Stability analysis for set-valued inverse mixed variational inequalities in reflexive Banach spaces

Abstract

This work is devoted to the analysis for a new class of set-valued inverse mixed variational inequalities (SIMVIs) in reflexive Banach spaces, when both the mapping and the constraint set are perturbed simultaneously by two parameters. Several equivalence characterizations are given for SIMVIs to have nonempty and bounded solution sets. Based on the equivalence conditions, under the premise of monotone mappings, the stability result for the SIMVIs is obtained in the reflexive Banach space. Furthermore, to illustrate the results, an example of the traffic network equilibrium control problem is provided at the end of this paper. The results presented in this paper generalize and extend some known results in this area.

1 Introduction

As an important part of nonlinear analysis, variational inequalities are widely applied in finance, transportation, economics, optimization, engineering science, and other fields. In the initial stage of the development of variational inequalities, researchers have mainly studied the nonemptiness and boundedness of variational inequality solution sets; see [1–3]. In recent years, the stability analysis of variational inequalities has been extensively developed because of the importance of investigating the properties of solutions to problems with perturbed data in practical applications; see [4–7]. McLinden [8] investigated the stability of a variational inequality with monotone operators and convex sets in reflexive Banach spaces and obtained a variety of results about stability involving a natural parameter. Addi et al. [9] investigated the stability of a finite semicoercive variational inequalities with respect to data perturbation by using recession analysis. He et al. [10] investigated the stability of generalized variational inequalities with either the mapping or the constraint set perturbed in reflexive Banach spaces. Fan et al. [11] studied the stability of a variational inequality where the mapping and the constraint set are perturbed simultaneously in reflexive Banach spaces. In addition, with the proposed mixed variational inequality, Zhong et al. [12] analyzed the stability of a class of Minty mixed variational inequalities in reflexive Banach spaces based on the analysis in [10] and extended the results in [10].

Inverse variational inequality (IVI) research has also made great progress, as an IVI is a special case of a variational inequality. Yang [13] considered the dynamic power price problem as defined by an IVI in finite dimensional spaces from the perspective of optimal control. He et al. [14] regarded congestion control problems in finite dimensional spaces as constrained black box inverse inequality problems and solved a class of constrained ‘black box’ inverse variational inequalities in finite dimensional spaces. Scrimali et al. [15] used an evolutionary IVI to study a time-dependent spatial pricing equilibrium control problem in finite dimensional spaces. Li et al. [16] used an inverse mixed variational inequality in Hilbert spaces to study the equilibrium control problem of a transportation network. Barbagallo et al. [17] proposed using an IVI in Hilbert spaces to solve an oligopolistic market equilibrium problem. Moreover, to address IVI problems, Luo [18] used the Tikhonov regularization method to study the perturbation analysis of the solution set of the regularized inverse variational inequality in finite dimensional spaces. Vuong [19] used the neural network to obtain a projection algorithm for solving the IVI in finite dimensional spaces. Xu [20] used the image space analysis to investigate an inverse variational inequality with a cone constraint. Hu and Fang [21] studied the Levitin–Polyak well-posedness of IVIs in finite dimensional spaces. Luo [22] studied the stability for the set-valued inverse variational inequality with both the mapping and the constraint set that are perturbed in a reflexive Banach space. Aussel et al. [23] studied a gap function and error bounds for the IVI in finite dimensional spaces. Jiang et al. [24] used ADMM to analyze structured IVIs to solve policy design difficulties in finite dimensional spaces. Very recently, Zhang [25] investigated error bounds of an inverse mixed quasi-variational inequality problem in Hilbert spaces. Tangkhawiwetkul [26] studied and analyzed the generalized inverse mixed variational inequality in Hilbert spaces and obtained the existence and uniqueness of the solution for the problem. For more related research works, we can see [27–29].

However, most results about IVIs are about the existence, well-posedness, and applications, there are very few studies on the stability of IVIs in infinite dimensional spaces. But the stability analysis of IVIs with perturbed parameters is very important because it can help in identifying relatively high accuracy, predicting the future changes of the equilibria as a result of the changes in the governing system, providing valuable information for designing various equilibrium systems. Moreover, most results about IVIs are discussed in finite dimensional spaces or Hilbert spaces where the mappings in IVIs are single-valued. Thus, it is worth studying the stability of a generalized inverse variational inequality, which is called a set-valued inverse mixed variational inequality (SIMVI), with the constraint set and the mapping perturbed simultaneously by different parameters in reflexive Banach spaces. To illustrate the results, some examples are provided. To the best of our knowledge, the results are new.

The paper is built up as follows. Section 2 provides a few useful definitions and lemmas. In Sect. 3, to make the SIMVI have a nonempty and bounded solution set, we offer a number of equivalent characterizations. In Sect. 4, the stability of the solutions for the SIMVI with the mapping and the constraint set perturbed simultaneously is obtained. In Sect. 5, to illustrate the results, we give an example. In Sect. 6, we give the conclusion.

2 Preliminaries

In this paper, we let E be a reflexive Banach space with its dual space $E^{*}$, and let Λ be a nonempty, convex, and closed subset of $E^{*}$. Let $\Gamma :E\to 2^{E^{*}}$ be a set-valued mapping and $\Phi :E^{*}\to \mathbb{R}\cup \{+\infty \}$ be a proper lower semicontinuous convex functional. We denote the set-valued inverse variational inequality by $\operatorname{SIMVI} (\Lambda ,\Gamma )$, which means finding $w\in E$ and $w^{*}\in \Gamma (w)\cap \Lambda$ such that

$$ \bigl\langle v-w^{*},w\bigr\rangle +\Phi (v)-\Phi \bigl(w^{*}\bigr)\geq 0 \quad \text{for all } v\in \Lambda . $$

Note that if $E=\mathbb{R}^{n}$ and Γ is single-valued, the $\operatorname{SIMVI} ( \Lambda ,\Gamma )$ may be simplified to the inverse mixed variational inequality (IMVI) shown below: find $w\in \mathbb{R}^{n}$ such that

$$ \Gamma (w)\in \Lambda , \bigl\langle \tilde{v}-\Gamma (w),w\bigr\rangle + \Phi ( \tilde{v})-\Phi \bigl(\Gamma (w)\bigr)\geq 0\quad \text{for all } \tilde{v}\in \Lambda . $$

The work about IMVIs can be found in [4, 5, 16]. If $\Phi \equiv 0$ on $R^{n}$, then IMVI can be transformed to an inverse variational inequality (IVI): find $w\in \mathbb{R}^{n}$ such that

$$ \Gamma (w)\in \Lambda , \bigl\langle \tilde{v}-\Gamma (w),w\bigr\rangle \geq 0 \quad \text{for all } \tilde{v}\in \Lambda . $$

We use the sign “→” for strong convergence and “⇀” to represent weak convergence. The barrier cone of Λ is defined by

$$ \operatorname{barr}(\Lambda ):=\Bigl\{ w\in E:\sup_{v\in \Lambda}\langle v,w\rangle < \infty \Bigr\} . $$

The recession cone of Λ is a closed and convex cone defined by

$$ \Lambda _{\infty}:=\bigl\{ d\in E^{*}:\exists t_{n} \downarrow 0,\exists w_{n} \in \Lambda ,t_{n}w_{n} \rightharpoonup d\bigr\} $$

or

$$ \Lambda _{\infty}:=\bigl\{ d\in E^{*}:w_{0}+\lambda d \in \Lambda , \text{for all } \lambda >0, w_{0}\in \Lambda \bigr\} . $$

The definition of negative polar cone of Λ is

$$ \Lambda ^{-}:=\bigl\{ v\in E^{*}:\langle v,w\rangle \leq 0, \text{for all } w\in \Lambda \bigr\} , $$

and $\operatorname{int} (\Lambda )$ represents the interior of Λ.

Assume that $\Phi :\Lambda \subset E^{*}\to \mathbb{R}\cup \{+\infty \}$ is a proper, convex, and lower semicontinuous functional. The recession function of Φ, denoted by $\Phi _{\infty}$, is defined by

$$ \Phi _{\infty}(w):=\lim_{t\to +\infty} \frac{\Phi (w_{0}+tw)-\Phi (w_{0})}{t}, $$

where $w_{0}$ is any point in $\Phi =\{v\in E^{*}:\Phi (v)<+ \infty \}$. Then it means that

$$ \Phi _{\infty}(w):=\lim_{t\to +\infty}\frac{\Phi (tw)}{t}. $$

The functional $\Phi _{\infty}(\cdot )$ has been proved to be a proper, convex, lower semicontinuous, and weakly lower semicontinuous with the property that

$$\begin{aligned} { \Phi (w+v)\leq \Phi (w)+\Phi _{\infty}(v)\quad \text{for all } w \in \operatorname{dom} \Phi ,v\in E^{*}.} \end{aligned}$$

(2.1)

Obviously, we know that $\Phi _{\infty}(\cdot )$ is positively homogeneous of degree 1, i.e.,

$$\begin{aligned} { \Phi _{\infty}(\lambda w)=\lambda \Phi _{\infty}(w) \quad \text{for all } w\in E,\lambda \geq 0.} \end{aligned}$$

(2.2)

The conjugate function $\Phi ^{*}(w):E\to \mathbb{R}\cup \{+\infty \}$ of Φ is defined by

$$ \Phi ^{*}(w):=\sup_{v\in \Lambda}\bigl\{ \langle v,w\rangle - \Phi (v)\bigr\} , $$

where the domain of $\Phi ^{*}$ is defined by $\operatorname{dom} \Phi ^{*}=\{w\in E:\Phi ^{*}(w)<+\infty \}$.

