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

- Areerat Arunchai
^{1}, - Somyot Plubtieng
^{1}Email author and - Ching-Feng Wen
^{2}

**2014**:442

https://doi.org/10.1186/1029-242X-2014-442

© Arunchai et al.; licensee Springer. 2014

**Received: **23 April 2014

**Accepted: **20 October 2014

**Published: **4 November 2014

## 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.

## Keywords

*η*-

*α*pseudomonotone operatorstrictly

*η*-quasimonotone operatorsolution existence

## 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 mappings in nonreflexive Banach spaces. Recently, Verma [15] studied a class variational inequality relaxed monotone mapping. Moreover, Fang and Huang [16] obtained the existence of solutions for variational-like inequalities with relaxed *η*-*α* monotone mappings 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 mappings 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 mappings. By using the KKM technique, they obtain some existence results for variational-like inequalities with relaxed *η*-*α* pseudomonotone mappings 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 mappings 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 mappings and prove the existence of solutions for variational-like inequality and generalized variational-like inequality with relaxed *η*-*α* pseudomonotone mappings and strictly *η*-quasimonotone mappings in Banach spaces by using KKM technique. The results presented in this paper improve and extend some corresponding results of several authors.

## 2 Preliminaries

*X*be a real reflexive Banach space with dual space ${X}^{\ast}$ and $\u3008\cdot ,\cdot \u3009$ 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 mappings. 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

*variational*-

*like inequality*and the abbreviation $VLI(K,F)$ is the problem of finding an $x\in K$ such that

We introduce the definition of relaxed *η*-*α* pseudomonotone for *α* mapping which comes from a family of functions which contains all mappings *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 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 mapping $F:K\to {X}^{\ast}$ 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 $x,y\in K$,$\u3008F(y),\eta (x,y)\u3009\ge 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),\eta (x,y)\u3009\ge \alpha (\eta (x,y)).$(2.3)If $\eta (x,y)=x-y$ for all distinct points*x*,*y*in*K*, then (2.3) becomes$\u3008F(y),x-y\u3009\ge 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),x-y\u3009\ge \alpha (x-y),$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 $x,y\in K$,$\u3008F(y),\eta (x,y)\u3009>0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),\eta (x,y)\u3009>0.$(2.4)If $\eta (x,y)=x-y$ for all distinct points*x*,*y*in*K*, then (2.4) becomes$\u3008F(y),x-y\u3009>0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),x-y\u3009>0,$and

*F*is said to be*strictly quasimonotone*.

**Definition 2.2**The mapping $\mathrm{\Phi}:K\to {2}^{{X}^{\ast}}$ 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 $\alpha \in A$,$\begin{array}{c}\u3008{y}^{\ast},\eta (x,y)\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\exists}{y}^{\ast}\in \mathrm{\Phi}(y)\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008{x}^{\ast},\eta (x,y)\u3009\ge \alpha (\eta (x,y)),\phantom{\rule{1em}{0ex}}\mathrm{\exists}{x}^{\ast}\in \mathrm{\Phi}(x),\mathrm{\forall}x,y\in X.\hfill \end{array}$ - (ii)
*Strictly**η*-*quasimonotone*if there exist $\eta :K\times K\to X$ such that$\begin{array}{c}\u3008F(y),\eta (x,y)\u3009>0,\phantom{\rule{1em}{0ex}}\mathrm{\exists}{y}^{\ast}\in \mathrm{\Phi}(y)\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),\eta (x,y)\u3009>0,\phantom{\rule{1em}{0ex}}\mathrm{\exists}{x}^{\ast}\in \mathrm{\Phi}(x),\mathrm{\forall}x,y\in X.\hfill \end{array}$

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

*F*is a relaxed

*η*-

*α*pseudomonotone mapping with

But it is not a relaxed *α*-pseudomonotone mapping. In fact, if we let $x=-1$, $y=0$, $\u3008F(y),x-y\u3009\ge 0$, but $\u3008F(x),x-y\u3009<\alpha (x-y)$, which is a contradiction.

