0):
(3.31)S(λ1,μ1)⊂S(λ2,μ2)+((m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2)+(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2))B(0,1).Proof.
Since the system of{N′(ξ¯)}ξ¯∈Be*, which are given by Theorem 3.3, is an open covering of the weak* compact set Be*, there exist a finite number of points (ξi)(i=1,2,…,n.) from Be* such that
(3.32)Be*⊂⋃i=1nN′(ξi).
Hence, let Ñ(λ0)=⋂i=1nN′ξi(λ0) and Ñ(μ0)=⋂i=1nN′ξi(μ0). Then Ñ(λ0) and Ñ(μ0) are desired neighborhoods of λ0 and μ0, respectively. Indeed, let (λ,μ)∈Ñ(λ0)×Ñ(μ0) be given arbitrarily. For any ξ∈Be*, by virtue of (3.32), there exists i0∈{1,2,…,n} such that ξ∈N′(ξi0). From the construction of the neighborhoods Ñ(λ0) and Ñ(μ0), one has
(3.33)(λ,μ)∈Nξi0′(λ0)×Nξi0′(μ0).
Then, from the assumption of existence for ξ-solution set and Lemma 3.2, S(λ,μ)=⋃ξ∈Be*Sξ(λ,μ) is nonempty.
Now, we show that (3.31) holds. Indeed, taking any (λ1,μ1),(λ2,μ2)∈Ñ(λ0)×Ñ(μ0), we need to show that for any x1∈S(λ1,μ1), there exists x2∈S(λ2,μ2) satisfying
(3.34)d(x1,x2)≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2)+(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2).
Sincex1∈S(λ1,μ1)=⋃ξ∈Be*Sξ(λ1,μ1), there exists ξ̂∈Be* such that
(3.35)x1=xξ̂(λ1,μ1)∈Sξ̂(λ1,μ1).
It follows from (3.32) that there exists i0∈{1,2,…,n} such that ξ̂∈N′(ξi0). Thus, by the construction of the neighborhoods Ñ(λ0) and Ñ(μ0), we have
(3.36)(λ1,μ1),(λ2,μ2)∈Nξi0(λ0)×Nξi0(μ0).
Obviously, thanks to Theorem 3.3, we have
(3.37)d(xξ̂(λ1,μ1),xξ̂(λ2,μ2))≤(m1h-2m2l1γ2)1/βdγ1/β(μ1,μ2)+(2m2l2γ2h-2m2l1γ2)1/βdα2γ2/β(λ1,λ2).
Let x2=xξ̂(λ2,μ2). Then, (3.34) holds, and the proof is complete.Remark 3.7.
Theorem3.6 generalizes, and improves the corresponding results of S. J. Li and X. B. Li [25] in the following three aspects.(i)
The vector-valued mappingF(x,y,μ) is extended to set-valued, and the parametric vector equilibrium problem is extended to the parametric vector quasiequilibrium problem.(ii)
The assumption (H4) of Theorem 3.1 in [25] is removed.(iii)
TheC-convexity of F(x,·,μ) (see Theorem 3.1 in [25]) is extended to C-convexlikeness.In addition, it is easy to see that the assumption (iv) of Theorem3.6 is different form the assumption (H1) of Theorem 3.1 in S. J. Li and X. B. Li [25].Moreover, we also can see that the obtained result extends the ones of [23]. Now, we give the following example to illustrate the case.Example 3.8.
LetX=Y=ℝ,Λ=M=[0,1],C=ℝ+, and e=2/2∈intC. Let K:X×M⇉Y be defined by K(x,λ)=[λ2,1], and let F:X×X×M⇉Y be a set-valued mapping defined by
(3.38)F(x,y,λ)=[(34+2λ)(y+3)(x-y),20-|x|1/2].
Consider that λ0=0.5 and N(λ0)=Λ. Then, E(λ)=[λ2,1] and E(Λ)=E(N(λ0))=[0,1].Obviously,K(·,·) is (0.1,2·1)-Hölder continuous in E(N(λ0))×N(μ0); for all x,y∈E(N(λ0)), F(x,y,·) is 72·1-Hölder continuous with respect to e=2/2∈intC at λ0∈M; for each x∈E(N(λ0)) and λ∈N(E(λ0)), F(x,·,λ) is 95·1-Hölder continuous with respect to e∈intC on E(N(λ0)). Here ℓ1=0,α1=1,ℓ2=2,α2=1,m1=72,γ1=1,m2=95,γ2=1. Take β=1 and h=3/4, for any ξ∈Be* and for all x,y∈E(N(λ0))(x≠y), we have(3.39)hdβ(x,y)≤d(inf⁡z∈F(x,y,μ)ξ(z),R+)+d(inf⁡z∈F(y,x,μ)ξ(z),R+)
and also have α1γ2=β and h=3/4>2m2ℓ1γ2. Therefore, all assumptions of Theorem 3.3 hold, and thus it is applicable.However, the assumption (ii) of Theorem 3.1 (or (ii′) of Theorem 4.1) in [23] does not hold. In fact, for any λ∈Λ, for any h>0 and β>0, there exists y0=0∈E(N(λ0))∖Sξ(λ,μ) such that(3.40)F(y0,x¯,λ)+hB(0,dβ(x¯,y0))=[-(34+2λ)(x¯+3)x¯,20-|x¯|1/2]+hB(0,dβ(0,x¯))⊈-R+
for all x¯∈Sξ(λ,μ). Thus, Theorems 3.1 and 4.1 in Li et al. [23] are not applicable.



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*Source: 236413-2012-02-15.xml*