Uniqueness and stability of the solution for strongly pseudomonotone variational inequalities in Banach spaces

Abstract

In this paper, we prove an existence and uniqueness theorem of the solution for strongly pseudomonotone variational inequalities in reflexive Banach spaces. Based on this result, and investigate the stability behavior of the perturbed variational inequalities. Moreover, we obtain an existence theorem of solutions for strongly quasimonotone variational inequalities in finite dimensional spaces.

Keywords

Variational inequality / strong pseudomonotonicity / strong quasimonotonicity / existence and uniqueness / stability

1 Introduction

Let \mathbb{B} be a real reflexive Banach space, and denote its dual space as {\mathbb{B}}^{\mathbb{\ast }}. Let K be a nonempty closed convex set in K, and F:K\to {\mathbb{B}}^{\mathbb{\ast }} be a nonlinear map. In this paper, we consider the variational inequality (abbreviated as VI(K,F)): find x\in K, such that

(1.1)⟨F\left(x\right),y-x⟩\ge 0,\mathrm{\forall }y\in K.

We denote the solution set of this problem by SOL(K,F). The model VI(K,F) unifies the optimality conditions of the optimization problem, the complementarity problem, nonlinear equations, and Nash equilibrium problems, etc. Therefore, it has been widely applied in natural and social sciences and attracted the attention of many researchers. For example, see [1-11] and many others.

The existence of solutions is one of the fundamental problems in the theory of variational inequalities. It is well-known that in a reflexive Banach space, a sufficient condition for the problem VI(K,F) to have a unique solution is that the operator F is a strongly monotone and Lipschitz-continuous map. The following example shows that this condition is not necessary.

Example 1.1　Let \mathbb{B} be a real reflexive Banach space, and A:\mathbb{B}\to {\mathbb{B}}^{\mathbb{\ast }} be a bounded linear operator satisfying the coercivity condition

(1.2)⟨A\left(x\right),x⟩\ge \gamma \Vert x{\Vert }^2,\mathrm{\forall }x\in \mathbb{B},

where \gamma >0 is a constant. Let K be a nonempty closed convex set in \mathbb{B}, and g:K\to \mathbb{R} be a continuous functional satisfying

(1.3)g\left(x\right)\ge M,\mathrm{\forall }x\in K,

where M>0 is a constant. Define the map F:K\to {\mathbb{B}}^{\mathbb{\ast }} as follows:

F\left(x\right):=g\left(x\right)\left(A\left(x\right)+b\right),\mathrm{\forall }x\in K,

where b\in {\mathbb{B}}^{\mathbb{\ast }} is a constant vector. It is verified that the operator x\mapsto A\left(x\right)+b is strongly monotone (with constant γ) and Lipschitz-continuous (with constant \Vert A\Vert). Therefore, VI(K,A+b)has a unique solution, denoted as x*. As g\left(x\right)>0, it can be deduced that x* is also the unique solution of VI(K,F). However, in general, F is not strongly monotone.

Example 1.1 inspires people to seek more relaxed sufficient conditions for the problem VI(K,F) to have a unique solution, which mainly motivates of the work in this paper.

In non-reflexive Banach spaces, Chang et al. [1] proved that when the constraint set K was a bounded closed convex set and the operator F was monotone and continuous on finite dimensional sub-spaces, then VI(K,F) had a solution. When the constraint set K is an unbounded weak* closed convex set and satisfies \theta \in K, and if the operator F further satisfies the coercivity condition {\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{f}}_{x\in K,\Vert x\Vert \to +\mathrm{\infty }}⟨F\left(x\right),x⟩>0, it can also guarantee that VI(K,F) has a solution. When the constraint set K is a bounded closed convex set, Verma [10] extended the conclusion of Chang et al. to strongly pseudomonotone operators (this concept is independent of monotonicity) and proved that the solution was unique. Lee et al. [6] further generalized the result of Verma [10] from variational inequalities to quasivariational inequalities. Watson [11] weakened the condition of the operator F from monotone to pseudomonotone and from continuous on finite dimensional sub-spaces to hemicontinuous, improving some conclusions of Chang et al. [1]. When Verma [10] proved the existence and uniqueness theorem of the solution of the problem VI(K,F), it relied on the boundedness of the constraint set K. Whether the boundedness restriction of the constraint set K can be removed remains a concern.

