On the Hölder continuity of solution maps to parametric generalized vector quasi-equilibrium problems via nonlinear scalarization

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 quasi-equilibrium problem with set-valued mappings. The results are different from the recent ones in the literature.

1 Introduction

The generalized vector quasi-equilibrium problem is a unified model of several problems, namely generalized vector quasi-variational inequalities, vector quasi-optimization 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 set-valued mappings. Anh and Khanh [15] further discussed the uniqueness and Hölder continuity of the solutions for perturbed equilibrium problems with set-valued mappings. Anh and Khanh [16] extended the results of [15] to the case of perturbed quasi-equilibrium problems with set-valued 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 set-valued solution mappings for two classes of parametric generalized vector quasi-equilibrium problems in general metric spaces. Li et al. [18] extended the results of [17] to perturbed generalized vector quasi-equilibrium 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 quasi-equilibrium problem with set-valued 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 set-valued solution mapping to a parametric vector variational inequality [27]. Using this method, Bianchi and Pini [28] obtained the Hölder continuity of the single-valued solution mapping to a parametric vector equilibrium problem. Recently, Chen and Li [29] studied Hölder continuity of the solution mapping for both set-valued and single-valued 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 quasi-equilibrium 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 quasi-equilibrium problem with set-valued 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 quasi-equilibrium problem with set-valued 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 quasi-equilibrium 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 quasi-equilibrium 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 $\parallel y^{\ast }\parallel :=sup\{\parallel 〈y^{\ast },y〉\parallel :\parallel y\parallel =1\}$, where $〈y^{\ast },y〉$ 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

$$C^{\ast }:=\{y^{\ast }\in Y^{\ast }:〈y^{\ast },y〉\ge 0,\mathrm{\forall }y\in C\}$$

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

$$B_e^{\ast }:=\{y^{\ast }\in C^{\ast }:〈y^{\ast },e〉=1\}$$

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

$${\xi }_q=min\{t\in \mathbb{R}:y\in tq-C\}.$$

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]:

$${\xi }_q(y)=\underset{1\le i\le l}{max}\{y_i\},\mathrm{\forall }y=(y_1,y_2,\dots ,y_l)\in {\mathbb{R}}^l.$$

(1)

Lemma 2.1 [[25], Proposition 1.43]

For any fixed $q\in intC$, $y\in Y$ and $r\in \mathbb{R}$,

• (i) ${\xi }_q<r\iff y\in rq-intC$ (that is, ${\xi }_q(y)\ge r\iff y\notin rq-intC$);

• (ii) ${\xi }_q(y)\le r\iff y\in rq-C$;

• (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

$${\xi }_q(y+rq)={\xi }_q(y)+r,\mathrm{\forall }y\in Y,r\in \mathbb{R}.$$

(2)

For any $q\in intC$, the set $C^q$ defined by

$$C^q:=\{y^{\ast }\in C^{\ast }:〈y^{\ast },q〉=1\}$$

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 $y\in Y$,

$${\xi }_q(y)=\underset{y^{\ast }\in C^q}{max}〈y^{\ast },y〉.$$

Proposition 2.3 [[30], Proposition 2.3]

${\xi }_q$ is Lipschitz on Y, and its Lipschitz constant is

$$L:=\underset{y^{\ast }\in C^q}{sup}\parallel y^{\ast }\parallel \in [\frac{1}{\parallel q\parallel },+\mathrm{\infty }).$$

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, $|{\xi }_q(x)-{\xi }_q(y)|=\frac{1}{q}|x-y|$ for all $x,y\in \mathbb{R}$.

• (ii) If $Y={\mathbb{R}}^2$ and $C=\{(y_1,y_2)\in {\mathbb{R}}^2:\frac{1}{4}{y}_1\le y_2\le 2y_1\}$. Take $q=(2,3)\in intC$, then

  $$C^q:=\{(y_1,y_2)\in \mathbb{R}:2y_1+3y_2=1,y_1\in [-0.1,2]\},$$

and the Lipschitz constant is $L={sup}_{y^{\ast }\in C^q}\parallel y^{\ast }\parallel =\parallel (-2,1)\parallel =\sqrt{5}$. Hence,

$$|{\xi }_q(y)-{\xi }_q\left(y^{\prime }\right)|\le \sqrt{5}\parallel y-y^{\prime }\parallel ,\mathrm{\forall }y,y^{\prime }\in {\mathbb{R}}^2.$$

