Existence theorems for relaxed η-α pseudomonotone and strictly η-quasimonotone generalized variational-like inequalities

Abstract

In this paper, we prove the existence of solutions for a variational-like inequality and a generalized variational-like inequality in the relaxed η-α pseudomonotone and strictly η-quasimonotone cases in Banach spaces by using the KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.

1 Introduction

The variational inequality was first introduced and studied in the finite-dimensional Euclidean space by Giannessi [1]. Variational inequality problems play a critical role in many fields of science, engineering, and economics. In the last four decades, since the time of the celebrated Hartman-Stampacchia theorem (see [2, 3]), the existence of a solution of a variational inequality, a generalized variational inequality, and other related problems has become a basic research topic which continues to attract the attention of researchers in applied mathematics (see for instance [4–13], and the references therein).

In 1995, Chang et al. [14] introduced and studied the problem of the existence of solutions and the perturbation problem for some kind of variational inequalities with monotone and semimonotone map**s in nonreflexive Banach spaces. Recently, Verma [15] studied a class variational inequality relaxed monotone map**. Moreover, Fang and Huang [16] obtained the existence of solutions for variational-like inequalities with relaxed η-α monotone map**s in reflexive Banach spaces. In 2003, Facchinei and Pang [17, 18] used the degree theory to obtain a necessary and sufficient condition of variational inequality problems for continuous pseudomonotone map**s in a finite-dimensional space. In 2008, Kien et al. [19] proposed some extensions of the results of Facchinei and Pang [17, 18] to the case of variational inequalities and generalized variational inequalities in infinite-dimensional reflexive Banach spaces.

On the other hand, Bai et al. [20] introduced the new concept of relaxed η-α pseudomonotone map**s. By using the KKM technique, they obtain some existence results for variational-like inequalities with relaxed η-α pseudomonotone map**s in reflexive Banach spaces. In 2007, Wu and Huang [21] introduced the two new concepts of relaxed η-α pseudomonotonicity and relaxed η-α demipseudomonotonicity in Banach spaces. In 2009, Pourbarat and Abbasi [22] tried to replace some conditions of the work of Wu and Huang [21] with some new conditions. Moreover, they present the solvability of variational-like inequalities with relaxed η-α monotone map**s in arbitrary Banach spaces (see also in [2, 15–20] and [23–28]).

Inspired and motivated by [19], we introduce a new definition of relaxed η-α pseudomonotone map**s and prove the existence of solutions for variational-like inequality and generalized variational-like inequality with relaxed η-α pseudomonotone map**s and strictly η-quasimonotone map**s in Banach spaces by using KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.

2 Preliminaries

Let X be a real reflexive Banach space with dual space $X^{\ast }$ and $〈\cdot ,\cdot 〉$ denoted the pairing between $X^{\ast }$ and X. Let K be a nonempty subset of X, and $2^X$ denote the family of all the nonempty subset of X and $\mathrm{\Phi }:K\to 2^{X^{\ast }}$ and $\eta :K\times K\to X$ be map**s. The generalized variational-like inequality defined by K and Φ, denoted by $GVLI(K,\mathrm{\Phi })$, is the problem of finding a point $x\in K$ such that

$$\mathrm{\exists }{x}^{\ast }\in \mathrm{\Phi }(x),〈x^{\ast },\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

(2.1)

The set of all $x\in K$ satisfying (2.1) is denoted by $SOL(K,\mathrm{\Phi })$. If $\mathrm{\Phi }(x)=\{F(x)\}$ for all $x\in K$, where $F:K\to X^{\ast }$ is a single-valued map**, then the problem $GVLI(K,\mathrm{\Phi })$ is called a variational-like inequality and the abbreviation $VLI(K,F)$ is the problem of finding an $x\in K$ such that

$$〈F(x),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

(2.2)

We introduce the definition of relaxed η-α pseudomonotone for α map** which comes from a family of functions which contains all map**s α given in [20]. In fact, the new definition is an extension of Definition 2.1 in [20]. Then we recall some definitions and results which are needed in the sequel.

We introduce the family

$$\mathbf{\text{A}}=\{\alpha \text{: }\alpha :X\to \mathbb{R};\underset{t\to 0^{+}}{lim\hspace{0.17em}sup}\frac{\alpha (t\eta (x,y))}{t}=0\mathrm{\forall }(x,y)\in K\times K\}.$$

We note that if $\alpha (tx)=k(t)\alpha (x)$, for all $x\in X$ where k is a function from $(0,\mathrm{\infty })$ to $(0,\mathrm{\infty })$ with ${lim}_{t\to 0}\frac{k(t)}{t}=0$, then $\alpha \in A$.