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

where $c>0$. Then the mapping *F* is strictly *η*-quasimonotone but fails to be strictly quasimonotone since if $x\in (-1,1)$ and $y<-1$, then we have $\u3008F(y),x-y\u3009\ge 0$ but $\u3008F(x),x-y\u3009<0$.

**Definition 2.5** ([20])

Let $F:K\to {X}^{\ast}$ and $\eta :K\times K\to X$ be two mappings. *F* is said to be *η*-*hemicontinuous* if, for any fixed $x,y\in K$, the mapping $f:[0,1]\to (-\mathrm{\infty},+\mathrm{\infty})$ defined by $f(t)=\u3008F(x+t(y-x)),\eta (y,x)\u3009$ 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 mapping $F:K\to {2}^{X}$ is said to be a *KKM mapping* 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 mapping*.

*If*$F(x)$

*is closed in*

*X*

*for every*

*x*

*in*

*K*

*and compact for some*${x}_{0}\in K$,

*then*

**Lemma 2.8** (Michael selection theorem [30])

*Let* *X* *be a paracompact space and* *Y* *be a Banach space*. *Then every lower semicontinuous multivalued mapping 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 mappings

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 mappings.

**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 mappings*.

*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*

*Proof*Suppose that $x\in K$ is a solution of $VLI(K,F)$. Since

*F*is relaxed

*η*-

*α*pseudomonotone, we have

*η*, we have

□

**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 mappings*.

*Assume that*:

- (i)
*F**is a relaxed**η*-*α**pseudomonotone mapping 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*${L}_{<}(F,{x}^{\mathrm{ref}}):=\{x\in K:\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009<\alpha \left(\eta (x,{x}^{\mathrm{ref}})\right)\}$*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*${L}_{\le}(F,{x}^{\mathrm{ref}}):=\{x\in K:\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009\le \alpha \left(\eta (x,{x}^{\mathrm{ref}})\right)\}$*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

*T*is a KKM mapping. Indeed, if

*T*is not a KKM mapping, then there exists $\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}\subset K$ such that $co\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}\u2288{\bigcup}_{i=1}^{n}T({x}_{i})$. 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

It is a contradiction and this implies that *T* is a KKM mapping. 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 $\u3008F(y),\eta (x,y)\u3009\ge 0$. Since *F* is a relaxed *η*-*α* pseudomonotone, we obtain $\u3008F(x),\eta (x,y)\u3009\ge \alpha (\eta (x,y))$. 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 mapping.

*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

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

*η*-

*α*pseudomonotonicity of

*F*, we have

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

*F*is relaxed

*η*-

*α*pseudomonotone, we obtain

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 mappings*.

*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 mapping*; - (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*${L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}}):=\{x\in K:\underset{{x}^{\ast}\in \mathrm{\Phi}(x)}{inf}\u3008{x}^{\ast},\eta (x,{x}^{\mathrm{ref}})\u3009<\alpha \left(\eta (x,{x}^{\mathrm{ref}})\right)\}$

*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 mapping $F:K\to {X}^{\ast}$ 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

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

*η*-

*α*pseudomonotone, we obtain

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

## 4 Generalized variational-like inequality with strictly *η*-quasimonotone mappings

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 mappings.

**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 mappings*.

*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 mapping*$x\mapsto \u3008Tz,\eta (x,y)\u3009$*is convex*.

*Then*$x\in K$

*is a solution of*$VLI(K,F)$

*if and only if*

*Proof* Suppose that $x\in K$ is a solution of $VLI(K,F)$. That is $\u3008F(x),\eta (y,x)\u3009\ge 0$ $\mathrm{\forall}y\in K$. To show that $\u3008F(y),\eta (y,x)\u3009\ge 0$ $\mathrm{\forall}y\in K$. Assume that there exists ${y}_{0}\in K$ such that $\u3008F({y}_{0}),\eta ({y}_{0},x)\u3009<0$. By the property of *η*, we have $\u3008F({y}_{0}),\eta (x,{y}_{0})\u3009>0$. Since *F* is strictly *η*-quasimonotone, we have $\u3008F(x),\eta (x,{y}_{0})\u3009>0$. By the property of *η* again, we get $\u3008F(x),\eta ({y}_{0},x)\u3009<0$. It is a contradiction. Hence $\u3008F(y),\eta (y,x)\u3009\ge 0$ $\mathrm{\forall}y\in K$.