According to Proposition 2.5 in [6], we have

$$\begin{aligned} { \Phi _{\infty}(w)\leq \liminf_{n\to \infty} \frac{\Phi (t_{n}w_{n})}{t_{n}},} \end{aligned}$$

(2.3)

where $w_{0}$ is any point in domΦ, $\{w_{n}\}$ is any sequence in E converging weakly to w, and $t_{n}$ is any real sequence converging to +∞.

Definition 2.1

[22] A set-valued mapping $\Gamma :E\to 2^{E^{*}}$ is said to be

• (i) upper semicontinuous at $w_{0}\in E$ if, for any neighborhood $N(\Gamma (w_{0}))$ of $\Gamma (w_{0})$, there exists a neighborhood $N(w_{0})$ of $w_{0}$ such that $\Gamma (w)\subset N(\Gamma (w_{0}))$ for all $w\in N(w_{0})$;

• (ii) lower semicontinuous at $w_{0}\in E$ if, for any $v_{0}\in \Gamma (w_{0})$ and any neighborhood $N(v_{0})$ of $v_{0}$, there exists a neighborhood $N(w_{0})$ of $w_{0}$ such that $\Gamma (w)\bigcap N(v_{0}) \neq \emptyset$ for all $w \in N(w_{0})$;

• (iii) upper hemicontinuous iff the restriction of Γ to every line segment of E is upper semicontinuous;

• (iv) monotone on E iff, for all $(w,w^{*})$, $(v,v^{*})$ in the graph Γ,

  $$ \bigl\langle v^{*}-w^{*},v-w\bigr\rangle \geq 0. $$

It is evident that Γ is lower semicontinuous at $w_{0}\in E$ if and only if, for any ${w_{n}}$ with $w_{n} \to w_{0}$ and $v_{0}\in \Gamma (w_{0})$, there exists $v_{n}\in \Gamma (w_{n})$ such that $v_{n}\to v_{0}$.

Lemma 2.1

[10] Let $K\subset E$ be a nonempty closed convex set. If barrK has nonempty interior, then there does not exist ${w_{n}}\subset K$ with $\Vert w_{n}\Vert \to \infty$ such that $\frac{w_{n}}{\Vert w_{n}\Vert}\rightharpoonup 0$. If additionally K is a cone, then there does not exist ${d_{n}}\subset K$ with each $\Vert d_{n}\Vert =1$ such that $d_{n}\rightharpoonup 0$.

Lemma 2.2

[11] Let $(Z,d)$ be a metric space, $\alpha _{0}\in Z$ be a given point. Let $L:Z\to 2^{E^{*}}$ be a set-valued mapping with nonempty values, and L is upper semicontinuous at $\alpha _{0}$, then there exists a neighborhood $\mathcal{W}$ of $\alpha _{0}$ such that $(L(\alpha ))_{\infty}\subset (L(\alpha _{0}))_{\infty}$ for all $\alpha \in \mathcal{W}$.

Lemma 2.3

[30] Let K be a nonempty convex subset of a Hausdorff topological vector space X and $G:K\to 2^{X}$ be a set-valued mapping from K into X satisfying the following properties:

• (a) G is a KKM mapping, i.e., for every finite subset A of K, $\operatorname{co}(A)\subset \bigcup_{w\in A}G(w)$;

• (b) $G(w)$ is a closed set in X for every $w\in K$;

• (c) $G(w_{0})$ is compact in X for some $w_{0}\in K$.

Then $\bigcap_{w\in K}G(w)\neq \emptyset$.

3 Boundedness of solution sets

In this section, we give some characterizations about the solutions of the $\operatorname{SIMVI} (\Lambda ,F)$. Theorem 3.1 is critical for demonstrating the equivalence of the nonemptiness and boundedness of the solution set. For the convenience of discussion, we let $G:=\Lambda \times E$ and propose the set-valued dual inverse mixed variational inequality (for short, $\operatorname{SDIMVI} (G, \Gamma )$), which means finding $(v,w)\in G$ such that

$$\begin{aligned} { \inf_{\mu ^{*}\in F(\mu )}\bigl\langle \mu ^{*}-z, \mu -w\bigr\rangle + \langle z-v,\mu \rangle +\Phi (z)-\Phi (v)\geq 0\quad \text{for all } (z, \mu )\in G,} \end{aligned}$$

(3.1)

which is closely related to $\operatorname{SIMVI} (\Lambda ,\Gamma )$.

Theorem 3.1

Assume that $\Lambda \subset E^{*}$ is a nonempty convex and closed set, $\Gamma :E\to 2^{E^{*}}$ is a set-valued mapping with nonempty values, and $\Phi :\Lambda \subset E^{*}\to \mathbb{R}$ is a lower semicontinuous convex functional. Then we have two conclusions as follows:

• (a) every solution of $\operatorname{SIMVI} (\Lambda ,\Gamma )$ can solve $\operatorname{SDIMVI} (G, \Gamma )$ when Γ is monotone;

• (b) every solution of $\operatorname{SDIMVI} (G,\Gamma )$ can solve $\operatorname{SIMVI} ( \Lambda ,\Gamma )$ when Γ is upper hemicontinuous.

Proof

Firstly, we prove conclusion (a). Assume that w is a solution of $\operatorname{SIMVI} (\Lambda ,\Gamma )$, then there exists $w^{*}\in \Gamma (w)\cap \Lambda$ such that $\langle v-w^{*},w \rangle +\Phi (v)-\Phi (w^{*})\geq 0$ for all $v\in \Lambda$. Because Γ is monotone, then for any $(z,\mu )\in G$ and any $\mu ^{*}\in \Gamma (\mu )$, we have

$$\begin{aligned} 0 \leq &\bigl\langle \mu ^{*}-w^{*},\mu -w\bigr\rangle \\ =&\bigl\langle \mu ^{*}-z+z-w^{*},\mu -w\bigr\rangle \\ =&\bigl\langle \mu ^{*}-z,\mu -w\bigr\rangle +\bigl\langle z-w^{*},\mu -w\bigr\rangle \\ =&\bigl\langle \mu ^{*}-z,\mu -w\bigr\rangle +\bigl\langle z-w^{*},\mu \bigr\rangle - \bigl\langle z-w^{*},w\bigr\rangle +\Phi (z)-\Phi \bigl(w^{*}\bigr)-(\Phi (z)-\Phi \bigl(w^{*} \bigr) \\ =&\bigl\langle \mu ^{*}-z,\mu -w\bigr\rangle +\bigl\langle z-w^{*},\mu \bigr\rangle + \Phi (z)-\Phi \bigl(w^{*}\bigr)- \bigl[\bigl\langle z-w^{*},u\bigr\rangle +\Phi (z)-\Phi \bigl(w^{*}\bigr)\bigr] \\ \leq &\bigl\langle \mu ^{*}-z,\mu -w\bigr\rangle +\bigl\langle z-w^{*},\mu \bigr\rangle + \Phi (z)-\Phi \bigl(w^{*}\bigr). \end{aligned}$$

It follows that

$$\begin{aligned} { \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w\bigr\rangle + \bigl\langle z-w^{*},\mu \bigr\rangle +\Phi (z)- \Phi \bigl(w^{*}\bigr)\quad \text{for all } (z,\mu )\in G.} \end{aligned}$$

(3.2)

We replace $w^{*}\in \Gamma (w)\cap \Lambda$ in (3.2) with v, so there exists $(v,w)\in G$ such that

$$ \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z,\mu -w\bigr\rangle + \langle z-v,\mu \rangle +\Phi (z)-\Phi (v)\geq 0\quad \text{for all } (z, \mu )\in G, $$

which means w solves $\operatorname{SDIMVI} (G,\Gamma )$.

Next, we prove (b). Assume that $(v,w)\in G$ is a solution of $\operatorname{SDIMVI} (G, \Gamma )$, then we have

$$\begin{aligned} { \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w\bigr\rangle + \langle z-v,\mu \rangle +\Phi (z)-\Phi (v)\geq 0\quad \text{for all } (z, \mu )\in G.} \end{aligned}$$

(3.3)

For any $\hat{w}\in E$, $\hat{v}\in \Lambda$ and $v\in \Lambda$, we let $w(\tau )=w+\tau (\hat{w}-w)$ and $v(\tau )=v+\tau (\hat{v}-v)\in \Lambda$ for all $\tau \in [0,1]$. Take $z=v(\tau )$, $\mu =w(\tau )$, then by virtue of (3.3), it follows that

$$ \inf_{\mu ^{*}\in \Gamma (w(\tau ))}\bigl\langle \mu ^{*}-v(\tau ),w( \tau )-w \bigr\rangle +\bigl\langle v(\tau )-v,w(\tau )\bigr\rangle +\Phi \bigl(v(\tau ) \bigr)- \Phi (v)\geq 0, $$

which means

$$ \inf_{\mu ^{*}\in \Gamma (w(\tau ))}\bigl\langle \mu ^{*}-v(\tau ),\tau ( \hat{w}-w)\bigr\rangle +\bigl\langle \tau (\hat{v}-v),w(\tau )\bigr\rangle +\Phi \bigl(v+ \tau (\hat{v}-v)\bigr)-\Phi (v)\geq 0. $$

Due to Φ is convex, we know that

$$ \inf_{\mu ^{*}\in \Gamma (w(\tau ))}\bigl\langle \mu ^{*}-v(\tau ),\tau ( \hat{w}-w)\bigr\rangle +\bigl\langle \tau (\hat{v}-v),w(\tau )\bigr\rangle +\tau \Phi (\hat{v})-\tau \Phi (v)\geq 0. $$