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 set-valued mapping $G:\mathrm{\Lambda }\to 2^X$ is said to be $l\cdot \alpha$-Hölder continuous at ${\lambda }_0$ on a neighborhood $N({\lambda }_0)$ of ${\lambda }_0$ if and only if

$$G({\lambda }_1)\subseteq G({\lambda }_2)+l{B}_X(0,d^{\alpha }({\lambda }_1,{\lambda }_2)),\mathrm{\forall }{\lambda }_1,{\lambda }_2\in N({\lambda }_0).$$

(3)

When X is a normed space, we say that the vector-valued mapping $g:\mathrm{\Lambda }\to X$ is $l\cdot \alpha$-Hölder continuous at ${\lambda }_0$ on a neighborhood $N({\lambda }_0)$ of ${\lambda }_0$ iff

$$\parallel g({\lambda }_1)-g({\lambda }_2)\parallel \le l{d}^{\alpha }({\lambda }_1,{\lambda }_2),\mathrm{\forall }{\lambda }_1,{\lambda }_2\in N({\lambda }_0).$$

(4)

Definition 2.6 Let $l_1,l_2\ge 0$ and ${\alpha }_1,{\alpha }_2>0$. A set-valued 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

$$G(x_1,{\lambda }_1)\subseteq G(x_2,{\lambda }_2)+(l_1{d}_X^{{\alpha }_1}(x_1,x_2)+l_2{d}_{\mathrm{\Lambda }}^{{\alpha }_2}({\lambda }_1,{\lambda }_2))B_X(0,1)$$

(5)

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 quasi-equilibrium 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 set-valued mappings. For each $\lambda \in N({\lambda }_0)$ and $\mu \in N({\mu }_0)$, we consider the following parametric generalized vector quasi-equilibrium problem (PGVQEP):

Find $x_0\in K(x_0,\lambda )$ such that

$$F(x_0,y,\mu )\subset Y\mathrm{\setminus }(-intC),\mathrm{\forall }y\in K(x_0,\lambda ).$$

(6)

For each $\lambda \in N({\lambda }_0)$ and $\mu \in N({\mu }_0)$, let

$$E(\lambda ):=\{x\in X|x\in K(x,\lambda )\}.$$

The weak solution set of (6) is denoted by

$$S_W(\lambda ,\mu ):=\{x\in E(\lambda ):F(x,y,\mu )\subset Y\mathrm{\setminus }(-intC),\mathrm{\forall }y\in K(x,\lambda )\}.$$

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

$$S({\xi }_q,\lambda ,\mu ):=\{x\in E(\lambda ):\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z)\ge 0,\mathrm{\forall }y\in K(x,\lambda )\}.$$

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 $q\in intC$,

$$S_W(\lambda ,\mu )=S({\xi }_q,\lambda ,\mu ).$$

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

$$x\in E(\lambda )\text{and}F(x,y,\mu )\subset Y\mathrm{\setminus }(-intC),\mathrm{\forall }y\in K(x,\lambda ).$$

Therefore, for each $y\in K(x,\lambda )$ and each $z\in F(x,y,\mu )$, we have

$$z\notin -intC=0q-intC.$$

By Lemma 2.1(i), we conclude that ${\xi }_q(z)\ge 0$. Since z is arbitrary, we have

$$\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z)\ge 0\text{for all }y\in K(x,\lambda ),$$

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

$$x\in E(\lambda )\text{and}\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z)\ge 0,\mathrm{\forall }y\in K(x,\lambda ).$$

(7)

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

$$x\in E(\lambda )\text{and}F(x,y,\mu )\subset Y\mathrm{\setminus }(-intC),\mathrm{\forall }y\in K(x,\lambda ).$$

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 ${\mu }_1,{\mu }_2\in N({\mu }_0)$,

$$F(x,y,{\mu }_1)\subset F(x,y,{\mu }_2)+m_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)B_Y.$$