Definition 2.1 The map** $F:K\to X^{\ast }$ is said to be:

• (i) Relaxed η-α pseudomonotone if there exist $\eta :K\times K\to X$ and $\alpha :X\to \mathbb{R}$ with $\alpha \in A$, such that for every distinct points $x,y\in K$,

  $$〈F(y),\eta (x,y)〉\ge 0\Rightarrow 〈F(x),\eta (x,y)〉\ge \alpha (\eta (x,y)).$$

  (2.3)

  If $\eta (x,y)=x-y$ for all distinct points x, y in K, then (2.3) becomes

  $$〈F(y),x-y〉\ge 0\Rightarrow 〈F(x),x-y〉\ge \alpha (x-y),$$

  and F is said to be relaxed α pseudomonotone.

• (ii) Strictly η-quasimonotone if there exist $\eta :K\times K\to X$ such that for every distinct points $x,y\in K$,

  $$〈F(y),\eta (x,y)〉>0\Rightarrow 〈F(x),\eta (x,y)〉>0.$$

  (2.4)

  If $\eta (x,y)=x-y$ for all distinct points x, y in K, then (2.4) becomes

  $$〈F(y),x-y〉>0\Rightarrow 〈F(x),x-y〉>0,$$

  and F is said to be strictly quasimonotone.

Definition 2.2 The map** $\mathrm{\Phi }:K\to 2^{X^{\ast }}$ is said to be:

• (i) Relaxed η-α pseudomonotone if there exist $\eta :K\times K\to X$ and $\alpha :X\to \mathbb{R}$ with $\alpha \in A$,

  $$\begin{array}{c}〈y^{\ast },\eta (x,y)〉\ge 0,\mathrm{\exists }{y}^{\ast }\in \mathrm{\Phi }(y)\hfill \\ \Rightarrow 〈x^{\ast },\eta (x,y)〉\ge \alpha (\eta (x,y)),\mathrm{\exists }{x}^{\ast }\in \mathrm{\Phi }(x),\mathrm{\forall }x,y\in X.\hfill \end{array}$$

• (ii) Strictly η-quasimonotone if there exist $\eta :K\times K\to X$ such that

  $$\begin{array}{c}〈F(y),\eta (x,y)〉>0,\mathrm{\exists }{y}^{\ast }\in \mathrm{\Phi }(y)\hfill \\ \Rightarrow 〈F(x),\eta (x,y)〉>0,\mathrm{\exists }{x}^{\ast }\in \mathrm{\Phi }(x),\mathrm{\forall }x,y\in X.\hfill \end{array}$$

Example 2.3 If $F:(-\mathrm{\infty },0]\to [0,+\mathrm{\infty })$ define by $F(x)=x^2$ and

$$\eta (x,y)=|x-y|\mathrm{\forall }x,y\in (-\mathrm{\infty },0],$$

where $c>0$, then the map** F is a relaxed η-α pseudomonotone map** with

$$\alpha (z)=\{\begin{array}{cc}-[|z|],\hfill & z>0,\hfill \\ [|z|],\hfill & z\le 0.\hfill \end{array}$$

But it is not a relaxed α-pseudomonotone map**. In fact, if we let $x=-1$, $y=0$, $〈F(y),x-y〉\ge 0$, but $〈F(x),x-y〉<\alpha (x-y)$, which is a contradiction.

Example 2.4 If $F:(-\mathrm{\infty },1)\to \mathbb{R}$ define by $F(x)=x^2-1$ and

$$\eta (x,y)=-c(x-y)\mathrm{\forall }x,y\in (-\mathrm{\infty },1),$$

where $c>0$. Then the map** F is strictly η-quasimonotone but fails to be strictly quasimonotone since if $x\in (-1,1)$ and $y<-1$, then we have $〈F(y),x-y〉\ge 0$ but $〈F(x),x-y〉<0$.

Definition 2.5 ([20])

Let $F:K\to X^{\ast }$ and $\eta :K\times K\to X$ be two map**s. F is said to be η-hemicontinuous if, for any fixed $x,y\in K$, the map** $f:[0,1]\to (-\mathrm{\infty },+\mathrm{\infty })$ defined by $f(t)=〈F(x+t(y-x)),\eta (y,x)〉$ is continuous at 0⁺.

If $\eta (x,y)=x-y$ $\mathrm{\forall }x,y\in K$, then F is said to be hemicontinuous.