Since $\tau \in [0,1]$, there is

$$ \inf_{\mu ^{*}\in \Gamma (w(\tau ))}\bigl\langle \mu ^{*}-v(\tau ), \hat{w}-w \bigr\rangle +\bigl\langle \hat{v}-v,w(\tau )\bigr\rangle +\Phi (\hat{v})- \Phi (v)\geq 0, $$

and so

$$ \sup_{\mu ^{*}\in \Gamma (w(\tau ))}\bigl\langle \mu ^{*}-v(\tau ), \hat{w}-w \bigr\rangle +\bigl\langle \hat{v}-v,w(\tau )\bigr\rangle +\Phi (\hat{v})- \Phi (v)\geq 0. $$

Because Γ is upper hemicontinuous and $\tau \in [0,1]$, it can be seen from (iii) of Definition 2.1 that Γ is upper semicontinuous. Let $\tau \to 0^{+}$, it follows from the definition of upper semicontinuity that

$$ \sup_{w^{*}\in F(w)}\bigl\langle w^{*}-v,\hat{w}-w\bigr\rangle +\langle \hat{v}-v,w\rangle +\Phi (\hat{v})-\Phi (v)\geq 0\quad \text{for all } \hat{w}\in E,\hat{v}\in \Lambda . $$

Since $\hat{w}\in E$ was chosen arbitrarily, we take $\hat{w}=w-ru$ for any $r\in \mathbb{R}$ and any $u\in E$, we know that there exists $w^{*}\in \Gamma (w)$ such that

$$ \bigl\langle w^{*}-v,-ru\bigr\rangle +\langle \hat{v}-v,w\rangle +\Phi (\hat{v})- \Phi (v)\geq 0\quad \text{for all } \hat{v}\in \Lambda , $$

which means

$$\begin{aligned} { r\bigl\langle w^{*}-v,u\bigr\rangle \leq \langle \hat{v}-v,w\rangle +\Phi ( \hat{v})-\Phi (v)}\quad \text{for all } \hat{v}\in \Lambda . \end{aligned}$$

(3.4)

If $\hat{v}\in \Lambda$ is fixed, then $\langle \hat{v}-v,w\rangle +\Phi (\hat{v})-\Phi (v)$ is a constant; therefore, we get for any $r\in \mathbb{R}$

$$ r\bigl\langle w^{*}-v,u\bigr\rangle \leq \text{constant}. $$

As a result, we can deduce that $w^{*}=v\in \Lambda$. Owing to $u\in E$ was chosen arbitrarily, from (3.4), there exists $w^{*}\in \Gamma (w)\cap \Lambda$ such that

$$ \bigl\langle \hat{v}-w^{*},w\bigr\rangle +\Phi (\hat{v})-\Phi \bigl(w^{*}\bigr)\geq 0 \quad \text{for all } \hat{v}\in \Lambda . $$

Thus, we conclude that w solves the $\operatorname{SIMVI} (\Lambda ,\Gamma )$. □

Remark 3.1

When $\Phi \equiv 0$, based on the same conditions, Luo [22] obtained the corresponding result of Theorem 3.1. Thus, we note that Theorem 3.1 extends the results in Theorem 3.1 in [22].

Theorem 3.2

Assume that $\Lambda \subset E^{*}$ is a nonempty, convex, and closed set, $\Gamma :E\to 2^{E^{*}}$ is a set-valued mapping with nonempty values, and $\Phi :\Lambda \subset E^{*}\to \mathbb{R}$ is a convex and lower semicontinuous functional, $\operatorname{int}\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\neq \emptyset$ and $\operatorname{int}({\operatorname{dom}}\Phi ^{*})\neq \emptyset$. Consider the following assertions:

• (a) $\Lambda _{\infty}\cap \{d\in E^{*}:\langle d,w\rangle +\Phi _{\infty}(d) \leq 0\quad \textit{for all } w\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\}=\{0\}$;

• (b) $D_{1}\subset E$ and $D_{2}\subset \Lambda$ are two bounded sets, where $D:=D_{2}\times D_{1}\subset G$, such that for any $w\in E/D_{1}$, $v\in \Lambda /D_{2}$, there exist some $\bar{\mu}\in D_{1}$, $\bar{z}\in D_{2}$ such that

  $$\begin{aligned} { \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}- \bar{z},\mu -w \bigr\rangle +\langle \bar{z}-v,\bar{\mu}\rangle +\Phi (\bar{z})- \Phi (v)< 0;} \end{aligned}$$

  (3.5)

• (c) The solution set of $\operatorname{SIMVI} (\Lambda ,\Gamma )$ is nonempty and bounded;

• (d) $\operatorname{int}\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\cap \operatorname{int}({-\operatorname{dom}}\Phi ^{*})\neq \emptyset$.

Then (a) ⇒ (b) if $\operatorname{int}(\operatorname{barr}\Lambda )\neq \emptyset$; (b) ⇒ (c) if F is upper hemicontinuous and monotone; (c) ⇒ (d); (d) ⇒ (a).

Proof

(a) ⇒ (b): If not, we suppose that (b) does not hold, then we can choose a sequence $\{(v_{n},w_{n})\}\subset G$, satisfying for any n, $\Vert v_{n}\Vert \geq n$, $\Vert w_{n}\Vert \geq n$, and

$$\begin{aligned} { \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w_{n} \bigr\rangle +\langle z-v_{n},\mu \rangle +\Phi (z)-\Phi (v_{n})\geq 0} \end{aligned}$$

(3.6)

for any $(z,\mu )\in G$, where $\Vert z\Vert < n$, $\Vert \mu \Vert < n$. Without losing the generality, we let $d_{n}=\frac{v_{n}}{\Vert v_{n}\Vert}$, and so $d_{n}$ weakly converges $d_{0}$ as $n\to \infty$. By the definition of the recession cone, we know that $d_{n}\in \Lambda _{\infty}$. Because $\Lambda _{\infty}$ is closed, we obtain $d_{0}\in \Lambda _{\infty}$. Since $\operatorname{int}(\operatorname{barr}\Lambda )\neq \emptyset$ and from Lemma 2.1, it can be seen that $d_{0}\neq 0$. Now, we let $\mu =\tilde{\mu}\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}$, $\mu ^{*}=\tilde{\mu}^{*}\in \Gamma (\tilde{\mu})\cap \Lambda$, and $z=\tilde{\mu}^{*}$ in (3.6), there exists a large number $M_{1}>0$, for $n\in [M_{1},+\infty )$ one has

$$ \bigl\langle \tilde{\mu}^{*}-\tilde{\mu}^{*},\tilde{ \mu}-w_{n}\bigr\rangle + \bigl\langle \tilde{\mu}^{*}-v_{n}, \tilde{\mu}\bigr\rangle +\Phi \bigl(\tilde{\mu}^{*}\bigr)- \Phi (v_{n})\geq 0, $$

and so

$$ \bigl\langle \tilde{\mu}^{*}-v_{n},\tilde{\mu}\bigr\rangle + \Phi \bigl(\tilde{\mu}^{*}\bigr)- \Phi (v_{n})\geq 0. $$

Multiplying both sides by $\frac{1}{\Vert v_{n}\Vert}$, we get

$$ \frac{1}{ \Vert v_{n} \Vert }\bigl\langle \tilde{\mu}^{*}-v_{n},\tilde{ \mu} \bigr\rangle +\frac{\Phi (\tilde{\mu}^{*})-\Phi (v_{n})}{ \Vert v_{n} \Vert } \geq 0, $$

it shows that

$$ \biggl\langle \frac{\tilde{\mu}^{*}}{ \Vert v_{n} \Vert },\tilde{\mu}\biggr\rangle + \frac{\Phi (\tilde{\mu}^{*})}{ \Vert v_{n} \Vert } \geq \biggl\langle \frac{v_{n}}{ \Vert v_{n} \Vert },\tilde{\mu}\biggr\rangle + \frac{\Phi (v_{n})}{ \Vert v_{n} \Vert }. $$

Then we have

$$ \liminf_{n\to \infty}\biggl[\biggl\langle \frac{\tilde{\mu}^{*}}{ \Vert v_{n} \Vert },\tilde{ \mu}\biggr\rangle + \frac{\Phi (\tilde{\mu}^{*})}{ \Vert v_{n} \Vert }\biggr]\geq \liminf_{n\to \infty} \biggl[\biggl\langle \frac{v_{n}}{ \Vert v_{n} \Vert },\tilde{\mu}\biggr\rangle + \frac{\Phi (v_{n})}{ \Vert v_{n} \Vert }\biggr], $$

it implies that

$$ \liminf_{n\to \infty}\biggl[\biggl\langle \frac{\tilde{\mu}^{*}}{ \Vert v_{n} \Vert },\tilde{ \mu}\biggr\rangle + \frac{\Phi (\tilde{\mu}^{*})}{ \Vert v_{n} \Vert }\biggr]\geq \liminf_{n\to \infty} \biggl\langle \frac{v_{n}}{ \Vert v_{n} \Vert },\tilde{\mu}\biggr\rangle + \liminf _{n\to \infty}\frac{\Phi (v_{n})}{ \Vert v_{n} \Vert }. $$

By (2.3), we get

$$ 0\geq \langle d_{0},\tilde{\mu}\rangle +\Phi _{\infty}(d_{0}), $$

which implies $d_{0}\in \{d\in E^{*}:\langle d,w\rangle +\Phi _{\infty}(d)\leq 0 \quad \text{for all } w\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\}$, and so $0\neq d_{0}\in \Lambda _{\infty}\cap \{d\in E^{*}:\langle d,w \rangle +\Phi _{\infty}(d)\leq 0\quad \text{for all } w\in \{w\in E: \Gamma (w)\cap \Lambda \neq \emptyset \}\}$, which contradicts (a).