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

$$z_1=z_2+m_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)e.$$

By using Proposition 2.3, we obtain

$$\begin{array}{rcl}|{\xi }_q(z_1)-{\xi }_q(z_2)|& \le & L\parallel z_1-z_2\parallel \\ =& L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)\parallel e\parallel \\ \le & L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2),\end{array}$$

(8)

which gives that

$$-L{m}_1{d}^{{\gamma }_1}({\mu }_1,{\mu }_2)\le {\xi }_q(z_1)-{\xi }_q(z_2).$$

Since $z_1$ is arbitrary and ${\xi }_q(z_2)\ge {inf}_{z\in F(x,y,{\mu }_2)}{\xi }_q(z)$, we have

$$-L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)\le \underset{z\in F(x,y,{\mu }_1)}{inf}{\xi }_q(z)-\underset{z\in F(x,y,{\mu }_2)}{inf}{\xi }_q(z).$$

Applying the symmetry between ${\mu }_1$ and ${\mu }_2$, we arrive at

$$-L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)\le \underset{z\in F(x,y,{\mu }_2)}{inf}{\xi }_q(z)-\underset{z\in F(x,y,{\mu }_1)}{inf}{\xi }_q(z).$$

It follows from the last two inequalities that

$$|{\psi }_{{\xi }_q}(x,y,{\mu }_1)-{\psi }_{{\xi }_q}(x,y,{\mu }_2)|\le L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2),\mathrm{\forall }{\mu }_1,{\mu }_2\in N({\mu }_0).$$

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 $h\cdot \beta$-Hölder 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) $\beta ={\alpha }_1{\gamma }_2$, $h>2m_{2}L{l}_1^{{\gamma }_1}$, 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 $({\lambda }_1,{\mu }_1),({\lambda }_2,{\mu }_2)\in N({\lambda }_0)\times N({\mu }_0)$,

$$\begin{array}{rl}{d}_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\le & {\left(\frac{2m_{2}L{l}_2^{{\gamma }_2}}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2/\beta }({\lambda }_1,{\lambda }_2)\\ +{\left(\frac{m_{1}L}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2),\end{array}$$

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

$$d_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\le d_X(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))+d_X(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2)),$$

where $x({\lambda }_i,{\mu }_i)\in S_W({\lambda }_i,{\mu }_i)$, $i=1,2$.

Step 1: We prove that

$$d_1:=d_X(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))\le {\left(\frac{m_{1}L}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2)$$

(9)

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

$$d_X(x({\lambda }_1,{\mu }_1),x_2)\le l_1{d}_X^{{\alpha }_1}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))=l_1{d}_1^{{\alpha }_1}$$

(10)

and

$$d_X(x({\lambda }_1,{\mu }_2),x_1)\le l_1{d}_X^{{\alpha }_1}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))=l_1{d}_1^{{\alpha }_1}.$$

(11)

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

$${\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x_1,{\mu }_1):=\underset{z\in F(x({\lambda }_1,{\mu }_1),x_1,{\mu }_1)}{inf}{\xi }_q(z)\ge 0$$

(12)

and

$${\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_2,{\mu }_2):=\underset{z\in F(x({\lambda }_1,{\mu }_2),x_2,{\mu }_2)}{inf}{\xi }_q(z)\ge 0.$$

(13)

By virtue of (iv), we have

$$\begin{array}{rcl}h{d}_1^{\beta }& =& h{d}_X^{\beta }(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))\\ \le & d({\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2),{\mu }_1),{\mathbb{R}}_{+})+d({\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_1,{\mu }_1),{\mu }_1),{\mathbb{R}}_{+}).\end{array}$$

By combining (12) and (13) with the last inequality, we have

$$\begin{array}{rcl}h{d}_1^{\beta }& \le & |{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2),{\mu }_1)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x_1,{\mu }_1)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_1,{\mu }_1),{\mu }_1)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_2,{\mu }_2)|\\ \le & |{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2),{\mu }_1)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_1),x_1,{\mu }_1)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_1,{\mu }_1),{\mu }_1)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_1,{\mu }_1),{\mu }_2)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_1,{\mu }_1),{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_2,{\mu }_2)|\\ \le & L{m}_2{d}_X^{{\gamma }_2}(x({\lambda }_1,{\mu }_2),x_1)+L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)+L{m}_2{d}_X^{{\gamma }_2}(x({\lambda }_1,{\mu }_1),x_2)\\ \le & L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))\\ +L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2)+L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))\\ =& 2L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))+L{m}_1{d}_M^{{\gamma }_1}({\mu }_1,{\mu }_2).\end{array}$$