Definition 2.6 ([29])

A map** $F:K\to 2^X$ is said to be a KKM map** if, for any $\{x_1,\dots ,x_n\}\subset K$, $co\{x_1,\dots ,x_n\}\subset {\bigcup }_{i=1}^{n}F(x_i)$, where $co\{x_1,\dots ,x_n\}$ denotes the convex hull of $x_1,\dots ,x_n$.

Lemma 2.7 ([29])

Let K be a nonempty subset of a Hausdorff topological vector space X and let $F:K\to 2^X$ be a KKM map**. If $F(x)$ is closed in X for every x in K and compact for some $x_0\in K$, then

$$\bigcap _{x\in K}F(x)\ne \mathrm{\varnothing }.$$

Lemma 2.8 (Michael selection theorem [30])

Let X be a paracompact space and Y be a Banach space. Then every lower semicontinuous multivalued map** from X to the family of nonempty, closed, convex subsets of Y admits a continuous selection.

3 Generalized variational-like inequality with relaxed η-α pseudomonotone map**s

In this section, we will discuss the existence of solutions for the following variational-like inequality and generalized variational-like inequality with relaxed η-α pseudomonotone map**s.

Theorem 3.1 Let K be a nonempty closed convex subset of a real reflexive Banach space X. Let $F:K\to X^{\ast }$ and $\eta :K\times K\to X$ be map**s. Assume that:

• (i) F is an η-hemicontinuous and relaxed η-α pseudomonotone;

• (ii) $\eta (x,x)=0$ for all $x\in K$;

• (iii) $\eta (tx+(1-t)z,y)=t\eta (x,y)+(1-t)\eta (z,y)$ for all $x,y,z\in K$, $t\in [0,1]$.

Then $x\in K$ is a solution of $VLI(K,F)$ if and only if

$$〈F(y),\eta (y,x)〉\ge \alpha (\eta (y,x))\mathrm{\forall }y\in K.$$

(3.1)

Proof Suppose that $x\in K$ is a solution of $VLI(K,F)$. Since F is relaxed η-α pseudomonotone, we have

$$〈F(y),\eta (y,x)〉\ge \alpha (\eta (y,x))\mathrm{\forall }y\in K,$$

and hence $x\in K$ is a solution of (3.1). Conversely, suppose that $x\in K$ is a solution of (3.1) and $y\in K$ be any point. Letting $x_t=ty+(1-t)x$, $t\in (0,1]$, we have $x_t\in K$. It follows from (3.1) that

$$〈F(x_t),\eta (x_t,x)〉\ge \alpha (\eta (x_t,x)).$$

(3.2)

By the conditions of η, we have

$$\begin{array}{rl}〈F(x_t),\eta (x_t,x)〉& =〈F(x_t),\eta (ty+(1-t)x,x)〉\\ =t〈F(x_t),\eta (y,x)〉+(1-t)〈F(x_t),\eta (x,x)〉\\ =t〈F(x_t),\eta (y,x)〉.\end{array}$$

(3.3)

It follows from (3.2) and (3.3) that

$$t〈F(x_t),\eta (y,x)〉=〈F(x_t),\eta (x_t,x)〉\ge \alpha (\eta (x_t,x))\mathrm{\forall }t\in (0,1].$$

So, we have

$$〈F(x_t),\eta (y,x)〉\ge \frac{\alpha (\eta (x_t,x))}{t}=\frac{\alpha (t\eta (y,x))}{t}\mathrm{\forall }t\in (0,1].$$

Letting $t\to 0^{+}$, we get

$$〈F(x),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

 □

Theorem 3.2 Let X be a real reflexive Banach space and $K\subset X$ be a closed convex set. Let $F:K\to X^{\ast }$ and $\eta :K\times K\to X$ be are map**s. Assume that:

• (i) F is a relaxed η-α pseudomonotone map** and η-hemicontinuous;

• (ii) $\eta (x,x)=0$ for all $x\in K$;

• (iii) $\eta (tx+(1-t)z,y)=t\eta (x,y)+(1-t)\eta (z,y)$ for all $x,y,z\in K$, $t\in [0,1]$ and η is lower semicontinuous;

• (iv) $\alpha :X\to \mathbb{R}$ is lower semicontinuous.

Then the following statements are equivalent:

• (a) There exists a reference point $x^{\mathrm{ref}}\in K$ such that the set

  $$L_{<}(F,x^{\mathrm{ref}}):=\{x\in K:〈F(x),\eta (x,x^{\mathrm{ref}})〉<\alpha \left(\eta (x,x^{\mathrm{ref}})\right)\}$$

  is bounded (possibly empty).

