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On the Hölder continuity of solution maps to parametric generalized vector quasiequilibrium problems via nonlinear scalarization
Journal of Inequalities and Applications volume 2014, Article number: 425 (2014)
Abstract
In this paper, by using a nonlinear scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasiequilibrium problem with setvalued mappings. The results are different from the recent ones in the literature.
1 Introduction
The generalized vector quasiequilibrium problem is a unified model of several problems, namely generalized vector quasivariational inequalities, vector quasioptimization problems, traffic network problems, fixed point and coincidence point problems, etc. (see, for example, [1, 2] and the references therein). It is well known that the stability analysis of a solution mapping for equilibrium problems is an important topic in optimization theory and applications. Stability may be understood as lower or upper semicontinuity, continuity, and Lipschitz or Hölder continuity. There have been many papers to discuss the stability of solution mapping for equilibrium problems when they are perturbed by parameters (also known the parametric (generalized) equilibrium problems). Last decade, many authors intensively studied the sufficient conditions of upper (lower) semicontinuity of various solution mappings for parametric (generalized) equilibrium problems, see [3–10]. Let us begin now, Yen [11] obtained the Hölder continuity of the unique solution of a classic perturbed variational inequality by the metric projection method. Mansour and Riahi [12] proved the Hölder continuity of the unique solution for a parametric equilibrium problem under the concepts of strong monotonicity and Hölder continuity. Bianchi and Pini [13] introduced the concept of strong pseudomonotonicity and got the Hölder continuity of the unique solution of a parametric equilibrium problem. Anh and Khanh [14] generalized the main results of [13] to two classes of perturbed generalized equilibrium problems with setvalued mappings. Anh and Khanh [15] further discussed the uniqueness and Hölder continuity of the solutions for perturbed equilibrium problems with setvalued mappings. Anh and Khanh [16] extended the results of [15] to the case of perturbed quasiequilibrium problems with setvalued mappings and obtained the Hölder continuity of the unique solutions. Li et al. [17] introduced an assumption, which is weaker than the corresponding ones of [13, 14], and established the Hölder continuity of the setvalued solution mappings for two classes of parametric generalized vector quasiequilibrium problems in general metric spaces. Li et al. [18] extended the results of [17] to perturbed generalized vector quasiequilibrium problems.
Among many approaches for dealing with the lower semicontinuity, continuity and Hölder continuity of the solution mapping for a parametric vector equilibrium problem in general metric spaces, the scalarization method is of considerable interest. The classical scalarization method using linear functionals has been already used for studying the lower semicontinuity of the solution mapping [19–21] and the Hölder continuity [22] of the solution mapping to parametric vector equilibrium problems. Wang et al. [23] established the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. Recently, by using this method, Peng [24] established the sufficient conditions for the Hölder continuity of the solution mapping to a parametric generalized vector quasiequilibrium problem with setvalued mappings.
On the other hand, a useful approach for analyzing a vector optimization problem is to reduce it to a scalar optimization problem. Nonlinear scalarization functions play an important role in this reduction in the context of nonconvex vector optimization problems. The nonlinear scalarization function {\xi}_{q}, commonly known as the Gerstewitz function in the theory of vector optimization [25, 26], has been also used to study the lower semicontinuity of the setvalued solution mapping to a parametric vector variational inequality [27]. Using this method, Bianchi and Pini [28] obtained the Hölder continuity of the singlevalued solution mapping to a parametric vector equilibrium problem. Recently, Chen and Li [29] studied Hölder continuity of the solution mapping for both setvalued and singlevalued cases to parametric vector equilibrium problems. The key role in their paper is a globally Lipschitz property of the Gerstewitz function. Very recently, by using the idea in [29], Chen [30] obtained Hölder continuity of the unique solution to a parametric vector quasiequilibrium problem based on nonlinear scalarization approach under three different kinds of monotonicity hypotheses. It is natural to raise and give an answer to the following question.
Question Can one establish the Hölder continuity of a solution mapping to the parametric generalized vector quasiequilibrium problem with setvalued mappings by using a nonlinear scalarization method?
Motivated and inspired by Peng [24] and Chen [30] and research going on in this direction, in this paper we aim to give positive answers to the above question. We first establish the sufficient conditions which guarantee the Hölder continuity of a solution mapping to the parametric generalized vector quasiequilibrium problem with setvalued mappings by using a nonlinear scalarization method. We further study several kinds of the monotonicity conditions to obtain the Hölder continuity of the solution mapping. The main results of this paper are different from the corresponding results in Peng [24] and Chen [30]. These results improve the corresponding ones in recent literature.
The structure of the paper is as follows. Section 2 presents the parametric generalized vector quasiequilibrium problem and materials used in the rest of this paper. We establish, in Section 3, a sufficient condition for the Hölder continuity of the solution mapping to a parametric generalized vector quasiequilibrium problem.
2 Preliminaries
Throughout the paper, unless otherwise specified, we denote by \parallel \cdot \parallel and d(\cdot ,\cdot ) the norm and the metric on a normed space and a metric space, respectively. A closed ball with center 0\in X and radius \delta >0 is denoted by B(0,\delta ). We always consider X, Λ, M as metric spaces, and Y as a linear normed space with its topological dual space {Y}^{\ast}. For any {y}^{\ast}\in {Y}^{\ast}, we define \parallel {y}^{\ast}\parallel :=sup\{\parallel \u3008{y}^{\ast},y\u3009\parallel :\parallel y\parallel =1\}, where \u3008{y}^{\ast},y\u3009 denotes the value of {y}^{\ast} at y. Let C\subset Y be a pointed, closed and convex cone with intC\ne \mathrm{\varnothing}, where intC stands for the interior of C. Let
be the dual cone of C. Since intC\ne \mathrm{\varnothing}, the dual cone {C}^{\ast} of C has a weak* compact base. Let e\in intC. Then
is a weak*compact base of {C}^{\ast}. Clearly, {C}^{q} is a weak^{∗}compact base of {C}^{\ast}, that is, {C}^{q} is convex and weak^{∗}compact such that 0\notin {C}^{q} and {C}^{\ast}={\bigcup}_{t\ge 0}t{C}^{q}.
Let q\in intC, the nonlinear scalarization function [25, 26]{\xi}_{q}:Y\to \mathbb{R} is defined by
It is well known that {\xi}_{q} is a continuous, positively homogeneous, subadditive and convex function on Y, and it is monotone (that is, {y}_{2}{y}_{1}\in C\Rightarrow {\xi}_{q}({y}_{1})\le {\xi}_{q}({y}_{2})) and strictly monotone (that is, {y}_{2}{y}_{1}\in intC\Rightarrow {\xi}_{q}({y}_{1})<{\xi}_{q}({y}_{2})) (see [25, 26]). In case, Y={R}^{l}, C={R}_{+}^{l} and q=(1,1,\dots ,1)\in int{R}_{+}^{l}, the nonlinear scalarization function can be expressed in the following equivalent form [[25], Corollary 1.46]:
Lemma 2.1 [[25], Proposition 1.43]
For any fixed q\in intC, y\in Y and r\in \mathbb{R},