(b) ⇒ (c): We define $H:G\to 2^{G}$ as follows:

$$\begin{aligned}& H(z,\mu ):=\Bigl\{ (v,w)\in G:\inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w\bigr\rangle +\langle z-v,\mu \rangle +\Phi (z)-\Phi (v) \geq 0\Bigr\} \\& \quad \text{for all } (z,\mu )\in G. \end{aligned}$$

Let $\{(v_{n},w_{n})\}\subset H(z,\mu )$ with $(v_{n},w_{n})\to (v_{0},w_{0})$, then

$$ \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z,\mu -w_{n} \bigr\rangle +\langle z-v_{n},\mu \rangle +\Phi (z)- \Phi (v_{n})\geq 0. $$

It turns out that

$$ \liminf_{n\to \infty}\Bigl[\inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w_{n}\bigr\rangle +\langle z-v_{n}, \mu \rangle +\Phi (z)\Bigr]\geq \liminf_{n\to \infty}\Phi (v_{n}), $$

because Φ is a lower semicontinuous functional, we have

$$ \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z,\mu -w_{0} \bigr\rangle +\langle z-v_{0},\mu \rangle +\Phi (z) \geq \Phi (v_{0}). $$

We deduce that $(v_{0},w_{0})\in H(z,\mu )$, which means $H(z,\mu )$ is closed.

Step 1. Next we will demonstrate that H is a KKM mapping. In fact, by contradiction, suppose that there exist $\gamma _{1},\gamma _{2},\ldots ,\gamma _{n}\in [0,1]$, $\sum_{i=1}^{n}\gamma _{i}=1$, and

$$ (\tilde{v},\tilde{w})=\gamma _{1}(z_{1},\mu _{1})+\gamma _{2}(z_{2}, \mu _{2})+ \cdots +\gamma _{n}(z_{n},\mu _{n})\in \operatorname{co}\bigl\{ (z_{1},\mu _{1}),(z_{2}, \mu _{2}),\ldots ,(z_{n},\mu _{n})\bigr\} $$

for any finite set $\{(z_{1},\mu _{1}),(z_{2},\mu _{2}),\ldots ,(z_{n},\mu _{n})\}\in G$ such that

$$ (\tilde{v},\tilde{w})\notin \bigcup H_{i\in \{1,2,\ldots ,n\}}(z_{i}, \mu _{i}). $$

Then, for any $i=1,2,\ldots ,n$,

$$ \inf_{\mu _{i}^{*}\in \Gamma (\mu _{i})}\bigl\langle \mu _{i}^{*}-z_{i}, \mu _{i}-\tilde{w}\bigr\rangle +\langle z_{i}-\tilde{v},\mu _{i}\rangle + \Phi (z_{i})-\Phi (\tilde{v})< 0. $$

Due to Γ is monotone, there is $\mu _{i}^{*}\in \Gamma (\mu _{i})$ such that for any $u^{*}\in \Gamma (\tilde{u})$ and $i=1,2,\ldots ,n$,

$$\begin{aligned} 0 >&\bigl\langle \mu _{i}^{*}-w^{*}+w^{*}-z_{i}, \mu _{i}-\tilde{w}\bigr\rangle + \langle z_{i}-\tilde{v},\mu _{i}\rangle +\Phi (z_{i})-\Phi (\tilde{v}) \\ \geq &\bigl\langle w^{*}-z_{i},\mu _{i}\bigr\rangle -\bigl\langle w^{*}-z_{i}, \tilde{w}\bigr\rangle + \langle z_{i}-\tilde{v},\mu _{i}\rangle +\Phi (z_{i})- \Phi (\tilde{v}) \\ =&\bigl\langle w^{*}-\tilde{v},\mu _{i}\bigr\rangle -\bigl\langle w^{*}-z_{i}, \tilde{w}\bigr\rangle +\Phi (z_{i})-\Phi (\tilde{v}). \end{aligned}$$

Since Φ is convex, then we get

$$\begin{aligned} 0 >&\Biggl\langle w^{*}-\tilde{v},\sum_{i=1}^{n} \gamma _{i}\mu _{i} \Biggr\rangle -\Biggl\langle w^{*}-\sum_{i=1}^{n}\gamma _{i}z_{i},\tilde{w} \Biggr\rangle +\Phi \Biggl(\sum _{i=1}^{n}\gamma _{i}z_{i} \Biggr)-\Phi (\tilde{v}) \\ =&\bigl\langle w^{*}-\tilde{v},\tilde{w}\bigr\rangle -\bigl\langle w^{*}-\tilde{v}, \tilde{w}\bigr\rangle +\Phi (\tilde{v})-\Phi ( \tilde{v}) \\ =&0, \end{aligned}$$

which is contradiction. Therefore, H is the KKM mapping.

Step 2. We can suppose that D is a bounded, convex, and closed subset(if not, we consider replacing D with the closed convex hull of D). Let $\{(z_{1},\mu _{1}),(z_{2},\mu _{2}),\ldots ,(z_{m},\mu _{m})\}$ be the definite number of points in G, and let $N:=\operatorname{co}(D\cup \{(z_{1},\mu _{1}),(z_{2},\mu _{2}),\ldots ,(z_{m},\mu _{m}) \})$. N is weakly compact convex. Next, we consider the set-valued mapping H̃, defined by $\widetilde{H}(z,\mu ):=H(z,\mu )\cap N$ for any $(z,\mu )\in N$.

Firstly, we prove that $H(z,\mu )$ is a convex set for any $(z,\mu )\in N$. Let $\lambda \in [0,1]$, for arbitrary $w_{1},w_{2}\in E$ and $v_{1},v_{2}\in \Lambda$, there is

$$\begin{aligned}& \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z,\mu -\bigl[ \lambda w_{1}+(1- \lambda ) w_{2}\bigr]\bigr\rangle +\bigl\langle z-\bigl[\lambda v_{1}+(1-\lambda ) v_{2}\bigr], \mu \bigr\rangle +\Phi (z) \\& \qquad {}-\Phi \bigl(\lambda v_{1}+(1-\lambda ) v_{2}\bigr) \\& \quad = \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z,\lambda (\mu -w_{1}) \bigr\rangle +\bigl\langle \mu ^{*}-z,(1-\lambda ) ( \mu -w_{2})\bigr\rangle +\bigl\langle \lambda (z-v_{1}),x \bigr\rangle \\& \qquad {}+\bigl\langle (1-\lambda ) (z-v_{2}),\mu \bigr\rangle +\lambda \Phi (z)+(1- \lambda ) \Phi (z)-\Phi \bigl(\lambda v_{1}+(1-\lambda ) v_{2}\bigr) \\& \quad \geq \lambda \Bigl[\inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w_{1}\bigr\rangle +\langle z-v_{1}, \mu \rangle +\Phi (z)-\Phi (v_{1})\Bigr] \\& \qquad {}+(1-\lambda )\Bigl[\inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-z, \mu -w_{2}\bigr\rangle +\langle z-v_{2}, \mu \rangle +\Phi (z)-\Phi (v_{2})\Bigr] \\& \quad \geq 0, \end{aligned}$$

so $H(z,\mu )$ is convex. It is easy to know that each $\widetilde{H}(z,\mu )$ is a weakly compact convex subset of N. Obviously, $\widetilde{H}(z,\mu )$ is a closed set for any $(z,\mu )\in N$.

Now we prove H̃ is a KKM mapping. On the contrary, suppose that there are $\eta _{1},\eta _{2},\ldots , \eta _{n}\in [0,1]$, $\sum_{i=1}^{n}\eta _{i}=1$, and

$$ (t,s)=\eta _{1}(z_{1},\mu _{1})+\eta _{2}(z_{2},\mu _{2})+\cdots + \eta _{n}(z_{n},\mu _{n})\in \operatorname{co}\bigl\{ (z_{1},\mu _{1}),(z_{2},\mu _{2}), \ldots ,(z_{n},\mu _{n})\bigr\} $$

for any finite set $\{(z_{1},\mu _{1}),(z_{2},\mu _{2}),\ldots ,(z_{n},\mu _{n})\}\in N$ such that

$$ (t,s)\notin \bigcup \widetilde{H}_{i\in \{1,2,\ldots ,n\}}(z_{i},\mu _{i}). $$

Then, for any $i=1,2,\ldots ,n$,

$$ \inf_{\mu _{i}^{*}\in \Gamma (\mu _{i})}\bigl\langle \mu _{i}^{*}z_{i}, \mu _{i}-s\bigr\rangle +\langle z_{i}-t,\mu _{i} \rangle +\Phi (z_{i})- \Phi (t)< 0. $$

Because Γ is monotone, there exists $\mu _{i}^{*}\in \Gamma (\mu _{i})$ such that for any $w^{*}\in \Gamma (s)$ and $i=1,2,\ldots ,n$,

$$\begin{aligned} 0 >&\bigl\langle \mu _{i}^{*}-w^{*}+w^{*}-v_{i}, \mu _{i}-s\bigr\rangle + \langle z_{i}-t,\mu _{i} \rangle +\Phi (z_{i})-\Phi (t) \\ \geq &\bigl\langle \mu _{i}^{*}-w^{*},\mu _{i}-s\bigr\rangle +\bigl\langle w^{*}-z_{i}, \mu _{i}-s\bigr\rangle +\langle z_{i}-t,\mu _{i} \rangle +\Phi (z_{i})- \Phi (t) \\ \geq &\bigl\langle w^{*}-z_{i},\mu _{i}\bigr\rangle -\bigl\langle w^{*}-z_{i},s \bigr\rangle +\langle z_{i}-t,\mu _{i}\rangle +\Phi (z_{i})-\Phi (t) \\ =&\bigl\langle w^{*}-t,\mu _{i}\bigr\rangle -\bigl\langle w^{*}-z_{i},s\bigr\rangle + \Phi (z_{i})-\Phi (t). \end{aligned}$$

Since Φ is convex, then we get

$$\begin{aligned} 0 >&\Biggl\langle w^{*}-t,\sum_{i=1}^{n} \eta _{i}\mu _{i}\Biggr\rangle -\Biggl\langle w^{*}- \sum_{i=1}^{n}\eta _{i}z_{i},s\Biggr\rangle +\Phi \Biggl(\sum _{i=1}^{n}\eta _{i}z_{i}\Biggr)- \Phi (t) \\ =&\bigl\langle w^{*}-t,s\bigr\rangle -\bigl\langle w^{*}-t,s\bigr\rangle +\Phi (t)-\Phi (t) \\ =&0, \end{aligned}$$

which leads to a contradiction. Therefore, H̃ is a KKM mapping.