• (b) The variational-like inequality $VLI(K,F)$ has a solution.

  Moreover, if there exists a vector $x^{\mathrm{ref}}\in K$ such that the set

  $$L_{\le }(F,x^{\mathrm{ref}}):=\{x\in K:〈F(x),\eta (x,x^{\mathrm{ref}})〉\le \alpha \left(\eta (x,x^{\mathrm{ref}})\right)\}$$

  is bounded and $\eta (x,y)+\eta (y,x)=0$ for all x, y in K, then the solution set $SOL(K,F)$ is nonempty and bounded.

Proof Suppose that there exists a reference point $x^{\mathrm{ref}}\in K$, which satisfies (a). Then there exists an open ball, denoted by Ω such that

$$L_{<}(F,x^{\mathrm{ref}})\cup \left\{x^{\mathrm{ref}}\right\}\subset \mathrm{\Omega }.$$

We combine this with the obvious property $\partial \mathrm{\Omega }\cap L_{<}(F,x^{\mathrm{ref}})=\mathrm{\varnothing }$. Thus $〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge \alpha (\eta (x,x^{\mathrm{ref}}))$ $\mathrm{\forall }x\in K\cap \partial \mathrm{\Omega }$. Define the set-valued map**s $T,S:K\to 2^X$, for any $x\in K$, by

$$T(x)=\{y\in K\cap \overline{\mathrm{\Omega }}:〈F(y),\eta (x,y)〉\ge 0\}$$

and

$$S(x)=\{y\in K\cap \overline{\mathrm{\Omega }}:〈F(x),\eta (x,y)〉\ge \alpha (\eta (x,y))\}.$$

We claim that T is a KKM map**. Indeed, if T is not a KKM map**, then there exists $\{x_1,x_2,\dots ,x_n\}\subset K$ such that $co\{x_1,x_2,\dots ,x_n\}⊈{\bigcup }_{i=1}^{n}T(x_i)$. That is, there exists a $x_0\in co\{x_1,x_2,\dots ,x_n\}$, $x_0={\sum }_{i=1}^n{t}_i{x}_i$, where $t_i\ge 0$, $i=1,2,\dots ,n$, ${\sum }_{i=1}^n{t}_i=1$, but $x_0\notin {\bigcup }_{i=1}^{n}T(x_i)$. By the definition of T, we have

$$〈F(x_0),\eta (x_i,x_0)〉<0,i=1,2,\dots ,n.$$

Since ${\sum }_{i=1}^n{t}_i=1$ for $t_i\ge 0$ ($i=1,2,\dots ,n$), it follows that

$$\sum _{i=1}^n{t}_i〈F(x_0),\eta (x_i,x_0)〉<0.$$

On the other hand, we note that

$$\begin{array}{rcl}\sum _{i=1}^n{t}_i〈F(x_0),\eta (x_i,x_0)〉& =& 〈F(x_0),\eta (\sum _{i=1}^n{t}_i{x}_i,x_0)〉\\ =& 〈F(x_0),\eta (x_0,x_0)〉\\ =& 0.\end{array}$$

It is a contradiction and this implies that T is a KKM map**. Now we show that $T(x)\subset S(x)$ for all $x\in K$. For any given $x\in K$, let $y\in T(x)$. Thus, we have $〈F(y),\eta (x,y)〉\ge 0$. Since F is a relaxed η-α pseudomonotone, we obtain $〈F(x),\eta (x,y)〉\ge \alpha (\eta (x,y))$. This implies that $y\in S(x)$ and so $T(x)\subset S(x)$ for all $x\in K$. It follows that S is also a KKM map**.

From the assumptions, we know that $S(x)$ is weakly closed. In fact, since η and α are lower semicontinuous, we see that $S(x)$ is a weakly closed subset of $K\cap \overline{\mathrm{\Omega }}$. Since $K\cap \overline{\mathrm{\Omega }}$ is a weakly compact and $S(x)$ is a weakly closed subset of $K\cap \overline{\mathrm{\Omega }}$, we see that $S(x)$ is weakly compact for each $x\in K$. Thus, the conditions of Lemma 2.7 are satisfied in the weak topology. By Lemma 2.7 and Theorem 3.1, we have

$$\bigcap _{x\in K}T(x)=\bigcap _{x\in K}S(x)\ne \mathrm{\varnothing }.$$

It follows that there exists $z\in K\cap \overline{\mathrm{\Omega }}$ such that

$$〈F(z),\eta (x,z)〉\ge 0\mathrm{\forall }x\in K.$$

Hence $z\in SOL(K,F)$.