(i)
{\xi}_{q}<r\iff y\in rqintC (that is, {\xi}_{q}(y)\ge r\iff y\notin rqintC);

(ii)
{\xi}_{q}(y)\le r\iff y\in rqC;

(iii)
{\xi}_{q}(y)=r\iff y\in rq\partial C, where ∂C denotes the boundary of C;

(iv)
{\xi}_{q}(rq)=r.
The property (i) of Lemma 2.1 plays an essential role in scalarization. From the definition of {\xi}_{q}, property (iv) in Lemma 2.1 could be strengthened as
For any q\in intC, the set {C}^{q} defined by
is a weak^{∗}compact set of {Y}^{\ast} (see [[19], Lemma 5.1]). The following equivalent form of {\xi}_{q} can be deduced from [[31], Corollary 2.1] or [[32], Proposition 2.2] ([[25], Proposition 1.53]).
Proposition 2.2 [[30], Proposition 2.2]
Let q\in intC. Then, for y\in Y,
Proposition 2.3 [[30], Proposition 2.3]
{\xi}_{q} is Lipschitz on Y, and its Lipschitz constant is
The following example can be found in [[30], Example 2.1].
Example 2.4

(i)
If Y=\mathbb{R} and C={\mathbb{R}}_{+}, then the Lipschitz constant of {\xi}_{q} is L=\frac{1}{q} (q>0). Indeed, {\xi}_{q}(x){\xi}_{q}(y)=\frac{1}{q}xy for all x,y\in \mathbb{R}.