On the other hand, due to the widespread existence of various perturbation factors, it is difficult for a mathematical model to accurately depict real-world problems. This has prompted people to further study how to use mathematical models to describe the perturbations of problems and the relationships between perturbed problems and the original problems. Facchinei and Pang presented the following perturbation result in finite dimensional spaces (see [2, Corollary 5.5.12]).

Theorem 1.1 [2]　 Let K be a nonempty closed convex set in {\mathbb{R}}^n, F:K\to {\mathbb{R}}^n be continuous and there exists a point x^{\mathrm{r}\mathrm{e}\mathrm{f}}\in K such that

(1.4)L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right):=\{x\in K:⟨F\left(x\right),x-x^{\mathrm{r}\mathrm{e}\mathrm{f}}⟩\le 0\}

is bounded, then for any open set \mathcal{U} containing the solution set SOL(K,F), there exists \delta >0, such that for an operator G satisfying the following formula:

VI(K,F) has a solution at \mathcal{U}.

Recently, Kim et al. [5] applied Theorem 1.1 to derive the existence theorem of perturbed solutions for strongly pseudomonotone variational inequalities in finite dimensional spaces. Li and He [7] studied the existence of perturbed solutions for set-valued variational inequalities based on the method of Theorem 1.1 for characterizing perturbations. Kim et al. [5] also studied the existence of perturbed solutions for strongly pseudomonotone variational inequalities in Hilbert spaces.

Inspired by the above-mentioned references and using the equivalent characterization of the nonempty solution set among variational inequalities obtained by Kien et al. [4] as a tool and drawing on the basic ideas used by Kim et al. [5] in Hilbert spaces, this paper studies the existence and uniqueness of solutions to variational inequalities with strongly pseudomonotone maps in reflexive Banach spaces. On this basis, the properties of perturbed solutions of such variational inequalities are studied. In addition, we also derive the existence theorem of solutions to variational inequalities with strongly quasimonotone maps in finite dimensional spaces. The main results of this paper generalize and improve the corresponding results in the references [5, 10].

2 Preliminaries

In this paper, B\left(x;\epsilon \right) and \overline{B}\left(x;\epsilon \right) denote the open ball and closed ball centered at the point x with a radius ε > 0, respectively. → and ⇀ represent strong convergence and weak convergence, respectively.

Definition 2.1　The operator F:K\to {\mathbb{B}}^{\mathbb{\ast }} is called

(i) strongly monotone, if there exists a constant γ > 0, such that

⟨F\left(y\right)-F\left(x\right),y-x⟩\ge \gamma \Vert y-x{\Vert }^2,\mathrm{\forall }x,y\in K;

(ii) strongly pseudomonotone, if there exists a constant \gamma >0, such that the following implication holds

⟨F\left(x\right),y-x⟩\ge 0\Rightarrow ⟨F\left(y\right),y-x⟩\ge \gamma \Vert y-x{\Vert }^2,\mathrm{\forall }x,y\in K;

(iii) strongly quasimonotone, if there exists a constant γ > 0, such that the following implication holds

⟨F\left(x\right),y-x⟩>0\Rightarrow ⟨F\left(y\right),y-x⟩\ge \gamma \Vert y-x{\Vert }^2,\mathrm{\forall }x,y\in K;

(iv) monotone, if

⟨F\left(y\right)-F\left(x\right),y-x⟩\ge 0,\mathrm{\forall }x,y\in K;

(v) pseudomonotone, if the following implication holds

⟨F\left(x\right),y-x⟩\ge 0\Rightarrow ⟨F\left(y\right),y-x⟩\ge 0,\mathrm{\forall }x,y\in K;

(vi) quasimonotone, if the following implication holds

⟨F\left(x\right),y-x⟩>0\Rightarrow ⟨F\left(y\right),y-x⟩\ge 0,\mathrm{\forall }x,y\in K.

Note 2.1　The various generalized monotonicities in Definition 2.1 have the following implications:

\begin{array}{ccccc}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}& \Rightarrow & \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}& \Rightarrow & \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{y}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}\\ \Downarrow & & \Downarrow & & \Downarrow \\ \mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}& \Rightarrow & \mathrm{p}\mathrm{s}\mathrm{e}\mathrm{u}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}& \Rightarrow & \mathrm{q}\mathrm{u}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}\end{array}

The reverse directions of these implications generally do not hold. Consider the following examples.