Assume that (b) holds. We take any $x^{\mathrm{ref}}\in SOL(K,F)$. That is,

$$〈F\left(x^{\mathrm{ref}}\right),\eta (x,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }x\in K.$$

By the relaxed η-α pseudomonotonicity of F, we have

$$〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge \alpha \left(\eta (x,x^{\mathrm{ref}})\right)\mathrm{\forall }x\in K.$$

Hence $L_{<}(F,x^{\mathrm{ref}})=\mathrm{\varnothing }$ and (a) is valid.

Finally, suppose that there is some $x^{\mathrm{ref}}\in K$ such that the set $L_{\le }(F,x^{\mathrm{ref}})$ is bounded. Then $SOL(K,F)$ is nonempty by virtue of the implication (a) ⇒ (b). To prove that $SOL(K,F)$ is bounded, it suffices to show that $SOL(K,F)\subset L_{\le }(F,x^{\mathrm{ref}})$. Assume that $x\in SOL(K,F)$, but $x\notin L_{\le }(F,x^{\mathrm{ref}})$. Thus, we have

$$〈F(x),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K$$

(3.4)

and

$$〈F(x),\eta (x,x^{\mathrm{ref}})〉>\alpha \left(\eta (x,x^{\mathrm{ref}})\right).$$

(3.5)

Substituting $y=x^{\mathrm{ref}}$ into the inequality in (3.4), we have

$$〈F(x),\eta (x^{\mathrm{ref}},x)〉\ge 0.$$

(3.6)

This implies that $〈F(x),\eta (x,x^{\mathrm{ref}})〉\le 0$. From (3.5) and (3.6), we have $\alpha (\eta (x,x^{\mathrm{ref}}))<0$. By (3.6) and F is relaxed η-α pseudomonotone, we obtain

$$〈F\left(x^{\mathrm{ref}}\right),\eta (x^{\mathrm{ref}},x)〉\ge \alpha \left(\eta (x^{\mathrm{ref}},x)\right).$$

It implies that $〈F(x^{\mathrm{ref}}),\eta (x,x^{\mathrm{ref}})〉\le \alpha (\eta (x,x^{\mathrm{ref}}))$. By Theorem 3.1 and (3.5), we get $〈F(x^{\mathrm{ref}}),\eta (x,x^{\mathrm{ref}})〉\ge 0$. Hence

$$0\le 〈F\left(x^{\mathrm{ref}}\right),\eta (x,x^{\mathrm{ref}})〉\le \alpha \left(\eta (x,x^{\mathrm{ref}})\right)<0.$$

It is a contradiction. Therefore $x\in L_{\le }(F,x^{\mathrm{ref}})$. □

Theorem 3.3 Let X be a real reflexive Banach space and $K\subset X$ be a closed convex set. Let $\mathrm{\Phi }:K\to 2^{X^{\ast }}$ and $\eta :K\times K\to X$ be are map**s. Assume that:

• (i) Φ is a lower semicontinuous multifunction with nonempty closed convex values, where $X^{\ast }$ is endowed with the norm topology;

• (ii) Φ is a relaxed η-α pseudomonotone map**;

• (iii) $\eta (x,x)=0$ for all $x\in K$;

• (iv) $\eta (tx+(1-t)z,y)=t\eta (x,y)+(1-t)\eta (z,y)$ for all $x,y,z\in K$, $t\in [0,1]$ and η is lower semicontinuous;

• (v) $\alpha :X\to \mathbb{R}$ is lower semicontinuous.

Then the following statements are equivalent:

• (a) There exists a reference point $x^{\mathrm{ref}}\in K$ such that the set

  $$L_{<}(\mathrm{\Phi },x^{\mathrm{ref}}):=\{x\in K:\underset{x^{\ast }\in \mathrm{\Phi }(x)}{inf}〈x^{\ast },\eta (x,x^{\mathrm{ref}})〉<\alpha \left(\eta (x,x^{\mathrm{ref}})\right)\}$$

is bounded (possibly empty).

• (b) The generalized variational-like inequality $GVLI(K,\mathrm{\Phi })$ has a solution.