(ii)
If Y={\mathbb{R}}^{2} and C=\{({y}_{1},{y}_{2})\in {\mathbb{R}}^{2}:\frac{1}{4}{y}_{1}\le {y}_{2}\le 2{y}_{1}\}. Take q=(2,3)\in intC, then
{C}^{q}:=\{({y}_{1},{y}_{2})\in \mathbb{R}:2{y}_{1}+3{y}_{2}=1,{y}_{1}\in [0.1,2]\},
and the Lipschitz constant is L={sup}_{{y}^{\ast}\in {C}^{q}}\parallel {y}^{\ast}\parallel =\parallel (2,1)\parallel =\sqrt{5}. Hence,
Now we recall some basic definitions and their properties which will be used in the sequel.
Definition 2.5 (Classical notion)
Let l\ge 0 and \alpha >0. A setvalued mapping G:\mathrm{\Lambda}\to {2}^{X} is said to be l\cdot \alphaHölder continuous at {\lambda}_{0} on a neighborhood N({\lambda}_{0}) of {\lambda}_{0} if and only if
When X is a normed space, we say that the vectorvalued mapping g:\mathrm{\Lambda}\to X is l\cdot \alphaHölder continuous at {\lambda}_{0} on a neighborhood N({\lambda}_{0}) of {\lambda}_{0} iff
Definition 2.6 Let {l}_{1},{l}_{2}\ge 0 and {\alpha}_{1},{\alpha}_{2}>0. A setvalued mapping G:X\times \mathrm{\Lambda}\to {2}^{X} is said to be ({l}_{1}\cdot {\alpha}_{1},{l}_{2}\cdot {\alpha}_{2})Hölder continuous at {x}_{0}, {\lambda}_{0} on neighborhoods N({x}_{0}) and N({\lambda}_{0}) of {x}_{0} and {\lambda}_{0} if and only if
for all {x}_{1},{x}_{2}\in N({x}_{0}), \mathrm{\forall}{\lambda}_{1},{\lambda}_{2}\in N({\lambda}_{0}).
3 Main results
By using a nonlinear scalarization technique, we present the sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasiequilibrium problem.
Let N({\lambda}_{0})\subset \mathrm{\Lambda} and N({\mu}_{0})\subset M be neighborhoods of {\lambda}_{0} and {\mu}_{0}, respectively, and let K:X\times \mathrm{\Lambda}\to {2}^{X} and F:X\times X\times M\to {2}^{Y} be setvalued mappings. For each \lambda \in N({\lambda}_{0}) and \mu \in N({\mu}_{0}), we consider the following parametric generalized vector quasiequilibrium problem (PGVQEP):
Find {x}_{0}\in K({x}_{0},\lambda ) such that
For each \lambda \in N({\lambda}_{0}) and \mu \in N({\mu}_{0}), let
The weak solution set of (6) is denoted by
For each \lambda \in N({\lambda}_{0}), \mu \in N({\mu}_{0}) and fixed q\in intC, the {\xi}_{q}solution set of (6) is denoted by
We first establish the following lemmas which will be used in the sequel.
Lemma 3.1 For each \lambda \in N({\lambda}_{0}), \mu \in N({\mu}_{0}) and fixed q\in intC,
Proof Let \lambda \in N({\lambda}_{0}), \mu \in N({\mu}_{0}) and fixed q\in intC. For any x\in {S}_{W}(\lambda ,\mu ), we have
Therefore, for each y\in K(x,\lambda ) and each z\in F(x,y,\mu ), we have
By Lemma 2.1(i), we conclude that {\xi}_{q}(z)\ge 0. Since z is arbitrary, we have
which gives that {S}_{W}(\lambda ,\mu )\subseteq S({\xi}_{q},\lambda ,\mu ).
On the other hand, for each x\in S({\xi}_{q},\lambda ,\mu ), we have that
Thus, for each y\in K(x,\lambda ) and each z\in F(x,y,\mu ), we have that {\xi}_{q}(z)\ge 0. By Lemma 2.1(i), we can obtain z\notin intC. Therefore, we have z\in Y\mathrm{\setminus}(intC), which implies that
Hence, S({\xi}_{q},\lambda ,\mu )\subseteq {S}_{W}(\lambda ,\mu ). The proof is completed. □
Lemma 3.2 Suppose that N({\lambda}_{0}) and N({\mu}_{0}) are the given neighborhoods of {\lambda}_{0} and {\mu}_{0}, respectively.

(a)
If for each x,y\in E(N({\lambda}_{0})), F(x,y,\cdot ) is {m}_{1}\cdot {\gamma}_{1}Hölder continuous at {\mu}_{0}\in M, then for any fixed q\in intC, the function
{\psi}_{{\xi}_{q}}(x,y,\cdot )=\underset{z\in F(x,y,\cdot )}{inf}{\xi}_{q}(z)is L{m}_{1}\cdot {\gamma}_{1}Hölder continuous at {\mu}_{0}.

(b)
If for each x\in E(N({\lambda}_{0})) and \mu \in N(E({\mu}_{0})), F(x,\cdot ,\mu ) is {m}_{2}\cdot {\gamma}_{2}Hölder continuous on E(N({\lambda}_{0})), then for any fixed q\in intC, the function
{\psi}_{{\xi}_{q}}(x,\cdot ,\mu )=\underset{z\in F(x,\cdot ,\mu )}{inf}{\xi}_{q}(z)is L{m}_{2}\cdot {\gamma}_{2}Hölder continuous on E(N({\lambda}_{0})).
Proof (a) Let x,y\in E(N({\lambda}_{0})). The {m}_{1}\cdot {\gamma}_{1}Hölder continuity of F(x,y,\cdot ) implies that there exists a neighborhood N({\mu}_{0}) of {\mu}_{0} such that for all {\mu}_{1},{\mu}_{2}\in N({\mu}_{0}),
So, for any {z}_{1}\in F(x,y,{\mu}_{1}), there exist {z}_{2}\in F(x,y,{\mu}_{2}) and e\in {B}_{Y} such that
By using Proposition 2.3, we obtain
which gives that
Since {z}_{1} is arbitrary and {\xi}_{q}({z}_{2})\ge {inf}_{z\in F(x,y,{\mu}_{2})}{\xi}_{q}(z), we have
Applying the symmetry between {\mu}_{1} and {\mu}_{2}, we arrive at
It follows from the last two inequalities that
Therefore, we conclude that {\psi}_{{\xi}_{q}}(x,y,\cdot )={inf}_{z\in F(x,y,\cdot )}{\xi}_{q}(z) is L{m}_{1}\cdot {\gamma}_{1}Hölder continuous at {\mu}_{0}.