*F*is

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

□

**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 mappings*.

*Assume that*:

- (i)
*F**is a strictly**η*-*quasimonotone mapping 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 mapping*$x\mapsto \u3008Tz,\eta (x,y)\u3009$*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*${L}_{<}(F,{x}^{\mathrm{ref}}):=\{x\in K:\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009<0\}$

*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*

*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

*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 mapping. Similar to the proof of Theorem 3.2 we show that

*T*is a KKM mapping. 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, $\u3008F(y),\eta (x,y)\u3009\ge 0$. Since

*F*is strictly

*η*-quasimonotone, we have $\u3008F(x),\eta (x,y)\u3009\ge 0$. 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 mapping. 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

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

*η*-quasimonotonicity of

*F*, we have

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

This implies that $\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009\le 0$. 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 mappings*.

*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 mapping*; - (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 mapping*$x\mapsto \u3008Tz,\eta (x,y)\u3009$*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*${L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}}):=\{x\in K:\underset{{x}^{\ast}\in \mathrm{\Phi}(x)}{inf}\u3008{x}^{\ast},\eta (x,{x}^{\mathrm{ref}})\u3009<0\}$

*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 mapping $F:K\to {X}^{\ast}$ 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

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

*η*-quasimonotone and Theorem 4.1, we obtain

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

## Declarations

### Acknowledgements

The first author would like to thank the Thailand Research Fund for financial support and the second author is also supported by the Royal Golden Jubilee Program under Grant PHD/0282/2550, Thailand.