Proof Since Φ is lower semicontinuous multifunction with nonempty closed convex values, by Michael’s selection theorem (see for instance [30]) it admits a continuous selection; that is, there exists a continuous map** $F:K\to X^{\ast }$ such that $F(x)\in \mathrm{\Phi }(x)$ for every $x\in K$. If (a) holds, then there exists an open ball, denoted by Ω such that

$$L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})\cup \left\{x^{\mathrm{ref}}\right\}\subset \mathrm{\Omega }.$$

We combine this with the obvious property $\partial \mathrm{\Omega }\cap L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})=\mathrm{\varnothing }$. Thus, we have

$$〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge \underset{x^{\ast }\in \mathrm{\Phi }(x)}{inf}〈x^{\ast },\eta (x,x^{\mathrm{ref}})〉\ge \alpha \left(\eta (x,x^{\mathrm{ref}})\right)\mathrm{\forall }x\in K\cap \partial \mathrm{\Omega }.$$

Applying Theorem 3.2, we get $SOL(K,F)\ne \mathrm{\varnothing }$. For any $x\in SOL(K,F)$, if we choose $x^{\ast }=F(x)$ then

$$〈x^{\ast },\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

It follows that $\mathrm{\varnothing }\ne SOL(K,F)\subset SOL(K,\mathrm{\Phi })$.

We prove that (b) ⇒ (a). Assume that (b) holds. We take any $x^{\mathrm{ref}}\in SOL(K,\mathrm{\Phi })$. Thus there exists $x^{\ast }\in \mathrm{\Phi }(x^{\mathrm{ref}})$ satisfying

$$〈x^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }y\in K.$$

Because Φ is a relaxed η-α pseudomonotone, we obtain

$$〈y^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge \alpha \left(\eta (y,x^{\mathrm{ref}})\right)\mathrm{\forall }y\in K,y^{\ast }\in \mathrm{\Phi }(y).$$

It follows that

$$\underset{y^{\ast }\in \mathrm{\Phi }(y)}{inf}〈y^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge \alpha \left(\eta (y,x^{\mathrm{ref}})\right)\mathrm{\forall }y\in K.$$

Hence $L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})=\mathrm{\varnothing }$ and (a) is valid. □

4 Generalized variational-like inequality with strictly η-quasimonotone map**s

In this section, we will discuss the existence of solutions for the following variational-like inequality and generalized variational-like inequality with strictly η-quasimonotone map**s.

Theorem 4.1 Let K be a nonempty closed convex subset of a real reflexive Banach space X. Let $F:K\to X^{\ast }$ and $\eta :K\times K\to X$ be map**s. Assume that:

• (i) F is η-hemicontinuous and strictly η-quasimonotone;

• (ii) $\eta (x,x)=0$ for all $x\in K$;

• (iii) $\eta (x,y)+\eta (y,x)=0$ for all $x,y\in K$;

• (iv) for any fixed $y,z\in K$, the map** $x\mapsto 〈Tz,\eta (x,y)〉$ is convex.

Then $x\in K$ is a solution of $VLI(K,F)$ if and only if

$$〈F(y),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

(4.1)

Proof Suppose that $x\in K$ is a solution of $VLI(K,F)$. That is $〈F(x),\eta (y,x)〉\ge 0$ $\mathrm{\forall }y\in K$. To show that $〈F(y),\eta (y,x)〉\ge 0$ $\mathrm{\forall }y\in K$. Assume that there exists $y_0\in K$ such that $〈F(y_0),\eta (y_0,x)〉<0$. By the property of η, we have $〈F(y_0),\eta (x,y_0)〉>0$. Since F is strictly η-quasimonotone, we have $〈F(x),\eta (x,y_0)〉>0$. By the property of η again, we get $〈F(x),\eta (y_0,x)〉<0$. It is a contradiction. Hence $〈F(y),\eta (y,x)〉\ge 0$ $\mathrm{\forall }y\in K$.

Conversely, suppose that $x\in K$ is a solution of (4.1) and $y\in K$ is arbitrary. Letting $x_t=ty+(1-t)x$, $t\in (0,1]$, we have $x_t\in K$. It follows from (4.1) that

$$〈F(x_t),\eta (x_t,x)〉\ge 0.$$

(4.2)

By assumption, we have

$$\begin{array}{rcl}〈F(x_t),\eta (x_t,x)〉& =& 〈F(x_t),\eta (ty+(1-t)x,x)〉\\ \le & t〈F(x_t),\eta (y,x)〉+(1-t)〈F(x_t),\eta (x,x)〉\\ =& t〈F(x_t),\eta (y,x)〉.\end{array}$$