(b)
It follows by a similar argument as in part (a). The proof is completed. □
Now, by using the nonlinear scalarization technique, we propose some sufficient conditions for Hölder continuity of the solution mapping for (PGVQEP).
Theorem 3.3 For each fixed q\in intC, let S({\xi}_{q},\lambda ,\mu ) be nonempty in a neighborhood N({\lambda}_{0})\times N({\mu}_{0}) of ({\lambda}_{0},{\mu}_{0})\in \mathrm{\Lambda}\times M. Assume that the following conditions hold.

(i)
K(\cdot ,\cdot ) is ({l}_{1}\cdot {\alpha}_{1},{l}_{2}\cdot {\alpha}_{2})Hölder continuous on E(N({\lambda}_{0}))\times N({\lambda}_{0});

(ii)
For each x,y\in E(N({\lambda}_{0})), F(x,y,\cdot ) is {m}_{1}\cdot {\gamma}_{1}Hölder continuous at {\mu}_{0}\in M;

(iii)
For each x\in E(N({\lambda}_{0})) and \mu \in N({\mu}_{0}), F(x,\cdot ,\mu ) is {m}_{2}\cdot {\gamma}_{2}Hölder continuous on E(N({\lambda}_{0}));

(iv)
F(\cdot ,\cdot ,\mu ) is h\cdot \betaHölder strongly monotone with respect to {\xi}_{q}, that is, there exist constants h>0, \beta >0 such that for every x,y\in E(N({\lambda}_{0})), x\ne y,
h{d}_{X}^{\beta}(x,y)\le d(\underset{z\in F(x,y,\mu )}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+})+d(\underset{z\in F(y,x,\mu )}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+}); 
(v)
\beta ={\alpha}_{1}{\gamma}_{2}, h>2{m}_{2}L{l}_{1}^{{\gamma}_{1}}, where L:={sup}_{\lambda \in {C}^{q}}\parallel \lambda \parallel \in [\frac{1}{\parallel q\parallel},+\mathrm{\infty}) is the Lipschitz constant of {\xi}_{q} on Y.
Then, for every (\lambda ,\mu )\in N({\lambda}_{0})\times N({\mu}_{0}), the solution x(\lambda ,\mu ) of (PVQGEP) is unique, and x(\lambda ,\mu ) as a function of λ and μ satisfies the Hölder condition: for all ({\lambda}_{1},{\mu}_{1}),({\lambda}_{2},{\mu}_{2})\in N({\lambda}_{0})\times N({\mu}_{0}),
where x({\lambda}_{i},{\mu}_{i})\in {S}_{W}({\lambda}_{i},{\mu}_{i}), i=1,2.
Proof Let ({\lambda}_{1},{\mu}_{1}),({\lambda}_{2},{\mu}_{2})\in N({\lambda}_{0})\times N({\mu}_{0}). The proof is divided into the following three steps based on the fact that
where x({\lambda}_{i},{\mu}_{i})\in {S}_{W}({\lambda}_{i},{\mu}_{i}), i=1,2.
Step 1: We prove that
for all x({\lambda}_{1},{\mu}_{1})\in {S}_{W}({\lambda}_{1},{\mu}_{1}) and x({\lambda}_{1},{\mu}_{2})\in {S}_{W}({\lambda}_{1},{\mu}_{2}).
If x({\lambda}_{1},{\mu}_{1})=x({\lambda}_{1},{\mu}_{2}), then we are done. So, we assume that x({\lambda}_{1},{\mu}_{1})\ne x({\lambda}_{1},{\mu}_{2}). Since x({\lambda}_{1},{\mu}_{1})\in K(x({\lambda}_{1},{\mu}_{1}),{\lambda}_{1}) and x({\lambda}_{1},{\mu}_{2})\in K(x({\lambda}_{1},{\mu}_{2}),{\lambda}_{1}), by the {l}_{1}\cdot {\alpha}_{1}Hölder continuity of K(\cdot ,{\lambda}_{1}), there exist {x}_{1}\in K(x({\lambda}_{1},{\mu}_{1}),{\lambda}_{1}) and {x}_{2}\in K(x({\lambda}_{1},{\mu}_{2}),{\lambda}_{1}) such that
and
Since x({\lambda}_{1},{\mu}_{1})\in {S}_{W}({\lambda}_{1},{\mu}_{1}) and x({\lambda}_{1},{\mu}_{2})\in {S}_{W}({\lambda}_{1},{\mu}_{2}), by Lemma 3.1, we obtain
and
By virtue of (iv), we have
By combining (12) and (13) with the last inequality, we have
Whence, assumption (iv) implies that
Step 2: We prove that
for all x({\lambda}_{1},{\mu}_{2})\in {S}_{W}({\lambda}_{1},{\mu}_{2}) and x({\lambda}_{2},{\mu}_{2})\in {S}_{W}({\lambda}_{2},{\mu}_{2}).
If x({\lambda}_{1},{\mu}_{2})=x({\lambda}_{2},{\mu}_{2}), then we are done. So, we assume that x({\lambda}_{1},{\mu}_{2})\ne x({\lambda}_{2},{\mu}_{2}). Since x({\lambda}_{1},{\mu}_{2})\in K(x({\lambda}_{1},{\mu}_{2}),{\lambda}_{1}) and x({\lambda}_{2},{\mu}_{2})\in K(x({\lambda}_{2},{\mu}_{2}),{\lambda}_{2}), by the {l}_{2}\cdot {\alpha}_{2}Hölder continuity of K(x({\lambda}_{1},{\mu}_{2}),\cdot ) and K(x({\lambda}_{2},{\mu}_{2}),\cdot ), there exist {x}_{1}^{\prime}\in K(x({\lambda}_{2},{\mu}_{2}),{\lambda}_{1}) and {x}_{2}^{\prime}\in K(x({\lambda}_{1},{\mu}_{2}),{\lambda}_{2}) such that
and
Again, by the Hölder continuity of K(\cdot ,\cdot ), there exist {x}_{1}^{\u2033}\in K(x({\lambda}_{1},{\mu}_{2}),{\lambda}_{1}) and {x}_{2}^{\u2033}\in K(x({\lambda}_{2},{\mu}_{2}),{\lambda}_{2}) such that
and
Since x({\lambda}_{1},{\mu}_{2})\in {S}_{W}({\lambda}_{1},{\mu}_{2}) and x({\lambda}_{2},{\mu}_{2})\in {S}_{W}({\lambda}_{2},{\mu}_{2}), by Lemma 3.1, we obtain the following:
and
By virtue of (iv), we have
By combining (20) and (21) with the last inequality, we have
By virtue of (16), (17), (18) and (19), we get
Whence, condition (v) implies that
Step 3: Let x({\lambda}_{1},{\mu}_{1})\in {S}_{W}({\lambda}_{1},{\mu}_{1}) and x({\lambda}_{2},{\mu}_{2})\in {S}_{W}({\lambda}_{2},{\mu}_{2}). It follows from (9) and (15) that
Thus,
Taking {\lambda}_{2}={\lambda}_{1} and {\mu}_{2}={\mu}_{1}, we see that the diameter of S({\lambda}_{1},{\mu}_{1}) is 0, that is, this set is a singleton \{x({\lambda}_{1},{\mu}_{1})\}. This implies that the (PGVQEP) has a unique solution in a neighborhood of ({\lambda}_{0},{\mu}_{0}). The proof is completed. □
Definition 3.4 Let F:X\times X\times M\to {2}^{Y} be a setvalued mapping. A setvalued mapping F(\cdot ,\cdot ,\mu )\mapsto {2}^{Y} is said to be