Example 2.1　Define F:\mathbb{B}\to {\mathbb{B}}^{\mathbb{\ast }} as F\left(x\right)=\theta, where \theta \in {\mathbb{B}}^{\mathbb{\ast }} is a functional \theta :\mathbb{B}\to \mathbb{R},x\mapsto 0. It is verified that F is monotone, and thus pseudomonotone. However, F is not strongly pseudomonotone, and thus not strongly monotone. This shows that the monotonicity of an operator does not imply strong monotonicity, and pseudomonotonicity does not imply strong pseudomonotonicity.

Example 2.2　Let \mathbb{B}=l_2. Take \alpha ,\beta \in \mathbb{R} satisfying \beta >\alpha >\frac{\beta }{2}>0, define

K:=\{x\in \mathbb{B}:\Vert x\Vert \le \alpha \},F\left(x\right):=\left(\beta -\Vert x\Vert \right)x.

Prove that F is strongly pseudomonotone but not monotone, and thus not strongly monotone.

Proof　Take any x,y\in K, and assume ⟨F\left(x\right),y-x⟩\ge 0. According to , there is ⟨x,y-x⟩\ge 0, so

(2.1)\begin{array}{rl}⟨F\left(y\right),y-x⟩& =\left(\beta -\Vert y\Vert \right)⟨y,y-x⟩\\ & \ge \left(\beta -\Vert y\Vert \right)\left[⟨y,y-x⟩-⟨x,y-x⟩\right]\\ & \ge \left(\beta -\alpha \right)\Vert y-x{\Vert }^2.\end{array}

F is (β-α)-strongly pseudomonotone. Take x=(\frac{\beta }{2},0,\dots ,0,\dots ),y=(\alpha ,0,\dots ,0,\dots )\in K, and after calculation, it is known that

Therefore, F is not monotone, and consequently not strongly monotone. □

Example 2.1 and Example 2.2 also illustrate that the strong pseudomonotonicity and monotonicity of an operator are two independent concepts.

Example 2.3　Let \mathbb{B}=\mathbb{R},K=\left[1,+\mathrm{\infty }\right). Define F:K\to \mathbb{R} as F\left(x\right)=-x. It can be verified that F is pseudomonotone, but not monotone. This shows that the pseudomonotonicity of an operator does not imply monotonicity.

Example 2.4　Let \mathbb{B}=\mathbb{R},K=\left[0,+\mathrm{\infty }\right). Define define F:K\to \mathbb{R} as F\left(x\right)=-x. It is verified that F is quasimonotone, but not pseudomonotone. In fact, take x=0,y=1, then ⟨F\left(x\right),y-x⟩=0, but ⟨F\left(y\right),y-x⟩=-1. This shows that the quasimonotonicity of an operator does not imply pseudomonotonicity. Take note to compare this example with Example 2.3. In addition, Hadjisavvas and Schaible also provide an example of a quasimonotone but non-pseudomonotone operator (see [3, Example 3.1]).

Example 2.5　Let \mathbb{B}=\mathbb{R},K=[0,+\mathrm{\infty }]. Define F:K\to \mathbb{R} as F\left(x\right)=-1. It is verified that F is monotone, and thus pseudomonotone and quasimonotone, but not strongly quasimonotone. In fact, for any positive number γ, take x,y\in [0,+\mathrm{\infty }):x=y+\frac{2}{\gamma }, then⟨F\left(x\right),y-x⟩=-\left(y-x\right)=\frac{2}{\gamma }>0. But according to ⟨F\left(y\right),y-x⟩=-\left(y-x\right)=\frac{2}{\gamma } and \gamma \Vert y-x{\Vert }^2=\gamma \cdot \frac{4}{{\gamma }^2}, there is. This shows that the quasimonotonicity of an operator does not imply strong quasimonotonicity.