(4.3)

It follows from (4.2) and (4.3) that

$$〈F(x_t),\eta (y,x)〉\ge 0\mathrm{\forall }t\in (0,1].$$

Since F is η-hemicontinuous and letting $t\to 0^{+}$, we get

$$〈F(x),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

 □

Theorem 4.2 Let X be a real reflexive Banach space and $K\subset X$ be a closed convex set. Let $F:K\to X^{\ast }$ and $\eta :K\times K\to X$ be are map**s. Assume that:

• (i) F is a strictly η-quasimonotone map** and η-hemicontinuous;

• (ii) $\eta (x,x)=0$ for all $x\in K$;

• (iii) $\eta (x,y)+\eta (y,x)=0$ for all $x,y\in K$;

• (iv) for any fixed $y,z\in K$, the map** $x\mapsto 〈Tz,\eta (x,y)〉$ is convex and η is lower semicontinuous.

Then the following statements are equivalent:

• (a) There exists a reference point $x^{\mathrm{ref}}\in K$ such that the set

  $$L_{<}(F,x^{\mathrm{ref}}):=\{x\in K:〈F(x),\eta (x,x^{\mathrm{ref}})〉<0\}$$

is bounded (possibly empty).

• (b) The variational-like inequality $VLI(K,F)$ has a solution.

Moreover, if there exists a vector $x^{\mathrm{ref}}\in K$ such that the set

$$L_{\le }(F,x^{\mathrm{ref}}):=\{x\in K:〈F(x),\eta (x,x^{\mathrm{ref}})〉\le 0\}$$

is bounded, then the solution set $SOL(K,F)$ is nonempty and bounded.

Proof Suppose that (a) holds. Then there exists a reference point $x^{\mathrm{ref}}\in K$ and an open ball, denoted by Ω such that

$$L_{<}(F,x^{\mathrm{ref}})\cup \left\{x^{\mathrm{ref}}\right\}\subset \mathrm{\Omega }.$$

We combine this with the obvious property $\partial \mathrm{\Omega }\cap L_{<}(F,x^{\mathrm{ref}})=\mathrm{\varnothing }$. Thus $〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge \alpha (\eta (x,x^{\mathrm{ref}}))$ $\mathrm{\forall }x\in K\cap \partial \mathrm{\Omega }$. Defined the set-valued map**s $T,S:K\to 2^X$, for any $x\in K$, by

$$T(x)=\{y\in K\cap \overline{\mathrm{\Omega }}:〈F(y),\eta (x,y)〉\ge 0\}$$

and

$$S(x)=\{y\in K\cap \overline{\mathrm{\Omega }}:〈F(x),\eta (x,y)〉\ge 0\}.$$

Since η is lower semicontinuous, we find that $T(x)$ and $S(x)$ are weakly closed subsets of $K\cap \overline{\mathrm{\Omega }}$. We claim that T is a KKM map**. Similar to the proof of Theorem 3.2 we show that T is a KKM map**. Now we show that $T(x)\subset S(x)$ for all $x\in K$. For any given $x\in K$, we let $y\in T(x)$. That is, $〈F(y),\eta (x,y)〉\ge 0$. Since F is strictly η-quasimonotone, we have $〈F(x),\eta (x,y)〉\ge 0$. This implies that $y\in S(x)$ and so $T(x)\subset S(x)$ for all $x\in K$. It follows that S is also a KKM map**. Since $K\cap \overline{\mathrm{\Omega }}$ is weakly compact and $S(x)$ is a weakly closed subset of $K\cap \overline{\mathrm{\Omega }}$, we find that $S(x)$ is weakly compact for each $x\in K$. Thus, the condition of Lemma 2.7 is satisfied in the weak topology. By Lemma 2.7 and Theorem 4.1, we have

$$\bigcap _{x\in K}T(x)=\bigcap _{x\in K}S(x)\ne \mathrm{\varnothing }.$$

It follows that there exists $z\in K\cap \overline{\mathrm{\Omega }}$ such that

$$〈F(z),\eta (x,z)〉\ge 0\mathrm{\forall }x\in K.$$

Hence $z\in SOL(K,F)$.

Assume that (b) holds. We take any $x^{\mathrm{ref}}\in SOL(K,F)$, that is,

$$〈F\left(x^{\mathrm{ref}}\right),\eta (x,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }x\in K.$$

By the strict η-quasimonotonicity of F, we have

$$〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }x\in K.$$

Hence $L_{<}(F,x^{\mathrm{ref}})=\mathrm{\varnothing }$ and (a) is valid.