(A)
h\cdot \betaHölder strongly monotone with respect to {\xi}_{q} if there exist q\in intC and h>0, \beta >0 such that for every x,y\in E(N(\lambda )) with x\ne y,
\underset{z\in F(x,y,\mu )}{inf}{\xi}_{q}(z)+\underset{z\in F(y,x,\mu )}{inf}{\xi}_{q}(z)+h{d}_{X}^{\beta}(x,y)\le 0; 
(B)
h\cdot \betaHölder strongly pseudomonotone with respect to q\in intC and h>0, \beta >0 such that for every x,y\in E(N({\lambda}_{0})) with x\ne y,
z\notin intC,\phantom{\rule{1em}{0ex}}\mathrm{\exists}z\in F(x,y,\mu )\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{z}^{\prime}+h{d}_{X}^{\beta}(x,y)q\in C,\phantom{\rule{1em}{0ex}}\mathrm{\exists}{z}^{\prime}\in F(y,x,\mu ). 
(C)
quasimonotone on E(N({\lambda}_{0})) if \mathrm{\forall}x,y\in E(N({\lambda}_{0})) with x\ne y,
z\in intC,\phantom{\rule{1em}{0ex}}\mathrm{\exists}z\in F(x,y,\mu )\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{z}^{\prime}\notin intC,\phantom{\rule{1em}{0ex}}\mathrm{\exists}{z}^{\prime}\in F(y,x,\mu ).
The following proposition provides the relation among monotonicity conditions defined above.
Proposition 3.5

(i)
(A) ⇒ (iv).