Step 3. Since $\widetilde{H}(z,\mu )$ is a weakly compact closed set for any $(z,\mu )\in N$ and H̃ is also a KKM mapping, so by Lemma 2.3 we have

$$ \emptyset \neq \bigcap_{(z,x)\in N}\widetilde{H}(z,\mu ). $$

Furthermore, if there exists some $(v_{0},w_{0})\in \bigcap_{(z,\mu )\in N}\widetilde{H}(z,\mu )$ but $(v_{0},w_{0})\notin D$, then from (3.5) we know that for some $(\bar{z},\bar{\mu})\in D_{2}\times D_{1}\subset G$, there is

$$ \inf_{\mu ^{*}\in \Gamma (\mu )}\bigl\langle \mu ^{*}-\bar{z},\bar{ \mu}-w_{0} \bigr\rangle +\langle \bar{z}-v_{0},\bar{\mu} \rangle +\Phi (\bar{\mu})- \Phi (v_{0})< 0, $$

therefore, $(v_{0},w_{0})\notin H(z,\mu )$, and so $(v_{0},w_{0})\notin \widetilde{H}(z,\mu )$, it is a contradiction. Then

$$\begin{aligned} { \emptyset \neq \bigcap_{(z,\mu )\in N} \widetilde{H}(z,\mu ) \subset D.} \end{aligned}$$

(3.7)

Let $(v,w)\in \bigcap_{(z,\mu )\in N}\widetilde{H}(z,\mu )$, it can be seen from (3.7) that $(v,w)\in D$, then we have

$$ \bigcap_{(z,\mu )\in N}\widetilde{H}(z,\mu )\subset \bigcap _{i=1}^{m} \widetilde{H}(z_{i},\mu _{i}), $$

and hence $(v,w)\in \bigcap_{i=1}^{m}(H(z_{i},\mu _{i})\cap D)$. It can be known that $\{H(z,\mu )\cap D:(z,\mu )\in G\}$ has the property of finite intersection. For each $(z,\mu )\in G$, it follows from the weak compactness of $H(z,\mu )\cap D$ that $\bigcap_{(z,\mu )\in G}(H(z,\mu )\cap D)$ is nonempty, which coincides with the solution set of $\operatorname{SDIMVI} (G,\Gamma )$. Thus, according to Theorem 3.1, we can obtain that the solution set of $\operatorname{SIMVI} (\Lambda , \Gamma )$ is nonempty and bounded.

(c) ⇒ (d): If (d) does not hold, then $\operatorname{int}(\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \})\cap \operatorname{int}(-\operatorname{dom}\Phi ^{*})=\emptyset$. There exists $w\in E$ and $\Gamma (w)\cap \Lambda \neq \emptyset$, but $w\notin \operatorname{int}(- \operatorname{dom}\Phi ^{*})$, then we have

$$\begin{aligned} { \sup_{v\in \Lambda}\bigl\{ \langle v,w\rangle -\Phi (v) \bigr\} \geq +\infty .} \end{aligned}$$

(3.8)

From (c) we know that the solution set of $\operatorname{SIMVI} (\Lambda ,\Gamma )$ is nonempty, then there exists $w\in E$, $w^{*}\in \Gamma (w)\cap \Lambda$ such that

$$ \bigl\langle \tilde{v}-w^{*},w\bigr\rangle +\Phi (\tilde{v})-\Phi \bigl(w^{*}\bigr)\geq 0 \quad \text{for all } \tilde{v}\in \Lambda . $$

Let $d_{0}\in \Lambda _{\infty}$, according to the definition of recession of cone, there is $w^{*}+\lambda d_{0}\in \Lambda$, where $\lambda >0$. Due to the arbitrariness of $\tilde{v}\in \Lambda$, taking $\tilde{v}=w^{*}+td\in \Lambda$, we have

$$\begin{aligned} \bigl\langle w^{*}+td-w^{*},u\bigr\rangle +\Phi \bigl(w^{*}+td\bigr)-\Phi \bigl(w^{*}\bigr) \geq &0 \end{aligned}$$

and

$$\begin{aligned} \bigl\langle w^{*}+td,-w\bigr\rangle -\Phi \bigl(w^{*}+td \bigr) \leq &\bigl\langle -w^{*},w \bigr\rangle -\Phi \bigl(w^{*}\bigr). \end{aligned}$$

Since $-\Phi (w^{*})<+\infty$. Thus, $\langle -w^{*},w\rangle -\Phi (w^{*})<+\infty$, and so

$$ \bigl\langle w^{*}+td,-w\bigr\rangle -\Phi \bigl(w^{*}+td \bigr)< +\infty , $$

which contradicts (3.8). The proof is complete.

(d) ⇒ (a): If (a) does not hold, then $\Lambda _{\infty}\cap \{d\in E^{*}:\langle d,w\rangle +\Phi _{\infty}(d) \leq 0 \text{ for all } w\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\}\neq \{0\}$. We know that there exists a sequence $\{d_{n}\}\subset \Lambda _{\infty}\cap \{d\in E^{*}:\langle d,w \rangle +\Phi _{\infty}(d)\leq 0 \text{ for all } w \in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\}$. Without losing the generality, we let $d_{n}=\frac{v_{n}}{\Vert v_{n}\Vert}$, and so $d_{n}$ weakly converges $d_{0}$ as $n\to \infty$. Since $\Lambda _{\infty}$ is a closed and convex cone, so $d_{0}\in \Lambda _{\infty}$. According to Lemma 2.1, it follows that $d_{0}\neq 0$.

For any $\tilde{w}\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}$, we have

$$ \langle d_{n},\tilde{w}\rangle +\Phi _{\infty}(d_{n}) \leq 0. $$

Combining with $d_{n}\rightharpoonup d_{0}$ and the weak lower semicontinuity of $\Phi _{\infty}(\cdot )$, it follows that

$$\begin{aligned} { \langle d_{0},-\tilde{w}\rangle -\Phi _{\infty}(d_{0})\geq 0.} \end{aligned}$$

(3.9)

Due to $\operatorname{int}(\{w\in E:\Gamma (w))\cap \Lambda \neq \emptyset \}\cap ( \operatorname{int}({-\operatorname{dom}}\Phi ^{*})\neq \emptyset$, then there exists $\xi \in \operatorname{int}(\{w\in E:\Gamma (w))\cap \Lambda \neq \emptyset \}\cap \operatorname{int}({-\operatorname{dom}}\Phi ^{*})$. Next we prove that

$$\begin{aligned} { \langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})=0.} \end{aligned}$$

(3.10)

Indeed, if (3.10) does not hold, then

$$ \langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})>0, $$

(e)

as (3.9) holds.

By $\xi \in \operatorname{int}(-\operatorname{dom}\Phi ^{*})$, we obtain

$$\begin{aligned} { \Phi ^{*}(-\xi )=\sup_{v\in \Lambda}{\langle v,-\xi \rangle -\Phi (v)}< + \infty .} \end{aligned}$$

(3.11)

Let $w_{0}\in \Lambda$. $d_{0}\in \Lambda _{\infty}$ implies that $w_{0}+td_{0}\in \Lambda$ for all $t>0$. It follows from (3.11) that

$$ \langle w_{0}+td_{0},-\xi \rangle -\Phi (w_{0}+td_{0})< +\infty . $$

From (2.1) and (2.2), we have

$$ \langle w_{0}+td_{0},-\xi \rangle -\Phi (w_{0})-t\Phi _{\infty}(d_{0})< + \infty , $$

which immediately implies that

$$ \langle w_{0},-\xi \rangle -\Phi (w_{0})+t\bigl(\langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})\bigr)< + \infty . $$

It is known that $w_{0}\in \Lambda$, and from (3.11) we can deduced that $\langle w_{0},-\xi \rangle -\Phi (w_{0})<+\infty$. Thus

$$\begin{aligned} { t\bigl(\langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})\bigr)< +\infty .} \end{aligned}$$

(3.12)

According to $(e)$, and letting $t\to +\infty$, we get $t(\langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0}))$ has no upper bound, which contradicts (3.12). Therefore, (3.10) is proved. Since $\xi \in \operatorname{int}\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\cap \operatorname{int}({-\operatorname{dom}}\Phi ^{*})$, for any $w_{1}\in E$, there exists $t\in (0,1)$ such that $\xi +(1-t)w_{1}\in \operatorname{int}\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\cap \operatorname{int}({-\operatorname{dom}}\Phi ^{*})$. From (3.9), we know that

$$ \bigl\langle d_{0},-\bigl(\xi +(1-t)w_{1}\bigr)\bigr\rangle -\Phi _{\infty}(d_{0})\geq 0, $$

because $\langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})=0$, then we have $\langle d_{0},-w_{1}\rangle \geq 0$.