(14)

Whence, assumption (iv) implies that

$$d_X(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))\le {\left(\frac{L{m}_1}{h-L{m}_2{l}_1^{{\gamma }_2}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2).$$

Step 2: We prove that

$$d_2:=d_X(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\le {\left(\frac{2L{m}_2{l}_2^{{\gamma }_2}}{h-2L{m}_2{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2/\beta }({\lambda }_1,{\lambda }_2)$$

(15)

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

$$d_X(x({\lambda }_1,{\mu }_2),x_2^{\prime })\le l_2{d}_{\mathrm{\Lambda }}^{{\alpha }_2}({\lambda }_1,{\lambda }_2)$$

(16)

and

$$d_X(x({\lambda }_2,{\mu }_2),x_1^{\prime })\le l_2{d}_{\mathrm{\Lambda }}^{{\alpha }_2}({\lambda }_1,{\lambda }_2).$$

(17)

Again, by the Hölder continuity of $K(\cdot ,\cdot )$, there exist $x_1^{″}\in K(x({\lambda }_1,{\mu }_2),{\lambda }_1)$ and $x_2^{″}\in K(x({\lambda }_2,{\mu }_2),{\lambda }_2)$ such that

$$d_X(x_1^{\prime },x_1^{″})\le l_1{d}_X^{{\alpha }_1}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))=l_1{d}_2^{{\alpha }_1}$$

(18)

and

$$d_X(x_2^{\prime },x_2^{″})\le l_1{d}_X^{{\alpha }_1}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))=l_1{d}_2^{{\alpha }_1}.$$

(19)

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:

$${\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_1^{″},{\mu }_2):=\underset{z\in F(x({\lambda }_1,{\mu }_2),x_1^{″},{\mu }_2)}{inf}{\xi }_q(z)\ge 0$$

(20)

and

$${\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x_2^{″},{\mu }_2):=\underset{z\in F(x({\lambda }_2,{\mu }_2),x_2^{″},{\mu }_2)}{inf}{\xi }_q(z)\ge 0.$$

(21)

By virtue of (iv), we have

$$\begin{array}{rcl}h{d}_2^{\beta }& =& h{d}_X^{\beta }(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\\ \le & d({\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2),{\mu }_2),{\mathbb{R}}_{+})+d({\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x({\lambda }_1,{\mu }_2),{\mu }_2),{\mathbb{R}}_{+}).\end{array}$$

By combining (20) and (21) with the last inequality, we have

$$\begin{array}{rcl}h{d}_2^{\beta }& \le & |{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2),{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_1^{″},{\mu }_2)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x({\lambda }_1,{\mu }_2),{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x_2^{″},{\mu }_2)|\\ \le & |{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2),{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_1^{\prime },{\mu }_2)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_1^{\prime },{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_1,{\mu }_2),x_1^{″},{\mu }_2)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x({\lambda }_1,{\mu }_2),{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x_2^{\prime },{\mu }_2)|\\ +|{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x_2^{\prime },{\mu }_2)-{\psi }_{{\xi }_q}(x({\lambda }_2,{\mu }_2),x_2^{″},{\mu }_2)|\\ \le & L{m}_2{d}_X^{{\gamma }_2}(x({\lambda }_2,{\mu }_2),x_1^{\prime })+L{m}_2{d}_X^{{\gamma }_2}(x_1^{\prime },x_1^{″})\\ +L{m}_2{d}_X^{{\gamma }_2}(x({\lambda }_1,{\mu }_2),x_2^{\prime },)+L{m}_2{d}_X^{{\gamma }_2}(x_2^{\prime },x_2^{″}).\end{array}$$

(22)