Example 2.6　Let \mathbb{B}=\mathbb{R},K=\mathbb{R}. Define F:K\to \mathbb{R} as

F\left(x\right)=\{\begin{array}{ll}0,& x\in \left(-\mathrm{\infty },1\right];\\ x-1,& x\in \left(1,+\mathrm{\infty }\right).\end{array}

Next, it requires to prove that F is strongly quasimonotone but not strongly pseudomonotone. In fact, based on ⟨F\left(x\right),y-x⟩>0, it can be inferred\left(x-1\right)\left(y-x\right)>0, which implies x>1 and y>x. At this time, ⟨F\left(y\right),y-x⟩=\left(y-1\right)\left(y-x\right)>{\left(y-x\right)}^2, so F is strongly quasimonotone (with the constant 1). On the other hand, for any positive number γ, take x = 0, y = 1, and there is ⟨F\left(x\right),y-x⟩=0, but from ⟨F\left(y\right),y-x⟩=0 and \gamma \Vert y-x{\Vert }^2=\gamma, there is . Therefore, F is not strongly pseudomonotone. This shows that the strong quasimonotonicity of an operator does not imply strong pseudomonotonicity.

Definition 2.2　The operator F:K\to {\mathbb{B}}^{\mathbb{\ast }} is called

(i) Lipschitz-continuous, if there exists a constant L > 0, such that

\Vert F\left(y\right)-F\left(x\right)\Vert \le L\Vert y-x\Vert ,\mathrm{\forall }x,y\in K;

(ii) is continuous at point x_0\in K, if the following implication holds:

x_n\to x_0\Rightarrow F\left(x_n\right)\to F\left(x_0\right);

(iii) is weakly continuous at point x_0\in K, if the following implication holds:

x_n\to x_0\Rightarrow F\left(x_n\right)\rightharpoonup F\left(x_0\right).

Note 2.2　The various continuities in Definition 2.2 have the following implications:

\mathrm{L}\mathrm{i}\mathrm{p}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{z}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\Rightarrow \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\Rightarrow \mathrm{t}\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}.

Generally, the reverse directions of these implications do not hold.

The following lemma is a powerful tool for proving the main results of this paper. Let L be an arbitrary finite dimensional subspace of a Banach space \mathbb{B}, and denote K_L:=K\cap L.

Lemma 2.1 [4, Theorem 3]　 Let K be a nonempty closed convex subset in a reflexive Banach space \mathbb{B}, and F:K\to {\mathbb{B}}^{\mathbb{\ast }} be a pseudomonotone and weakly continuous nonlinear map on any K_L. Then the following statements are equivalent:

(a) There exists a point x^{\mathrm{r}\mathrm{e}\mathrm{f}}\in K, such that the set

is bounded (probably empty);

(b) There exist open ball Ω and point x^{\mathrm{r}\mathrm{e}\mathrm{f}}\in \mathrm{\Omega }\cap K, such that

⟨F\left(x\right),x-x^{\mathrm{r}\mathrm{e}\mathrm{f}}⟩\ge 0,\mathrm{\forall }x\in K\cap \mathrm{\partial }\mathrm{\Omega };

(c) Problem \mathrm{V}\mathrm{I}(K,F)has a solution.

In addition, if there exists a point x^{\mathrm{r}\mathrm{e}\mathrm{f}}\in K, such that the set L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right) (the definition is given in (1.4)) is bounded, then the solution set SOL(K,F) is nonempty and bounded.

Note 2.3　Regarding the inclusion , Kien et al. [4] point out that the closure of may be a proper subset of L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right). To understand the conditions of Lemma 2.1, it is noted that even if is bounded, L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right) can be unbounded. For example, for the K and F in Example 2.6, take x^{\mathrm{r}\mathrm{e}\mathrm{f}}=2, then there are and L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)=(-\mathrm{\infty },2].

3 Existence and uniqueness theorems of solutions

In this section, we establish the existence and uniqueness theorems for solutions of variational inequalities with strongly pseudomonotone maps in reflexive Banach spaces.

Theorem 3.1　 Let K be a nonempty closed convex subset in a reflexive Banach space \mathbb{B}, and F:K\to {\mathbb{B}}^{\mathbb{\ast }} be a γ-strongly pseudomonotone and weakly continuous nonlinear map on any K_L. Then the variational inequality problem VI(K,F) has a unique solution.