For any $w_{1}\in E$, there exists $t\in (0,1)$ such that $\xi -(1-t)w_{1}\in \operatorname{int}\{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\cap \operatorname{int}({-\operatorname{dom}}\Phi ^{*})$. By (3.9), we get

$$ \bigl\langle d_{0},-\bigl(\xi -(1-t)w_{1}\bigr)\bigr\rangle -\Phi _{\infty}(d_{0})\geq 0. $$

By $\langle d_{0},-\xi \rangle -\Phi _{\infty}(d_{0})=0$, we can obtain $\langle d_{0},w_{1}\rangle \geq 0$. Then it can be deduced that $\langle d_{0},w_{1}\rangle =0$, which contradicts $d_{0}\neq 0$. Thus $\Lambda _{\infty}\cap \{d\in E^{*}:\langle d,w\rangle +\Phi _{\infty} \leq 0\quad \text{for all } w\in \{w\in E:\Gamma (w)\cap \Lambda \neq \emptyset \}\}=\{0\}$ is verified. □

4 Stability for the SIMVI

Our goal in this section is to establish the stability of the solutions for $\operatorname{SIMVI} (\Lambda ,\Gamma )$ with monotone and upper hemicontinuous mappings. The following Theorem 4.1 is of great help in obtaining the stability results.

Theorem 4.1

Let $\alpha _{0}\in Z_{1}$ and $\beta _{0}\in Z_{2}$ be given points, $(Z_{1},d_{1})$ and $(Z_{2},d_{2})$ be metric spaces, $\Gamma :E\times Z_{2}\to 2^{E^{*}}$ be a lower semicontinuous set-valued mapping for the second variable on $Z_{2}$, $\Phi :L(\alpha )\subset E^{*}\to \mathbb{R}$ be a convex and lower semicontinuous functional, and $L:Z_{1}\to 2^{E^{*}}$ be a continuous set-valued mapping. Assume there is a neighborhood $\mathcal{W\times U}$ of $(\alpha _{0},\beta _{0})$ such that $\Gamma (w,\beta )$ has nonempty, closed values for every $w\in E$ and $\beta \in \mathcal{U}$, and $L(\alpha )$ has nonempty, convex, closed values for any $\alpha \in \mathcal{W}$. If

$$\begin{aligned}& \bigl(L(\alpha _{0})\bigr)_{\infty}\cap \bigl\{ d\in E^{*}:\langle d,w\rangle +\Phi _{ \infty}(d)\leq 0 \textit{ for all } w\in \bigl\{ w\in E:\Gamma (w,\beta _{0}) \cap L(\alpha _{0})\bigr\} \neq \emptyset \bigr\} \\& \quad =\{0\}, \end{aligned}$$

then there exists a neighborhood $\mathcal{\bar{W}\times \bar{U}}$ of $(\alpha _{0},\beta _{0})$ with $\mathcal{\bar{W}\times \bar{U}}\subset \mathcal{W\times U}$, such that for any $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$,

$$\begin{aligned}& \bigl(L(\alpha )\bigr)_{\infty}\cap \bigl\{ d\in E^{*}:\langle d,w\rangle +\Phi _{ \infty}(d)\leq 0, \textit{for all } w\in \bigl\{ w\in E: \Gamma (w,\beta ) \cap L(\alpha )\bigr\} \neq \emptyset \bigr\} \\& \quad =\{0\}. \end{aligned}$$

Proof

Suppose the conclusion is not true, then for any neighborhood $\mathcal{\bar{W}\times \bar{U}}$ of $(\alpha _{0},\beta _{0})$, there is $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$ such that $(L(\alpha ))_{\infty}\cap \{d\in E^{*}:\langle d,w\rangle +\Phi _{ \infty}(d)\leq 0\quad \text{for all } w\in \{w\in E:\Gamma (w,\beta ) \cap L(\alpha )\}\neq \emptyset \}\neq \{0\}$ holds. Then, we can select a sequence ${(\alpha _{n},\beta _{n})}$ in $Z_{1}\times Z_{2}$ with $(\alpha _{n},\beta _{n})$ converging to $(\alpha _{0},\beta _{0})$ such that for any n, $(L(\alpha _{n}))_{\infty}\cap \{d\in E^{*}:\langle d,w\rangle +\Phi _{ \infty}(d)\leq 0\quad \text{for all } w\in \{w\in E:\Gamma (w,\beta _{n}) \cap L(\alpha _{n})\}\neq \emptyset \}\neq \{0\}$. Thus, there exists a sequence $\{d_{n}\}$ such that for any n, $\Vert d_{n}\Vert =1$ and $d_{n}\in (L(\alpha _{n}))_{\infty}\cap \{d\in E^{*}:\langle d,w \rangle +\Phi _{\infty}(d)\leq 0\quad \text{for all } w\in \{w\in E: \Gamma (w,\beta _{n})\cap L(\alpha _{n})\}\neq \emptyset \}$. Without losing the generality, we can suppose that $d_{n}\rightharpoonup d_{0}$ as $n\to \infty$. Furthermore, according to Lemma 2.1 it can be known that $d_{0}\neq 0$. Since $d_{n}\in (L(\alpha _{n}))_{\infty}$ and L on $Z_{1}$ is upper semicontinuous, then from Lemma 2.2 we have $d_{n}\in (L(\alpha _{0}))_{\infty}$ for sufficiently large n. Because $(L(\alpha _{0}))_{\infty}$ is a closed cone, so $d_{0}\in (L(\alpha _{0}))_{\infty}$.

For any given $\bar{w}\in \{w\in E:\Gamma (w,\beta _{0})\cap L(\alpha _{0})\neq \emptyset \}$, there is $\delta _{0}\in E^{*}$ satisfying $\delta _{0}\in \Gamma (\bar{w},\beta _{0})\cap L(\alpha _{0})$. Next we prove that for any n, there is a sequence $\{\bar{\delta _{n}}\}$ such that $\bar{\delta _{n}}\to \delta _{0}$, $\bar{\delta}_{n}\in \Gamma (\bar{u},\beta _{n})\cap L(\alpha _{n})$. If not, we prove the contrary conclusion holds then for any sequence $\{\delta _{n}\}$ such that $\delta _{n}\to \delta _{0}$. However, $\delta _{n}\notin \Gamma (\bar{w},\beta _{n})\cap L(\alpha _{n})$. Since $\beta _{n}\to \beta _{0}$ and $\delta _{0}\in \Gamma (\bar{w},\beta _{0})$, and Γ on $Z_{2}$ is lower semicontinuous, so there is a sequence $\{\tilde{\delta}_{n}\}$ such that $\{\tilde{\delta}_{n}\}\to \delta _{0}$ and $\tilde{\delta}_{n}\in \Gamma (\bar{w},\beta _{n})$ for any n. Therefore, $\tilde{\delta}_{n}\notin L(\alpha _{n})$. Because Γ and L have nonempty closed values and L is lower semicontinuous, $\delta _{0}\notin L(\alpha _{0})$. Therefore, we get a contradiction. Now we know that there exists $\bar{\delta}_{n}\in \Gamma (\bar{w},\beta _{n})\cap L(\alpha _{n})$, it means that $\Gamma (\bar{w},\beta _{n})\cap L(\alpha _{n})\neq \emptyset$. Moreover, w̄ is fixed, then we get $\bar{w}\in \{w\in E:\Gamma (w,\beta _{n})\cap L(\alpha _{n})\neq \emptyset \}$. From $d_{n}\in \{d\in E^{*}:\langle d,w\rangle +\Phi _{\infty}(d)\leq 0 \quad \text{for all } w\in \{w\in E:\Gamma (w,\beta _{n})\cap L( \alpha _{n})\neq \emptyset \}\}$, we can obtain

$$ \langle d_{n},\bar{w}\rangle +\Phi _{\infty}(d_{n}) \leq 0. $$

Combining with $d_{n}\rightharpoonup d_{0}$ and the weak lower semicontinuity of $\Phi _{\infty}(\cdot )$, it follows that

$$ \langle d_{0},\bar{w}\rangle +\Phi _{\infty}(d_{0}) \leq 0. $$

Then we have

$$ d_{0}\in \bigl\{ d\in E^{*}:\langle d,w\rangle +\Phi _{\infty}(d)\leq 0, \text{for all } w\in \bigl\{ w\in E:\Gamma (w,\beta _{0})\cap L( \alpha _{0})\neq \emptyset \bigr\} \bigr\} , $$

and so

$$\begin{aligned} d_{0}\in{}& \bigl(L(\alpha _{0})\bigr)_{\infty} \\ &{}\cap \bigl\{ d\in E^{*}:\langle d,w \rangle +\Phi _{\infty}(d)\leq 0, \text{for all } w\in \bigl\{ w\in E: \Gamma (w,\beta _{0})\cap L( \alpha _{0})\neq \emptyset \bigr\} \bigr\} \end{aligned}$$

with $d_{0}\neq 0$, which contradicts the assumption and ends the proof of the theorem. □

Remark 4.1

If $\Phi \equiv 0$, then $\Phi _{\infty}\equiv 0$. Consequently, Theorem 4.1 reduces to Theorem 4.1 of [22]. Thus, Theorem 4.1 is a generalization of Theorem 4.1 in [22].

If K is bounded, we know $K_{\infty}=\{0\}$, so from Theorem 4.1 we have the following result.