By virtue of (16), (17), (18) and (19), we get

$$\begin{array}{r}h{d}_X^{\beta }(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\\ \le L{m}_2{l}_2^{{\gamma }_2}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2}({\lambda }_1,{\lambda }_2)+L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\\ +L{m}_2{l}_2^{{\gamma }_2}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2}({\lambda }_1,{\lambda }_2)+L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\\ =2L{m}_2{l}_2^{{\gamma }_2}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2}({\lambda }_1,{\lambda }_2)+2L{m}_2{l}_1^{{\gamma }_2}{d}_X^{{\alpha }_1{\gamma }_2}(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2)).\end{array}$$

(23)

Whence, condition (v) implies that

$$d_X^{\beta }(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\le {\left(\frac{2L{m}_2{l}_2^{{\gamma }_2}}{h-2L{m}_2{l}_1^{{\gamma }_2}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2}({\lambda }_1,{\lambda }_2).$$

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

$$\begin{array}{r}d(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\\ \le d(x({\lambda }_1,{\mu }_1),x({\lambda }_1,{\mu }_2))+d(x({\lambda }_1,{\mu }_2),x({\lambda }_2,{\mu }_2))\\ \le {\left(\frac{m_{1}L}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2)+{\left(\frac{2L{m}_2{l}_2^{{\gamma }_2}}{h-2L{m}_2{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2/\beta }({\lambda }_1,{\lambda }_2).\end{array}$$

Thus,

$$\begin{array}{r}\rho (S_W({\lambda }_1,{\mu }_1),S_W({\lambda }_2,{\mu }_2))\\ =\underset{x({\lambda }_1,{\mu }_1)\in S_W({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2)\in S_W({\lambda }_2,{\mu }_2)}{sup}{d}_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\\ \le {\left(\frac{m_{1}L}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2)+{\left(\frac{2L{m}_2{l}_2^{{\gamma }_2}}{h-2L{m}_2{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2/\beta }({\lambda }_1,{\lambda }_2).\end{array}$$

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 set-valued mapping. A set-valued mapping $F(\cdot ,\cdot ,\mu )\mapsto 2^Y$ is said to be

• (A) $h\cdot \beta$-Hölder 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) $h\cdot \beta$-Hölder 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 $x\ne y$,

  $$z\notin -intC,\mathrm{\exists }z\in F(x,y,\mu )\Rightarrow z^{\prime }+h{d}_X^{\beta }(x,y)q\in -C,\mathrm{\exists }{z}^{\prime }\in F(y,x,\mu ).$$

• (C) quasi-monotone on $E(N({\lambda }_0))$ if $\mathrm{\forall }x,y\in E(N({\lambda }_0))$ with $x\ne y$,

  $$z\in -intC,\mathrm{\exists }z\in F(x,y,\mu )\Rightarrow z^{\prime }\notin -intC,\mathrm{\exists }{z}^{\prime }\in F(y,x,\mu ).$$

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

$$\begin{array}{rcl}h{d}_X^{\beta }(x,y)& \le & -\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z)-\underset{z\in F(y,x,\mu )}{inf}{\xi }_q(z)\\ \le & d(\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z),{\mathbb{R}}_{+})+d(\underset{z\in F(y,x,\mu )}{inf}{\xi }_q(z),{\mathbb{R}}_{+}).\end{array}$$

(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

$${\xi }_q\left(z^{\prime }\right)+h{d}_X^{\beta }(x,y)={\xi }_q(z^{\prime }+h{d}_X^{\beta }(x,y)q)\le 0,$$

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,

$$h{d}_X^{\beta }(x,y)\le -\underset{z\in F(y,x,\mu )}{inf}{\xi }_q(z)\le d(\underset{z\in F(x,y,\mu )}{inf}{\xi }_q(z),{\mathbb{R}}_{+})+d(\underset{z\in F(y,x,\mu )}{inf}{\xi }_q(z),{\mathbb{R}}_{+}).$$

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 vector-valued mapping. Then (PGVQEP) becomes the following parametric vector quasi-equilibrium problem (PVQEP):

Find $x_0\in K(x_0,\lambda )$ such that

$$f(x_0,y,\mu )\notin -intC,\mathrm{\forall }y\in K(x_0,\lambda ).$$

(24)

Remark 3.9 In the case of a vector-valued 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 vector-valued mapping $f(\cdot ,\cdot ,\cdot )$ is extended to a set-valued one.