Proof　First, it requires to prove the existence. Arbitrarily take x^{\mathrm{r}\mathrm{e}\mathrm{f}}\in K. For each x\in L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right), there is ⟨F\left(x\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩\ge 0. According to the strong pseudomonotonicity of F, there is

(3.1)⟨F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩\ge \gamma \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x{\Vert }^2.

From Cauchy-Schwarz inequality, there is

(3.2)⟨F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩\le \Vert F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)\Vert \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x\Vert .

By combining (3.1) and (3.2), it is known that \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x\Vert \le \frac{1}{\gamma }\Vert F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)\Vert, so L_{\le }(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}})\subset \overline{B}(x^{\mathrm{r}\mathrm{e}\mathrm{f}};\frac{1}{\gamma }\Vert F(x^{\mathrm{r}\mathrm{e}\mathrm{f}})\Vert ). From Lemma 3.1, it can be inferred that SOL(K,F) is nonempty and bounded.

Then it requires to prove the uniqueness. Assume x_1,x_2\in \mathrm{S}\mathrm{O}\mathrm{L}\left(K,F\right), and there are

(3.3)⟨F\left(x_1\right),x_2-x_1⟩\ge 0,

and

⟨F\left(x_2\right),x_1-x_2⟩\ge 0.

According to the strong pseudomonotonicity of F, there is

(3.4)⟨F\left(x_1\right),x_1-x_2⟩\ge \gamma \Vert x_1-x_2{\Vert }^2.

Adding the two inequalities (3.3) and (3.4) together, there is

0\ge \gamma \Vert x_1-x_2{\Vert }^2,

so x_1=x_2. □

Note 3.1　(i) Theorem 3.1 improves the classical result: The strong monotonicity and Lipschitz continuity of the operator guarantee a unique solution to VI(K,F). Here, strong monotonicity is weakened to strong pseudomonotonicity, and Lipschitz continuity is weakened to weak continuity on finite dimensional subspaces.

(ii) Theorem 3.1 generalizes Theorem 2.1 of Kim et al. [5] from Hilbert spaces to reflexive Banach spaces.

(iii) When Theorem 2.2 of Verma [10] is restricted to reflexive Banach spaces, the conditions of Theorem 3.1 are weaker: it is not required that the constraint set K to be bounded. It requires the operator F to be weakly continuous on finite dimensional subspaces, while Theorem 2.2 of Verma [10] requires the constraint set K to be bounded and the operator F to be continuous on finite dimensional subspaces.

Example 3.1　Use Theorem 3.1 to prove that the VI(K,F)in Example 2.1 has a unique solution.

Proof　Due to the continuity of the functional g and the operator A, F is continuous on K. Next, it requires to prove that F is strongly pseudomonotone. In fact, take any x,y\in K, and assume ⟨F\left(x\right),y-x⟩\ge 0. From g\left(x\right)\ge M>0, it is inferred that⟨A\left(x\right)+b,y-x⟩\ge 0, and thus g\left(y\right)⟨A\left(x\right)+b,y-x⟩\ge 0. As a result, there is

(3.5)\begin{array}{rl}⟨F\left(y\right),y-x⟩& =g\left(y\right)⟨A\left(y\right)+b,y-x⟩\\ & \ge g\left(y\right)⟨A\left(y\right)+b,y-x⟩-g\left(y\right)⟨A\left(x\right)+b,y-x⟩\\ & =g\left(y\right)⟨A\left(y-x\right),y-x⟩\\ & \ge M\gamma \Vert y-x{\Vert }^2.\end{array}

F is Mγ-strongly pseudomonotone. According to Theorem 3.1, VI(K,F) has a unique solution.

Example 3.2　Let \mathbb{B}=l_2, take \alpha ,\beta \in \mathbb{R} satisfying \beta >\alpha >\frac{\beta }{2}>0, define

K:=\{x\in \mathbb{B}:\Vert x\Vert \le \alpha \},F\left(x\right):=\left(\beta -\Vert x\Vert \right)x.

It is known that F is continuous. As known from Example 2.2, F is strongly pseudomonotone. By applying Theorem 3.1, it can be seen that VI(K,F) has a unique solution.