Finally, suppose that there is some $x^{\mathrm{ref}}\in K$ such that the set $L_{\le }(F,x^{\mathrm{ref}})$ is bounded. Then $SOL(K,F)$ is nonempty by virtue of the implication (a) ⇒ (b). To prove that $SOL(K,F)$ is bounded, it suffices to show that $SOL(K,F)\subset L_{\le }(F,x^{\mathrm{ref}})$. Assume that $x\in SOL(K,F)$. Thus, we have

$$〈F(x),\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

(4.4)

Substituting $y=x^{\mathrm{ref}}$ into the inequality in (4.4), we have

$$〈F(x),\eta (x^{\mathrm{ref}},x)〉\ge 0.$$

(4.5)

This implies that $〈F(x),\eta (x,x^{\mathrm{ref}})〉\le 0$. Therefore $x\in L_{\le }(F,x^{\mathrm{ref}})$. □

Theorem 4.3 Let X be a real reflexive Banach space and $K\subset X$ be a closed convex set. Let $\mathrm{\Phi }:K\to 2^{X^{\ast }}$ and $\eta :K\times K\to X$ be are map**s. Assume that:

• (i) Φ is a lower semicontinuous multifunction with nonempty closed convex values, where $X^{\ast }$ is endowed with the norm topology;

• (ii) Φ is a strictly η-quasimonotone map**;

• (iii) $\eta (x,x)=0$ for all $x\in K$;

• (iv) $\eta (x,y)+\eta (y,x)=0$ for all $x,y\in K$;

• (v) for any fixed $y,z\in K$, the map** $x\mapsto 〈Tz,\eta (x,y)〉$ is convex and η is lower semicontinuous.

Then the following statements are equivalent:

• (a) There exists a reference point $x^{\mathrm{ref}}\in K$ such that the set

  $$L_{<}(\mathrm{\Phi },x^{\mathrm{ref}}):=\{x\in K:\underset{x^{\ast }\in \mathrm{\Phi }(x)}{inf}〈x^{\ast },\eta (x,x^{\mathrm{ref}})〉<0\}$$

is bounded (possibly empty).

• (b) The generalized variational-like inequality $GVLI(K,\mathrm{\Phi })$ has a solution.

Proof Since Φ is a lower semicontinuous multifunction with nonempty closed convex values, by Michael’s selection theorem (see for instance [30]) it admits a continuous selection; that is, there exists a continuous map** $F:K\to X^{\ast }$ such that $F(x)\in \mathrm{\Phi }(x)$ for every $x\in K$. If (a) holds, then there exists an open ball, denoted by Ω, such that

$$L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})\cup \left\{x^{\mathrm{ref}}\right\}\subset \mathrm{\Omega }.$$

We combine this with the obvious property $\partial \mathrm{\Omega }\cap L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})=\mathrm{\varnothing }$. Then we have

$$〈F(x),\eta (x,x^{\mathrm{ref}})〉\ge \underset{x^{\ast }\in \mathrm{\Phi }(x)}{inf}〈x^{\ast },\eta (x,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }x\in K\cap \partial \mathrm{\Omega }.$$

Applying Theorem 4.2, we get $SOL(K,F)\ne \mathrm{\varnothing }$. For any $x\in SOL(K,F)$, if we choose $x^{\ast }=F(x)$ then

$$〈x^{\ast },\eta (y,x)〉\ge 0\mathrm{\forall }y\in K.$$

It follows that $\mathrm{\varnothing }\ne SOL(K,F)\subset SOL(K,\mathrm{\Phi })$.

We prove that (b) ⇒ (a). Assume that (b) holds. We take any $x^{\mathrm{ref}}\in SOL(K,\mathrm{\Phi })$. Thus there exists $x^{\ast }\in \mathrm{\Phi }(x^{\mathrm{ref}})$ satisfying

$$〈x^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }y\in K.$$

Because Φ is strictly η-quasimonotone and Theorem 4.1, we obtain

$$〈y^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }y\in K,y^{\ast }\in \mathrm{\Phi }(y).$$

It follows that

$$\underset{y^{\ast }\in \mathrm{\Phi }(y)}{inf}〈y^{\ast },\eta (y,x^{\mathrm{ref}})〉\ge 0\mathrm{\forall }y\in K.$$

Hence $L_{<}(\mathrm{\Phi },x^{\mathrm{ref}})=\mathrm{\varnothing }$ and (a) is valid. □