4 Applications

Since the parametric generalized vector quasi-equilibrium problem (PGVQEP) contains as special cases many optimization-related problems, including quasi-variational inequalities, traffic equilibrium problems, quasi-optimization 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 Quasi-variational 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 set-valued mappings, where $B^{\ast }(X,Y)$ denotes the space of all bounded linear mappings of X into Y. Setting $F(x,y,\mu )=〈T(x,\mu ),y-x〉:={\bigcup }_{t\in T(x,\mu )}〈t,y-x〉$ in (6), we obtain parametric generalized vector quasi-variational inequalities (PGVQVI) in the case of set-valued mappings as follows:

$$\text{Find }{x}_0\in K(x_0,\lambda )\text{ such that }〈T(x_0,\mu ),y-x_0〉\subseteq Y\mathrm{\setminus }-intC,\mathrm{\forall }y\in K(x_0,\lambda ).$$

(25)

For each $\lambda \in N({\lambda }_0)$ and $\mu \in N({\mu }_0)$, let

$$E(\lambda ):=\{x\in X:x\in K(x,\lambda )\}.$$

The solution set of (25) is denoted by

$$S_{QVI}^V(\lambda ,\mu ):=\{x\in E(\lambda ):〈T(x,\mu ),y-x〉\subseteq Y\mathrm{\setminus }-intC,\mathrm{\forall }y\in K(x,\lambda )\}.$$

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

$$S_{QVI}^V({\xi }_q,\lambda ,\mu ):=\{x\in E(\lambda ):\underset{z\in 〈T(x,\mu ),y-x〉}{inf}{\xi }_q(z)\ge 0,\mathrm{\forall }y\in K(x,\lambda )\}.$$

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 $h\cdot \beta$-Hölder strongly monotone with respect to ${\xi }_q$, i.e., there exist constants $h>0$, $\beta >0$ such that for every $x,y\in E(N({\lambda }_0))$: $x\ne y$,

  $$h{\parallel x-y\parallel }^{\beta }\le d(\underset{z\in 〈T(x,\mu ),y-x〉}{inf}{\xi }_q(z),{\mathbb{R}}_{+})+d(\underset{z\in 〈T(y,\mu ),x-y〉}{inf}{\xi }_q(z),{\mathbb{R}}_{+});$$

• (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 $({\lambda }_1,{\mu }_1),({\lambda }_2,{\mu }_2)\in N({\lambda }_0)\times N({\mu }_0)$,

$$\begin{array}{rl}{d}_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\le & {\left(\frac{2ML{l}_2}{h-2ML{l}_1^{{\gamma }_3}}\right)}^{\frac{1}{\beta }}{d}_{\lambda }^{{\alpha }_2/\beta }({\lambda }_1,{\lambda }_2)\\ +{\left(\frac{N{m}_{3}L}{h-2ML{l}_1^{{\gamma }_3}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_3/\beta }({\mu }_1,{\mu }_2),\end{array}$$

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

$$\parallel T(x,\mu )\parallel \le M,\mathrm{\forall }(x,\mu )\in E(N({\lambda }_0))\times N({\mu }_0)$$

and

$$\parallel x-y\parallel \le \tilde{M},\mathrm{\forall }x,y\in E(N({\lambda }_0)).$$

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

$$T(x,{\mu }_1)\subseteq T(x,{\mu }_2)+m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)B_{B^{\ast }(X,Y)},\mathrm{\forall }{\mu }_1,{\mu }_2\in N({\mu }_0).$$

Then

$$\begin{array}{rl}〈T(x,{\mu }_1),y-x〉& \subseteq 〈T(x,{\mu }_2)+m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)B_{B^{\ast }(X,Y)},y-x〉\\ =〈T(x,{\mu }_2),y-x〉+〈m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)B_{B^{\ast }(X,Y)},y-x〉\\ =〈T(x,{\mu }_2),y-x〉+m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)〈B_{B^{\ast }(X,Y)},y-x〉\\ =〈T(x,{\mu }_2),y-x〉+m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)\bigcup _{g\in B_{B^{\ast }(X,Y)}}〈g,y-x〉\\ \subseteq 〈T(x,{\mu }_2),y-x〉+m_3{d}^{{\gamma }_3}({\mu }_1,{\mu }_2)\tilde{M}{B}_Y.\end{array}$$