In the remaining part of this section, the research in this paper is confined to finite dimensional spaces. It is noted that in a finite dimensional space, when only considering the existence part of the solution of VI(K,F) in Theorem 3.1, the strong pseudomonotonicity of the map in the required conditions can be weakened to strong quasimonotonicity. First, review the following lemma.

Lemma 3.1 [2, Proposition 2.2.3]　 Let K be a nonempty closed convex subset of {\mathbb{R}}^n, and F:K\to {\mathbb{R}}^n be a continuous map. Consider statements (a), (b), and (c) in Lemma 2.1, then (\mathrm{a})\Rightarrow (\mathrm{b})\Rightarrow (\mathrm{c}). Moreover, if the set L_{\le }\left(F,x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right) is bounded, the solution set SOL(K,F)is nonempty and compact.

In the existence theory of variational inequality solutions, when the constraint set K is unbounded, it is necessary to introduce the so-called coercivity conditions. For example, as can be seen from Lemma 3.1, for a continuous map F, the coercivity condition (a) or (b) is a sufficient condition for the existence of a solution to VI(K,F), which triggers research on coercivity conditions. On the one hand, people are committed to finding weaker coercivity conditions. On the other hand, the identifiability of coercivity conditions is not high, which hinders the application. Therefore, it is significant to derive certain easily-identifiable sufficient conditions for the establishment of coercivity conditions. Theorem 3.2 below gives a sufficient condition for the establishment of coercivity condition (a) or (b).

Theorem 3.2　 Let K be a nonempty closed convex subset in {\mathbb{R}}^n, and F:K\to {\mathbb{R}}^n be a γ-strongly quasimonotone and continuous nonlinear map. Then the coercivity condition (a) holds, and thus the coercivity condition (b) also holds. The problem VI(K,F) has a solution.

Proof　Arbitrarily takex^{\mathrm{r}\mathrm{e}\mathrm{f}}\in K. For each , there is ⟨F\left(x\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩>0. By the strong quasimonotonicity of F, it is known

(3.6)⟨F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩\ge \gamma \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x{\Vert }^2.

According to the Cauchy-Schwarz inequality, there is

(3.7)⟨F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right),x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x⟩\le \Vert F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)\Vert \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x\Vert .

By combining (3.6) and (3.7), it is known \Vert x^{\mathrm{r}\mathrm{e}\mathrm{f}}-x\Vert \le \frac{1}{\gamma }\Vert F\left(x^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)\Vert, so . The coercivity condition (a) holds. According to Lemma 3.1, it can be deduced that the coercivity condition (b) also holds and the problem VI(K,F) has a solution.

Note 3.2　Compare Theorem 3.2 with Theorem 3.1. After weakening the operator from strongly pseudomonotone to strongly quasimonotone, we cannot guarantee the uniqueness or even the boundedness of the solution to the problem VI(K,F). For example, consider the VI(K,F) in Example 2.6. It can be seen that F satisfies strong quasimonotonicity and continuity, while \mathrm{S}\mathrm{O}\mathrm{L}(K,F)=(-\mathrm{\infty },1] is unbounded.

4 Stability of perturbed solutions

In this section, based on Theorem 3.1, we study the nonemptiness and boundedness of the solutions of the perturbed variational inequalities, thus obtaining the stability of the perturbed solutions.

Theorem 4.1　 Let K be a nonempty closed convex subset in a reflexive Banach space \mathbb{B}, and F:K\to {\mathbb{B}}^{\mathbb{\ast }} be a γ-strongly pseudomonotone and weakly continuous nonlinear map on any K_L. Then for any positive number ε, there exists a positive number δ, such that for every pseudomonotone and weakly continuous nonlinear map \tilde{F} on any K_L satisfying the following inequality:

(4.1)

the problem \mathrm{V}\mathrm{I}(K,\tilde{F}) always has a solution and \mathrm{S}\mathrm{O}\mathrm{L}(K,\tilde{F})\subset B(x^{\ast },\epsilon ), where x* is the unique solution of VI(K,F).

Proof　According to Theorem 3.1, VI(K,F) has a unique solution x*. For any ε > 0, take \delta \in \left(0,\gamma \epsilon \right). For every pseudomonotone and weakly continuous, nonlinear map \tilde{F} on any K_L, consider the set

L_{\le }\left(\tilde{F},x^{\ast }\right):=\{x\in K:⟨\tilde{F}\left(x\right),x-x^{\ast }⟩\le 0\}.