## Authors’ Affiliations

## References

- Giannessi F:
**Theorems of alternative, quadratic programs and complementarity problems.**In*Variational Inequalities and Complementarity Problems*. Edited by: Cottle RW, Giannessi F, Lions JL. Wiley, New York; 1980:151–186.Google Scholar - Hartmann P, Stampacchia G:
**On some nonlinear elliptic differential functional equations.***Acta Math.*1966,**115:**153–188.View ArticleGoogle Scholar - Kinderlehrer D, Stampacchia G:
*An Introduction to Variational Inequalities and Their Applications*. Academic Press, New York; 1980.MATHGoogle Scholar - Aussel D, Hadjisavvas N:
**On quasimonotone variational inequalities.***J. Optim. Theory Appl.*2004,**121:**445–450.MathSciNetView ArticleMATHGoogle Scholar - Bianchi M, Hadjisavvas N, Shaible S:
**Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities.***J. Optim. Theory Appl.*2004,**122:**1–17.MathSciNetView ArticleMATHGoogle Scholar - Crouzeix JP:
**Pseudomonotone variational inequality problems: existence of solutions.***Math. Program.*1997,**78:**305–314.MathSciNetMATHGoogle Scholar - Daniilidis A, Hadjisavvas N:
**Coercivity conditions and variational inequalities.***Math. Program.*1999,**86:**433–438. 10.1007/s101070050097MathSciNetView ArticleMATHGoogle Scholar - Konnov IV:
**Generalized monotone equilibrium problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity*. Edited by: Hadjisavvas N, Komlósi S, Schaible S. Springer, Berlin; 2005:559–618.View ArticleGoogle Scholar - Yao JC:
**Variational inequalities with generalized monotone operators.***Math. Oper. Res.*1994,**19:**691–705. 10.1287/moor.19.3.691MathSciNetView ArticleMATHGoogle Scholar - Yao JC:
**Multi-valued variational inequalities with**K**-pseudomonotone operators.***J. Optim. Theory Appl.*1994,**80:**63–74. 10.1007/BF02196593MathSciNetView ArticleMATHGoogle Scholar - Yao JC, Chadli O:
**Pseudomonotone complementarity problems and variational inequalities.**In*Handbook of Generalized Convexity and Generalized Monotonicity*. Edited by: Hadjisavvas N, Komlósi S, Schaible S. Springer, Berlin; 2005:501–558.View ArticleGoogle Scholar - Ricceri B:
**Basic existence theorems for generalized variational and quasi-variational inequalities.**In*Variational Inequalities and Network Equilibrium Problems*. Edited by: Giannessi F, Maugeri A. Plenum, New York; 1995:251–255.Google Scholar - Yen ND:
**On a problem of B. Ricceri on variational inequalities. 5.**In*Fixed Point Theory and Applications*. Edited by: Cho YJ, Kim JK, Kang SM. Nova Science Publishers, New York; 2004:163–173.Google Scholar - Chang SS, Lee BS, Chen YQ:
**Variational inequalities for monotone operators in nonreflexive Banach spaces.***Appl. Math. Lett.*1995,**8:**29–34.MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators.***J. Math. Anal. Appl.*1997,**213:**387–392. 10.1006/jmaa.1997.5556MathSciNetView ArticleMATHGoogle Scholar - Fang YP, Huang NJ:
**Existence results for systems of strong implicit vector variational inequalities.***Acta Math. Hung.*2004,**103:**265–277.MathSciNetView ArticleMATHGoogle Scholar - Facchinei F, Pang JS
**I.**In*Finite-Dimensional Variational Inequalities and Complementarity Problems*. Springer, Berlin; 2003.Google Scholar - Facchinei F, Pang JS
**II.**In*Finite-Dimensional Variational Inequalities and Complementarity Problems*. Springer, Berlin; 2003.Google Scholar - Kien BT, Yao JC, Yen ND:
**On the solution existence of pseudomonotone variational inequalities.***J. Glob. Optim.*2008,**41:**135–145. 10.1007/s10898-007-9251-0MathSciNetView ArticleMATHGoogle Scholar - Bai MR, Zhoua SZ, Ni GY:
**Variational-like inequalities with relaxed**η**-**α**pseudomonotone mappings in Banach spaces.***Appl. Math. Lett.*2006,**19:**547–554. 10.1016/j.aml.2005.07.010MathSciNetView ArticleGoogle Scholar - Wu KQ, Huang NJ:
**Vector variational-like inequalities with relaxed**η**-**α**pseudomonotone mappings in Banach spaces.***J. Math. Inequal.*2007,**1:**281–290.MathSciNetView ArticleGoogle Scholar - Pourbarat K, Abbasi M:
**On the vector variational-like inequalities with relaxed**η**-**α**pseudomonotone mappings.***Iran. J. Math. Sci. Inform.*2009,**4**(1):37–42.MathSciNetMATHGoogle Scholar - Goeleven D, Motreanu D:
**Eigenvalue and dynamic problems for variational and hemivariational inequalities.***Commun. Appl. Nonlinear Anal.*1996,**3:**1–21.MathSciNetMATHGoogle Scholar - Siddiqi AH, Ansari QH, Kazmi KR:
**On nonlinear variational inequalities.***Indian J. Pure Appl. Math.*1994,**25:**969–973.MathSciNetMATHGoogle Scholar - Verma RU:
**On monotone nonlinear variational inequalities problems.***Comment. Math. Univ. Carol.*1998,**39:**91–98.MATHGoogle Scholar - Cottle RW, Yao JC:
**Pseudomonotone complementarity problems in Hilbert spaces.***J. Optim. Theory Appl.*1992,**78:**281–295.MathSciNetView ArticleMATHGoogle Scholar - Karamardian S:
**Complementarity problems over cones with monotone and pseudomonotone maps.***J. Optim. Theory Appl.*1976,**18:**445–454. 10.1007/BF00932654MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Lee GM, Yao JC:
**Generalized variational-like inequalities with compositely monotone multifunctions.***J. Korean Math. Soc.*2008,**45**(3):841–858. 10.4134/JKMS.2008.45.3.841MathSciNetView ArticleMATHGoogle Scholar - Yuan GXZ:
*KKM Theory and Applications in Nonlinear Analysis*. Dekker, New York; 1999.MATHGoogle Scholar - Michael EA:
**Continuous selections, I.***Ann. Math.*1956,**63:**361–382. 10.2307/1969615MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.