Hence

$$〈T(x,{\mu }_1),y-x〉\subseteq 〈T(x,{\mu }_2),y-x〉+m_1{d}^{{\gamma }_1}({\mu }_1,{\mu }_2)\tilde{M}{B}_Y.$$

Also, we put $m_2=M$ and ${\gamma }_2=1$. We need to show that

$$〈T(x,\mu ),y_1-x〉\subseteq 〈T(x,\mu ),y_2-x〉+\tilde{M}\parallel y_1-y_2\parallel B_Y.$$

For each fixed $x\in E(N({\lambda }_0))$ and $\mu \in N({\mu }_0)$,

$$\begin{array}{rcl}〈T(x,\mu ),y_1-x〉& =& \bigcup _{t\in T(x,\mu )}〈t,y_1-x〉\\ =& \bigcup _{t\in T(x,\mu )}〈t,y_1-x+y_2-y_2〉\\ =& \bigcup _{t\in T(x,\mu )}〈t,y_2-x〉+\bigcup _{t\in T(x,\mu )}〈t,y_1-y_2〉\\ \subseteq & 〈T(x,\mu ),y-x〉+M\parallel y_1-y_2\parallel B_Y.\end{array}$$

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 quasi-variational inequality problem in the case of scalar-valued one:

$$\text{Find }{x}_0\in K(x_0,\lambda )\text{ such that }〈t,y-x_0〉\ge 0,\mathrm{\forall }y\in K(x_0,\lambda ),\mathrm{\forall }t\in T(x_0,\mu ).$$

(26)

For each $\lambda \in N({\lambda }_0)$ and $\mu \in N({\mu }_0)$, let

$$E(\lambda ):=\{x\in X:x\in K(x,\lambda )\}.$$

The solution set of (26) is denoted by

$$S_{QVI}^S(\lambda ,\mu ):=\{x\in E(\lambda ):〈t,y-x〉\ge 0,\mathrm{\forall }y\in K(x,\lambda ),\mathrm{\forall }t\in T(x,\mu )\}.$$

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

$$S_{QVI}^S({\xi }_1,\lambda ,\mu ):=\{x\in E(\lambda ):\underset{z\in 〈T(x,\mu ),y-x〉}{inf}{\xi }_1(z)\ge 0,\mathrm{\forall }y\in K(x,\lambda )\}.$$

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 $h\cdot \beta$-Hölder strongly monotone, i.e., there exist constants $h>0$, $\beta >0$, such that for every $x,y\in E(N({\lambda }_0))$: $x\ne y$,

$$〈u-v,x-y〉\ge h{\parallel x-y\parallel }^{\beta },\mathrm{\forall }u\in T(x),\mathrm{\forall }v\in T(y).$$

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 $({\lambda }_1,{\mu }_1),({\lambda }_2,{\mu }_2)\in N({\lambda }_0)\times N({\mu }_0)$,

$$d_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\le {\left(\frac{2M{l}_2}{h-2M{l}_1^{{\gamma }_3}}\right)}^{\frac{1}{\beta }}{d}_{\lambda }^{{\alpha }_2/\beta }({\lambda }_1,{\lambda }_2)+{\left(\frac{N{m}_3}{h-2M{l}_1^{{\gamma }_3}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_3/\beta }({\mu }_1,{\mu }_2),$$

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 $x\ne y$,

$$\begin{array}{rcl}h{\parallel x-y\parallel }^{\beta }& \le & 〈u-v,x-y〉\\ =& 〈u,x-y〉+〈v,y-x〉\\ \le & \underset{u\in T(x)}{sup}〈u,x-y〉+\underset{v\in T(y)}{sup}〈v,y-x〉\\ =& \underset{u\in T(x)}{sup}-〈u,y-x〉+\underset{v\in T(y)}{sup}-〈v,x-y〉\\ =& -\underset{u\in T(x)}{inf}〈u,y-x〉-\underset{v\in T(y)}{inf}〈v,x-y〉\\ \le & d(\underset{u\in T(x)}{inf}〈u,y-x〉,{\mathbb{R}}_{+})+d(\underset{v\in T(y)}{inf}〈v,x-y〉,{\mathbb{R}}_{+}).\end{array}$$

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 user-optimized 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 quasi-variational inequalities, see [36, 37].