Next it requires to prove

(4.2)L_{\le }\left(\tilde{F},x^{\ast }\right)\subset B\left(x^{\ast },\epsilon \right).

For every x\in L_{\le }\left(\tilde{F},x^{\ast }\right), there is ⟨\tilde{F}\left(x\right),x-x^{\ast }⟩\le 0, so

(4.3)⟨F\left(x\right),x-x^{\ast }⟩\le ⟨F\left(x\right)-\tilde{F}\left(x\right),x-x^{\ast }⟩.

From x^{\ast }\in \mathrm{S}\mathrm{O}\mathrm{L}(K,F), it is known that ⟨F\left(x^{\ast }\right),x-x^{\ast }⟩\ge 0, and based on the strong pseudomonotonicity of F, there is

(4.4)⟨F\left(x\right),x-x^{\ast }⟩\ge \gamma \Vert x-x^{\ast }{\Vert }^2.

According to the Cauchy-Schwarz inequality and (3.1), there is

(4.5)⟨F\left(x\right)-\tilde{F}\left(x\right),x-x^{\ast }⟩\le \Vert F\left(x\right)-\tilde{F}\left(x\right)\Vert \Vert x-x^{\ast }\Vert \le \delta \Vert x-x^{\ast }\Vert .

According to (4.3), (4.4), (4.5), and \delta \in \left(0,\gamma \epsilon \right), there is

From this, it is known that L_{\le }\left(\tilde{F},x^{\ast }\right)\subset B\left(x^{\ast },\epsilon \right). According to Lemma 2.1, it can be deduced that \mathrm{S}\mathrm{O}\mathrm{L}(K,\tilde{F}) is nonempty and bounded.

Now it requires to prove that \mathrm{S}\mathrm{O}\mathrm{L}(K,\tilde{F})\subset L_{\le }(\tilde{F},x^{\ast }). In fact, for each \overline{x}\in \mathrm{S}\mathrm{O}\mathrm{L}\left(K,\tilde{F}\right), from x^{\ast }\in K, it is known ⟨\tilde{F}\left(\overline{x}\right),\overline{x}-x^{\ast }⟩\le 0, that is \overline{x}\in L_{\le }(\tilde{F},x^{\ast }). Combining with (4.2), there are

Note 3.1　It can be deduced from the conclusion \mathrm{S}\mathrm{O}\mathrm{L}(K,\tilde{F})\subset B(x^{\ast },\epsilon ) of Theorem 3.1 that when \epsilon \to 0, for any {\tilde{x}}_{\epsilon } taken from the solution set \mathrm{S}\mathrm{O}\mathrm{L}(K,\tilde{F}) of the perturbed variational inequality \mathrm{V}\mathrm{I}(K,\tilde{F}), there is {\tilde{x}}_{\epsilon }\to x^{\ast }.

Note 3.2　After comparison with Theorem 3.1 in reference [7], the way of characterizing the perturbation is similar. It has not only proved the existence of solutions to the perturbed variational inequality, but also demonstrated the boundedness of the solution set. This serves as the basis for obtaining the stability of the perturbed solutions.

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[5] Kim D.S., Vuong, P.T. , Khanh, P.D. . Qualitative properties of strongly pseudomonotone variational inequalities. Optim. Lett. 2016; 10: 1669–1679

[6] Lee B.S. , Lee, G.M. . Variational inequalities for (η,θ)-pseudomonotone operators in nonreflexive Banach spaces. Appl. Math. Lett. 1999; 12: 13–17

[7] Li F.L. , He, Y.R. . Solvability of a perturbed variational inequality. Pac. J. Optim. 2014; 10: 105–111

[8] NagurneyA., Network Economics: A Variational Inequality Approach, Dordrecht: Kluwer Academic Publishers, 1999

[9] Tang G.J. . and Wang, X., An extragradient-type method for variational inequalities on Banach Spaces. Acta Math. Sin. Chin. Ser. 2016; 59(2): 187–198

[10] Verma R.U. . Variational inequalities involving strongly pseudomonotone hemicontinuous mappings in non-reflexive Banach spaces. Appl. Math. Lett. 1998; 11(2): 41–43

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