(ii)
(B) and (C) ⇒ (iv).
Proof (i) From the definition of (A), we have
(ii) Assume that F satisfies definitions (B) and (C). We consider two cases.
Case 1. z\notin intC, \mathrm{\exists}z\in F(x,y,\mu ), then there exists {z}^{\prime}\in F(y,x,\mu ) such that {z}^{\prime}+h{d}_{X}^{\beta}(x,y)q\in C. From Lemma 2.1, we have
which implies that {inf}_{z\in F(y,x,\mu )}{\xi}_{q}(z)\le {\xi}_{q}({z}^{\prime})\le h{d}_{X}^{\beta}(x,y). Hence,
Case 2. z\in intC, \mathrm{\exists}z\in F(x,y,\mu ), then there exists {z}^{\prime}\in F(y,x,\mu ) such that z\notin intC. By a similar argument as in the previous case, we have the desired result. □
Remark 3.6 The converse of Proposition 3.5 does not hold in general, even in the special case X=Y=\mathbb{R} and C={\mathbb{R}}_{+}. See, for example, Examples 1.1 and 1.2 in [15]. Therefore, Theorem 3.3 still holds when condition (iv) is replaced by condition (A) or conditions (B) and (C). We can immediately obtain the following two theorems.
Theorem 3.7 Theorem 3.3 still holds when condition (iv) is replaced by condition (A).
Theorem 3.8 Theorem 3.3 still holds when condition (iv) is replaced by conditions (B) and (C).
Let f:X\times X\times M\to Y be a vectorvalued mapping. Then (PGVQEP) becomes the following parametric vector quasiequilibrium problem (PVQEP):
Find {x}_{0}\in K({x}_{0},\lambda ) such that
Remark 3.9 In the case of a vectorvalued mapping, condition (iv) in Theorem 3.3 and condition (ii^{′′}) coincide. Also, condition (A) and conditions (B) and (C) are the same as conditions (ii) and (ii′) in [30], respectively. It is obvious that Theorems 3.3, 3.7 and 3.8 extend Theorems 3.3, 3.1 and 3.2 in [30], respectively, in the case that the vectorvalued mapping f(\cdot ,\cdot ,\cdot ) is extended to a setvalued one.
4 Applications
Since the parametric generalized vector quasiequilibrium problem (PGVQEP) contains as special cases many optimizationrelated problems, including quasivariational inequalities, traffic equilibrium problems, quasioptimization problems, fixed point and coincidence point problems, complementarity problems, vector optimization, Nash equilibria, etc., we can derive from Theorem 3.3 a direct consequence for such special cases. We discuss now only some applications of our results.
4.1 Quasivariational inequalities
In this section, we assume that X is a normed space. Let K:X\times \mathrm{\Lambda}\rightrightarrows X and T:X\times M\rightrightarrows {B}^{\ast}(X,Y) be setvalued mappings, where {B}^{\ast}(X,Y) denotes the space of all bounded linear mappings of X into Y. Setting F(x,y,\mu )=\u3008T(x,\mu ),yx\u3009:={\bigcup}_{t\in T(x,\mu )}\u3008t,yx\u3009 in (6), we obtain parametric generalized vector quasivariational inequalities (PGVQVI) in the case of setvalued mappings as follows:
For each \lambda \in N({\lambda}_{0}) and \mu \in N({\mu}_{0}), let
The solution set of (25) is denoted by
For each \lambda \in N({\lambda}_{0}), \mu \in N({\mu}_{0}) and fixed q\in intC, the {\xi}_{q}solution set of (25) is
Theorem 4.1 Assume that for each fixed q\in intC, {S}_{QVI}^{V}({\xi}_{q},\lambda ,\mu ) is nonempty in a neighborhood N({\lambda}_{0})\times N({\mu}_{0}) of the considered point ({\lambda}_{0},{\mu}_{0})\in \mathrm{\Lambda}\times M. Assume further that the following conditions hold.

(i')
K(\cdot ,\cdot ) is ({l}_{1}\cdot {\alpha}_{1},{l}_{2}\cdot {\alpha}_{2})Hölder continuous on E(N({\lambda}_{0}))\times N({\lambda}_{0});

(ii')
For each x\in E(N({\lambda}_{0})), T(x,\cdot ) is {m}_{3}\cdot {\gamma}_{3}Hölder continuous at {\mu}_{0}\in M;

(iii')
T(\cdot ,\cdot ) is bounded in x\in E(N({\lambda}_{0})), and E(N({\lambda}_{0})) is bounded;