Let a set N of nodes, a set L of links, a set $W:=(W_1,\dots ,W_l)$ of origin-destination 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

$$F\in A:=\{F\in {\mathbb{R}}_{+}^m:F_s\le {\mathrm{\Gamma }}_s,s=1,\dots ,m\},$$

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

$$\begin{array}{r}\mathrm{\forall }{W}_j,\mathrm{\forall }{R}_q\in P_j,R_s\in P_j,\text{ there exists }t\in T(H)\text{ such that}\\ t_q<t_s\Rightarrow H_q={\mathrm{\Gamma }}_q\text{ or }{H}_s=0,\end{array}$$

(27)

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

$$\mathrm{\forall }{W}_j,\mathrm{\forall }{R}_q\in P_j,R_s\in P_j,\text{ for all }t\in T(H)\text{ such that }{t}_q<t_s\Rightarrow H_q={\mathrm{\Gamma }}_q\text{ or }{H}_s=0.$$

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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

$${\varphi }_{js}=\{\begin{array}{cc}1\hfill & \text{if }s\in P_j,\hfill \\ 0\hfill & \text{if }s\notin P_j.\hfill \end{array}$$

Then the matrix

$$\varphi =\{{\varphi }_{js}\},j=1,\dots ,l,s=1,\dots ,m,$$

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,

$$K(H,\lambda ):=\{F\in A:\varphi F=\rho (H,\lambda )\}.$$

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 quasi-variational 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 quasi-variational inequality:

$$\begin{array}{r}\mathit{\text{Find }}H\in K(H,\lambda )\mathit{\text{ such that there exists }}t\in T(H,\lambda )\mathit{\text{ satisfying }}〈t,F-H〉\ge 0,\\ \mathrm{\forall }F\in K(H,\lambda ).\end{array}$$

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 quasi-variational inequality:

$$\begin{array}{r}\mathit{\text{Find }}H\in K(H,\lambda )\mathit{\text{ such that for all }}t\in T(H,\lambda )\mathit{\text{ it satisfies }}〈t,F-H〉\ge 0,\\ \mathrm{\forall }F\in K(H,\lambda ).\end{array}$$

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 Quasi-optimization 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:

$$\begin{array}{r}x\le y\text{iff}y-x\in C;\\ x<y\text{iff}y-x\in intC.\end{array}$$

The orderings ≥ and > are defined similarly. Let $g:X\times M\to Y$ be a vector-valued mapping. For each $(\lambda ,\mu )\in \mathrm{\Lambda }\times M$, consider the problem of parametric quasi-optimization problem (PQOP) finding $x_0\in K(x_0,\lambda )$ such that

$$g(x_0,\mu )=\underset{y\in K(x_0,\lambda )}{min}g(y,\mu ).$$

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Since the constraint set depends on the minimizer $x_0$, this is a quasi-optimization 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 $h\cdot \beta$-Hölder 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) $\beta ={\alpha }_1{\gamma }_2$, $h>2m_{2}L{l}_1^{{\gamma }_1}$, 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 $({\lambda }_1,{\mu }_1),({\lambda }_2,{\mu }_2)\in N({\lambda }_0)\times N({\mu }_0)$,

$$\begin{array}{rl}{d}_X(x({\lambda }_1,{\mu }_1),x({\lambda }_2,{\mu }_2))\le & {\left(\frac{2m_{2}L{l}_2^{{\gamma }_2}}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_{\mathrm{\Lambda }}^{{\alpha }_2{\gamma }_2/\beta }({\lambda }_1,{\lambda }_2)\\ +{\left(\frac{m_{1}L}{h-2m_{2}L{l}_1^{{\gamma }_1}}\right)}^{\frac{1}{\beta }}{d}_M^{{\gamma }_1/\beta }({\mu }_1,{\mu }_2),\end{array}$$

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 quasi-equilibrium problem in the case where the mapping F is a general set-valued one. As applications, we derived this Hölder continuity for some quasi-variational inequalities, traffic network problems and quasi-optimization problems.