Corollary 4.1

Let $\alpha _{0}\in Z_{1}$ and $\beta _{0}\in Z_{2}$ be given points, $(Z_{1},d_{1})$ and $(Z_{2},d_{2})$ be metric spaces, $\Gamma :E\times Z_{2}\to 2^{E^{*}}$ be a set-valued mapping with nonempty value, $\Phi :L(\alpha )\subset E^{*}\to \mathbb{R}$ be a convex and lower semicontinuous functional, and $L:Z_{1}\to 2^{E^{*}}$ be a set-valued mapping with nonempty bounded value. Then, for any $(\alpha ,\beta )\in{Z_{1}\times Z_{2}}$,

$$\begin{aligned}& \bigl(L(\alpha )\bigr)_{\infty}\cap \bigl\{ d\in E^{*}:\langle d,w\rangle +\Phi _{ \infty}(d)\leq 0 \textit{ for all } w\in \bigl\{ w\in E: \Gamma (w,\beta ) \cap L(\alpha )\neq \emptyset \bigr\} \bigr\} \\& \quad =\{0\}. \end{aligned}$$

Theorem 4.2

Let $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$, $S(\alpha ,\beta )$ and $S(\alpha _{0},\beta _{0})$ represent the solution sets of $\operatorname{SIMVI} (L(\alpha ),\Gamma (\cdot ,\beta ))$ and $\operatorname{SIMVI} (L( \alpha _{0}),\Gamma (\cdot ,\beta _{0}))$, respectively. If the conditions in Theorem 4.1hold and

• (a) for every $\beta \in \mathcal{U}$, the mapping $w\mapsto \Gamma (w,\beta )$ is monotone and upper hemicontinuous on E;

• (b) the solution set of $\operatorname{SIMVI} (L(\alpha _{0}),\Gamma (\cdot , \beta _{0}))$ is nonempty and bounded;

• (c) $\operatorname{int}\{w\in E:\Gamma (w,\beta )\cap L(\alpha )\neq \emptyset \}\neq \emptyset$ and $\operatorname{int}({-\operatorname{dom}}\Phi ^{*})\neq \emptyset$.

Then

• (A) there exists a neighborhood $\mathcal{\bar{W}\times \bar{U}}$ of $(\alpha _{0},\beta _{0})$ with $\mathcal{\bar{W}\times \bar{U}}\subset \mathcal{W\times U}$ such that for every $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$, the solution set of $\operatorname{SIMVI} (L(\alpha ),\Gamma (\cdot ,\beta ))$ is nonempty and bounded;

• (B) if Φ is continuous on $L(\alpha )$ with $\alpha \in \mathcal{\bar{W}}$ and $\bigcup_{\alpha \in \mathcal{\bar{W}}}L(\alpha )$ is compact, then $\limsup_{(\alpha ,\beta )\to (\alpha _{0},\beta _{0})}S(\alpha , \beta )\subset S(\alpha _{0},\beta _{0})$.

Proof

• (A) By Theorem 3.2, it follows from condition (b) in Theorem 4.2 that

  $$\begin{aligned}& \bigl(L(\alpha _{0})\bigr)_{\infty}\cap \{d\in E^{*}: \langle d,w\rangle +\Phi _{ \infty}(d)\leq 0 \text{ for all } \\& \quad w\in \bigl\{ w \in E:\Gamma (w,\beta _{0}) \cap L(\alpha _{0})\neq \emptyset \bigr\} =\{0\}. \end{aligned}$$

  Next, applying Theorem 4.1, we know that there is a neighborhood $\mathcal{\bar{W}\times \bar{U}}$ of $(\alpha _{0},\beta _{0})$ with $\mathcal{\bar{W}\times \bar{U}}\subset \mathcal{W\times U}$ such that for any $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$,

  $$\begin{aligned}& \bigl(L(\alpha )\bigr)_{\infty}\cap \{d\in E^{*}:\langle d,w \rangle +\Phi _{ \infty}(d)\leq 0 \text{ for all } \\& \quad w\in \bigl\{ w\in E:\Gamma (w,\beta ) \cap L(\alpha )\neq \emptyset \bigr\} =\{0\}. \end{aligned}$$

  Utilizing Theorem 3.2 once more, we can now derive the solution set of $\operatorname{SIMVI} (L(\alpha ),\Gamma (\cdot ,\beta ))$ is nonempty and bounded for any $(\alpha ,\beta )\in \mathcal{\bar{W}\times \bar{U}}$.

• (B) For any sequence $\{(\alpha _{n},\beta _{n})\}\subset \mathcal{\bar{W}\times \bar{U}}$ with $(\alpha _{n},\beta _{n})\to (\alpha _{0},\beta _{0})$ when $n\to \infty$, we will verify that

  $$ \limsup_{n\to \infty}S(\alpha _{n},\beta _{n}) \subset S(\alpha _{0}, \beta _{0}). $$

  Let $w_{0}\in \lim \sup_{n\to \infty}S(\alpha _{n},\beta _{n})$, then there exists a sequence $\{w_{n}\}\subset S(\alpha _{n},\beta _{n})$ such that $w_{n}\to w_{0}$ as $n\to \infty$. From Theorem 3.1 and Theorem 4.2 (A), it can be known that the solution set of $\operatorname{SDIMVI} (G( \alpha _{n}),\Gamma (\cdot ,\beta _{n}))$ is nonempty and bounded, where $G(\alpha _{n}):=L(\alpha _{n})\times E$. So there exists $v_{n}\in L(\alpha _{n})$ such that $v_{n}\to v_{0}$ as $n\to \infty$ and

  $$\begin{aligned}& \inf_{\mu ^{*}\in \Gamma (\mu ,\beta _{n})}\bigl\langle \mu ^{*}-z,\mu -w_{n} \bigr\rangle +\langle z-v_{n},\mu \rangle +\Phi (z)-\Phi (v_{n}) \\& \quad \geq 0, \quad \text{for all }(z,\mu )\in L(\alpha _{n})\times E. \end{aligned}$$

  (4.1)

  On the other side, for any $z_{0}\in L(\alpha _{0})$ and ${\mu _{0}}^{*}\in \Gamma (\mu ,\beta _{0})$, L is lower semicontinuous on $Z_{1}$, and $\alpha _{n}\to \alpha _{0}$, then there exists $z_{n}\in L(\alpha _{n})$ such that $z_{n}\to z_{0}$ as $n\to \infty$. Since Γ is lower semicontinuous on $Z_{2}$ and $\beta _{n}\to \beta _{0}$, there exists ${\mu _{n}}^{*}\in \Gamma (\mu ,\beta _{n})$ such that ${\mu _{n}}^{*}\to{\mu _{0}}^{*}$ as $n\to \infty$. According to $v_{n}\in L(\alpha _{n})$ and upper semicontinuity of L, we get $v_{0}\in L(\alpha _{0})$. Therefore, (4.1) means that

  $$ \bigl\langle {\mu _{n}}^{*}-z_{n},\mu -w_{n}\bigr\rangle +\langle z_{n}-v_{n}, \mu \rangle +\Phi (z_{n})-\Phi (v_{n})\geq 0\quad \text{for all } \mu \in E. $$

  Since Φ is continuous on $L(\alpha )$, letting $n\to +\infty$, we obtain $(v_{0},w_{0})\in L(\alpha _{0})\times E$ and

  $$\begin{aligned}& \inf_{{\mu _{0}}^{*}\in \Gamma (\mu ,\beta _{0})}\bigl\langle {\mu _{0}}^{*}-z_{0}, \mu -w_{0}\bigr\rangle +\langle z_{0}-v_{0},w \rangle +\Phi (z_{0})-\Phi (v_{0}) \\& \quad \geq 0\quad \text{for all } z_{0}\in L(\alpha _{0}),\mu \in E. \end{aligned}$$

  Using Theorem 3.1 once more, there exist $w_{0}\in E$ and ${w_{0}}^{*}\in \Gamma (w_{0},\beta _{0})\cap L(\alpha _{0})$ such that

  $$ \bigl\langle \tilde{v}-{w_{0}}^{*},w_{0}\bigr\rangle +\Phi (\tilde{v})-\Phi \bigl({w_{0}}^{*}\bigr) \geq 0,\quad \text{for all } \tilde{v}\in L(\alpha _{0}), $$

  so $w_{0}\in S(\alpha _{0},\beta _{0})$.

 □

Next, we will give Examples 4.1, 4.2, 4.3, 4.4 to demonstrate the necessity of the conditions in Theorem 4.2.

Example 4.1

Let $Z_{1}=Z_{2}=[-1,1]$, $\alpha _{0}=\beta _{0}=0$, $\Phi (w)=\frac{1}{2}w$

$$\begin{aligned}& L(\alpha )=[0,2],\\& \Gamma (w,\beta )= \textstyle\begin{cases} \{0\}, &\beta \neq 0, \\ \{2\}, & \beta =0. \end{cases}\displaystyle \end{aligned}$$

It is easy to note that $L(\alpha )$ is continuous, $\Gamma (\cdot ,\beta )$ is both monotone and upper hemicontinuous on $[0,2]$, and $\Gamma (w,\cdot )$ is not lower semicontinuous at $\beta =0$. By calculation, we get $S(0,0)=(-\infty ,-\frac{1}{2}]$ and $S(0,\beta )=[-\frac{1}{2},+\infty )$ for any $\beta \neq 0$. Consequently, $\limsup_{\beta \to 0}S(0,\beta )=[-\frac{1}{2},+\infty )\not \subset S(0,0)$.