(iv')
T(\cdot ,\mu ) is h\cdot \betaHölder strongly monotone with respect to {\xi}_{q}, i.e., there exist constants h>0, \beta >0 such that for every x,y\in E(N({\lambda}_{0})): x\ne y,
h{\parallel xy\parallel}^{\beta}\le d(\underset{z\in \u3008T(x,\mu ),yx\u3009}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+})+d(\underset{z\in \u3008T(y,\mu ),xy\u3009}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+}); 
(v')
\beta ={\alpha}_{1}, h>2ML{l}_{1}^{{\gamma}_{1}}, where L:={sup}_{\lambda \in {C}^{q}}\parallel \lambda \parallel \in [\frac{1}{\parallel q\parallel},+\mathrm{\infty}) is the Lipschitz constant of {\xi}_{q} on Y.
Then, for every (\lambda ,\mu )\in N({\lambda}_{0})\times N({\mu}_{0}), the solution of (PGVQVI) is unique, x(\lambda ,\mu ), and this function satisfies the Hölder condition: for all ({\lambda}_{1},{\mu}_{1}),({\lambda}_{2},{\mu}_{2})\in N({\lambda}_{0})\times N({\mu}_{0}),
where x({\lambda}_{i},{\mu}_{i})\in {S}_{QVI}({\lambda}_{i},{\mu}_{i}), i=1,2.
Proof We verify that all the assumptions of Theorem 3.3 are fulfilled. First, (i′), (iv′) and (v′) are the same as (i), (iv) and (v) in Theorem 3.3. We need only to verify conditions (ii) and (iii). Taking M,\tilde{M}>0 such that
and
We put {m}_{1}=\tilde{M}{m}_{3} and {\gamma}_{1}={\gamma}_{3}. For any fixed x,y\in E(N({\lambda}_{0})), by assumption (ii′), we have
Then
Hence
Also, we put {m}_{2}=M and {\gamma}_{2}=1. We need to show that
For each fixed x\in E(N({\lambda}_{0})) and \mu \in N({\mu}_{0}),
Hence, condition (iii) is verified, and so we obtain the result. □
For (PGVQVI), if we put Y=\mathbb{R}, C=[0,+\mathrm{\infty}), then (25) becomes the following parametric generalized quasivariational inequality problem in the case of scalarvalued one:
For each \lambda \in N({\lambda}_{0}) and \mu \in N({\mu}_{0}), let
The solution set of (26) is denoted by
For each \lambda \in N({\lambda}_{0}), \mu \in N({\mu}_{0}) and fixed 1\in intC, the {\xi}_{q}solution set of (25) is
It follows from Lemma 2.1 that {S}_{QVI}^{S}({\xi}_{1},\lambda ,\mu ) coincides with {S}_{QVI}^{S}(\lambda ,\mu ).
Corollary 4.2 Assume that {S}_{QVI}^{S}(\lambda ,\mu ) is nonempty in a neighborhood N({\lambda}_{0})\times N({\mu}_{0}) of the considered point ({\lambda}_{0},{\mu}_{0})\in \mathrm{\Lambda}\times M. Assume further that conditions (i′)(iii′) and (v′) in Corollary 4.1 hold. Replace (iv′) by (iv^{′′}).
(iv^{′′}) T(\cdot ,\mu ) is h\cdot \betaHölder strongly monotone, i.e., there exist constants h>0, \beta >0, such that for every x,y\in E(N({\lambda}_{0})): x\ne y,
Then, for every (\lambda ,\mu )\in N({\lambda}_{0})\times N({\mu}_{0}), the solution of (PGVQVI) is unique, x(\lambda ,\mu ), and this function satisfies the Hölder condition: for all ({\lambda}_{1},{\mu}_{1}),({\lambda}_{2},{\mu}_{2})\in N({\lambda}_{0})\times N({\mu}_{0}),
where x({\lambda}_{i},{\mu}_{i})\in {S}_{QVI}^{S}({\lambda}_{i},{\mu}_{i}), i=1,2.
Proof It is not hard to show that (iv^{′′}) implies (iv′). Indeed, for any x,y\in E(N({\lambda}_{0})) with x\ne y,
Therefore, (iv′) is satisfied. □
Remark 4.3 Corollary 4.2 extends Corollary 3.1 in [33] since the mapping T is a multivalued mapping.
4.2 Traffic equilibrium problems
The foundation of the study of traffic network problems goes back to Wardrop [34], who stated the basic equilibrium principle in 1952. Over the past decades, a large number of efforts have been devoted to the study of traffic assignment models, with emphasis on efficiency and optimality, in order to improve practicability, reduce gas emissions and contribute to the welfare of the community. The variational inequality approach to such problems begins with the seminal work of Smith [35] who proved that the useroptimized equilibrium can be expressed in terms of a variational inequality. Thus, the possibility of exploiting the powerful tools of variational analysis has led to dealing with a large variety of models, reaching valuable theoretical results and providing applications in practical situations. In this paper, we are concerned with a class of equilibrium problems which can be studied in the framework of quasivariational inequalities, see [36, 37].
Let a set N of nodes, a set L of links, a set W:=({W}_{1},\dots ,{W}_{l}) of origindestination pairs (O/D pairs for short) be given. Assume that there are {r}_{j}\ge 1 paths connecting the pairs {W}_{j}, j=1,\dots ,l, whose set is denoted by {P}_{j}. Set m:={r}_{1}+\cdots +{r}_{l}; i.e., there are in whole m paths in the traffic network. Let F:=({F}_{1},\dots ,{F}_{m}) stand for the path flow vector. Assume that the travel cost of the path {R}_{s}, s=1,\dots ,m, is a set {T}_{s}(F)\subset {\mathbb{R}}_{+}. So, we have a multifunction T:{\mathbb{R}}_{+}^{m}\rightrightarrows {\mathbb{R}}_{+}^{m} with T(F):=({T}_{1}(F),\dots ,{T}_{m}(F)). Let the capacity restriction be
where {\mathrm{\Gamma}}_{s} are given real numbers. Extending the Wardrop definition to the case of multivalued costs, we propose the following definition.
A path flow vector H is said to be a weak equilibrium flow vector if
where j=1,\dots ,l and q,s\in \{1,\dots ,m\} are among {r}_{j} indices corresponding to {P}_{j}.
A path flow vector H is said to be a strong equilibrium flow vector if
Suppose that the travel demand {\rho}_{j} of the O/D pair {W}_{j}, j=1,\dots ,l, depends on the weak (or strong) equilibrium problem flow H. So, considering all the O/D pairs, we have a mapping \rho :{\mathbb{R}}_{+}^{m}\to {\mathbb{R}}_{+}^{l}. We use the Kronecker notation
Then the matrix
is called an O/D pair/path incidence matrix. The path flow vectors meeting the travel demands are called the feasible path flow vectors and form the constraint set, for a given weak (or strong) equilibrium flow H,
Assume further that the path costs are also perturbed, i.e., depend on a perturbation parameter μ of a metric space M: {T}_{s}(F,\mu ), s=1,\dots ,m.
Our traffic equilibrium problem is equivalent to a quasivariational inequality as follows (see [38]).
Lemma 4.4 A path vector flow H\in K(H,\lambda ) is a weak equilibrium flow if and only if it is a solution of the following quasivariational inequality:
Lemma 4.5 A path vector flow H\in K(H,\lambda ) is a strong equilibrium flow if and only if it is a solution of the following quasivariational inequality:
Corollary 4.6 Assume that solutions of the traffic network equilibrium problem exist and all the assumptions of Corollary 4.2 are satisfied. Then, in a neighborhood of ({\lambda}_{0},{\mu}_{0}), the solution is unique and satisfies the same Hölder condition as in Corollary 4.2.
4.3 Quasioptimization problem
For the normed linear space Y and pointed, closed and convex cone C with nonempty interior, we denote the ordering induced by C as follows:
The orderings ≥ and > are defined similarly. Let g:X\times M\to Y be a vectorvalued mapping. For each (\lambda ,\mu )\in \mathrm{\Lambda}\times M, consider the problem of parametric quasioptimization problem (PQOP) finding {x}_{0}\in K({x}_{0},\lambda ) such that
Since the constraint set depends on the minimizer {x}_{0}, this is a quasioptimization problem. Setting f(x,y,\mu )=g(y,\mu )g(x,\mu ), (PVQEP) becomes a special case of (PQOP).
The following results are derived from Theorem 3.8 (Theorem 3.3 cannot be applied since f(x,y,\mu )+f(y,x,\mu )=0, \mathrm{\forall}x,y\in A and \mu \in M).
Theorem 4.7 For (PQOP), assume that the solution exists in a neighborhood N({\lambda}_{0})\times N({\mu}_{0}) of the considered point ({\lambda}_{0},{\mu}_{0})\in \mathrm{\Lambda}\times M. Assume further that the following conditions hold.

(i)
K(\cdot ,\cdot ) is ({l}_{1}\cdot {\alpha}_{1},{l}_{2}\cdot {\alpha}_{2})Hölder continuous on E(N({\lambda}_{0}))\times N({\lambda}_{0});

(ii)
For each x,y\in E(N({\lambda}_{0})), F(x,y,\cdot ) is {m}_{1}\cdot {\gamma}_{1}Hölder continuous at {\mu}_{0}\in M;

(iii)
For each x\in E(N({\lambda}_{0})) and \mu \in N({\mu}_{0}), F(x,\cdot ,\mu ) is {m}_{2}\cdot {\gamma}_{2}Hölder continuous on E(N({\lambda}_{0}));

(iv)
F(\cdot ,\cdot ,\mu ) is h\cdot \betaHölder strongly monotone with respect to {\xi}_{q}, i.e., there exist constants h>0, \beta >0 such that for every x,y\in E(N({\lambda}_{0})): x\ne y,
h{d}_{X}^{\beta}(x,y)\le d(\underset{z\in F(x,y,\mu )}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+})+d(\underset{z\in F(y,x,\mu )}{inf}{\xi}_{q}(z),{\mathbb{R}}_{+}); 
(v)
\beta ={\alpha}_{1}{\gamma}_{2}, h>2{m}_{2}L{l}_{1}^{{\gamma}_{1}}, where L:={sup}_{\lambda \in {C}^{q}}\parallel \lambda \parallel \in [\frac{1}{\parallel q\parallel},+\mathrm{\infty}) is the Lipschitz constant of {\xi}_{q} on Y.
Then, for every (\lambda ,\mu )\in N({\lambda}_{0})\times N({\mu}_{0}), the solution of (PVQGEP) is unique, x(\lambda ,\mu ), and this function satisfies the Hölder condition:
for all ({\lambda}_{1},{\mu}_{1}),({\lambda}_{2},{\mu}_{2})\in N({\lambda}_{0})\times N({\mu}_{0}),
where x({\lambda}_{i},{\mu}_{i})\in {S}_{W}({\lambda}_{i},{\mu}_{i}), i=1,2.
5 Conclusions
In this paper, by using a nonlinear scalarization technique, we obtain sufficient conditions for Hölder continuity of the solution mapping for a parametric generalized vector quasiequilibrium problem in the case where the mapping F is a general setvalued one. As applications, we derived this Hölder continuity for some quasivariational inequalities, traffic network problems and quasioptimization problems.
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Acknowledgements
The authors were partially supported by the Thailand Research Fund and Naresuan University, Grant No. RSA5780003. The authors would like to thank the referees for their remarks and suggestions, which helped to improve the paper.
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Wangkeeree, R., Preechasilp, P. On the Hölder continuity of solution maps to parametric generalized vector quasiequilibrium problems via nonlinear scalarization. J Inequal Appl 2014, 425 (2014). https://doi.org/10.1186/1029242X2014425
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DOI: https://doi.org/10.1186/1029242X2014425
Keywords
 parametric generalized vector quasiequilibrium problem
 solution mapping
 Hölder continuity
 nonlinear scalarization