Example 4.2

Let $Z_{1}=Z_{2}=[-1,1]$, $\alpha _{0}=\beta _{0}=0$, $\Phi (w)=\frac{1}{2}w$

$$\begin{aligned}& L(\alpha )=[-1,1],\\& \Gamma (w,\beta )= \textstyle\begin{cases} \{w\},& \beta =0, \\ \{e^{w}\}, & \beta \neq 0,w\geq 0, \\ \{\ln w\}, &\beta \neq 0,w>0. \end{cases}\displaystyle \end{aligned}$$

Obviously, $L(\alpha )$ is continuous at $\alpha =0$, $\Gamma (\cdot ,\beta )$ is not monotone on $[-1,1]$, and $\Gamma (w,\cdot )$ is not lower semicontinuous at $\beta =0$. By calculation, we get $S(0,0)=\{-\frac{1}{2}\}$ and $S(0,\beta )=\{-\frac{1}{2},\frac{1}{e}\}$ for any $\beta \neq 0$. Consequently, $\limsup_{\beta \to 0}S(0,\beta )=\{-\frac{1}{2},\frac{1}{e}\}\not \subset S(0,0)$.

Example 4.3

Let $Z_{1}=Z_{2}=[-1,1]$, $\alpha _{0}=\beta _{0}=0$, $\Phi (w)=\frac{1}{2}w$

$$\begin{aligned}& L(\alpha )= \textstyle\begin{cases} [-2,0], & \alpha \neq 0, \\ [-3,0], & \alpha =0, \end{cases}\displaystyle \\& \Gamma (w,\beta )\equiv \{-2\}. \end{aligned}$$

Note that $L(\alpha )$ is not lower semicontinuous when $\alpha =0$, but it is upper semicontinuous. Moreover, $\Gamma (\cdot ,\beta )$ is not only monotone on $[-3,0]$, but also upper hemicontinuous. And $\Gamma (w,\cdot )$ is lower semicontinuous at $\beta =0$. By calculations, we get $S(0,0)=\{-\frac{1}{2}\}$ and $S(\alpha ,0)=[-\frac{1}{2},+\infty )$ for any $\alpha \neq 0$. Consequently, $\limsup_{\alpha \to 0}S(\alpha ,0)=[-\frac{1}{2},+\infty )\not \subset S(0,0)$.

Example 4.4

Let $Z_{1}=Z_{2}=[-1,1]$, $\alpha _{0}=\beta _{0}=0$, $\Phi (w)=\frac{1}{2}w$

$$\begin{aligned}& L(\alpha )= \textstyle\begin{cases} [-1,0], & \alpha \neq 0, \\ [1,2],& \alpha =0, \end{cases}\displaystyle \\& \Gamma (w,\beta )= \textstyle\begin{cases} [0,2],& \beta \neq 0, \\ [0,1], & \beta =0. \end{cases}\displaystyle \end{aligned}$$

At $\alpha =0$, it is obvious that $L(\alpha )$ is neither lower semicontinuous nor upper semicontinuous. Moreover, $\Gamma (\cdot ,\beta )$ is not monotone on $[-1,2]$, and $\Gamma (w,\cdot )$ is lower semicontinuous at $\beta =0$. By calculation, we obtain $S(0,0)=[-\frac{1}{2},+\infty )$ and $S(\alpha ,\beta )=(-\infty ,-\frac{1}{2}]$ for any $\alpha \neq 0$ and $\beta \neq 0$. Consequently, $\limsup_{\alpha \to 0,\beta \to 0}S(\alpha ,\beta )=(-\infty ,- \frac{1}{2}]\not \subset S(0,0)$.

5 An example

In this section, we will give an example about the stability of IMVI in the traffic network equilibrium control problem.

5.1 The traffic network equilibrium control problem

As an application of our main results, we shall give an example similar to Example 2.2 in [16]. We describe it simply.

Let a network have n parallel links linking a basic origin-destination pair, each link represent a feasible path and $x_{i}$ denote the flow on each link i, $t_{i}$ be the user cost associated with traversing the link i, d represent the travel demand of customers traveling between origin-destination pairs. We denote

$$ \Omega =\Biggl\{ x\Big|x\geq 0,\sum_{i=1}^{n}x_{i}=d \Biggr\} . $$

Now from traffic management authorities’ point of view, we consider the traffic network equilibrium control problem. Assume that the total loss of vehicle accidents and road damage is determined by the flow of all network links, that is,

$$ f(x)=\sum_{i=1}^{n}f_{i}(x_{i}), $$

where $f_{i}:R_{+}\to R$ is a convex and continuous function. The goal of the traffic management authorities is to control the traffic flow $x_{i}$ within predetermined intervals by adjusting the link toll $y_{i}$ and to minimize the total loss of vehicle accidents and road damage in the network. For a given adjustment of $y\in R^{n}$, we know that the resultant traffic network equilibrium flow $x(y)$ is a solution of the following parametric variational inequality:

$$\begin{aligned} \langle z-x,t+y\rangle \leq 0 \quad \text{for all } z\in \Omega . \end{aligned}$$

(5.1)

As a control approach, traffic management authorities could change link tolls to reduce loss and prevent traffic jams. We regard (5.1) as a ‘black-box’ procedure that returns a value of x at the point y. Consequently, the path flow $x(y)$ can be revealed. Assume that the desired link flows are constrained with the feasible link flow set $K=\{x|0\leq x\leq b\}$. Thus, by Example 2.2 in [16], the problem faced by the authority can be interpreted as follows:

$$\begin{aligned}& \min f\bigl[x(y)\bigr],\\& \text{s.t.}\quad x(y)\in K, \end{aligned}$$

where $x(y)$ is a solution of (5.1). Based on the KKT theory, from [16] we know the above problem can be transformed as an inverse mixed variational inequality as follows: find $y\in C$ such that $x(y)\in K$ and

$$\begin{aligned} \bigl\langle z-x(y),y\bigr\rangle +f(z)-f\bigl[x(y)\bigr]\geq 0, \quad \text{for all } z \in K, \end{aligned}$$

(5.2)

where $C=\{y\in R^{n}|\exists x(y)\in K,\langle z-x(y),y\rangle \leq 0\}$. We use the interval $[a,b]$ to represent the fluctuation range of gasoline prices, where a represents the lowest oil price and b represents the highest oil price. However, with the large fluctuation of gasoline prices, it will affect the traffic of vehicles on the road, which means that the feasible link flow set K and the traffic network equilibrium flow $x(y)$ will be influenced by a parameter q, where $q\in [a,b]$. Hence, we can easily see that K should be a set-valued mapping of q and x should be a set-valued mapping of y and q. Then (5.2) will be transformed as follows:

$$\begin{aligned} \bigl\langle z-x(y,q),y\bigr\rangle +f(z)-f\bigl[x(y,q)\bigr]\geq 0,\quad \text{for all } z \in K(q). \end{aligned}$$

(5.3)

So, the traffic network equilibrium control problem influenced by holidays and weekdays will lead to a stability problem for a class of inverse mixed variational inequality.

Corollary 5.1

Let $q_{0}\in [a,b]$, $x(y,\cdot ):[a,b]\to R$ be a lower semicontinuous mapping on $[a,b]$, and $K:[a,b]\to R$ be a continuous set-valued mapping with nonempty, convex, closed bounded value, where $y\in R^{n}$. If there is a neighborhood $\mathcal{U}$ of $q_{0}$ such that $x(y,q)$ has nonempty, closed values for any $q\in \mathcal{U}$. If the following conditions hold:

• (a) for every $q\in \mathcal{U}$, the mapping $y\mapsto x(q,y)$ is monotone and upper hemicontinuous on R;

• (b) the solution set of $\operatorname{SIMVI} (K(q_{0}),x(\cdot ,q_{0}))$ is nonempty and bounded;

• (c) $\operatorname{int}\{(y,q)\in R^{n}\times [a,b]:x(y,q)\cap K(q)\}\bigcap \operatorname{int}(-\operatorname{dom} f^{*})\neq \emptyset$;

• (d) if f is continuous and convex on $K(q)$ with $q\in \mathcal{{U}}$ and $\cup_{q\in \mathcal{{U}}}K(q)$ is compact.

Then $\limsup_{q\to q_{0}}S(q)\subset S(q_{0})$, where $q\in \mathcal{{U}}$, $S(q)$ and $S(q_{0})$ represent solution sets of $\operatorname{SIMVI} (K(q),x(\cdot ,q))$ and $\operatorname{SIMVI} (K(q_{0}),x(\cdot ,q_{0})))$, respectively.

Proof

Since $K(q)$ is a bounded set for any $q\in \mathcal{{U}}$, then from Corollary 4.1 we know that the conclusions of Theorem 4.1 hold. Then, using Theorem 3.2, there exists a neighborhood $\mathcal{\bar{U}}$ of $q_{0}$ with $\mathcal{\bar{U}}\subset \mathcal{U}$ such that for every $q\in \mathcal{\bar{U}}$, the solution set of $\operatorname{SIMVI} (K(q),x(\cdot ,q))$ is nonempty and bounded. By using Theorem 4.2, we can obtain $\limsup_{q\to q_{0}}S(q)\subset S(q_{0})$, where $q\in \mathcal{\bar{U}}$. □

6 Conclusion

In this paper, we introduced a new class of set-valued inverse mixed variational inequalities (SIMVI) in reflexive Banach spaces. We gave some equivalent characterizations such that the solution set of $\operatorname{SIMVI} (\Lambda , \Gamma )$ is nonempty and bounded in Theorem 3.2. The stability of SIMVI was obtained in Theorem 4.2 by using equivalent conditions when the mapping and the constraint set are perturbed simultaneously by different parameters in reflexive Banach spaces. We also gave some examples to show the conditions were necessary in Theorem 4.2. At the end of the paper, an example of the traffic network equilibrium control problem was provided to illustrate the application of the stability of IMVI. For further research, we can apply the theorem of set-valued analysis and inverse variational inequalities to study the stability of set-valued inverse quasi-mixed variational inequalities in Banach spaces.