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Existence theorems for relaxed ηα pseudomonotone and strictly ηquasimonotone generalized variationallike inequalities
Journal of Inequalities and Applications volume 2014, Article number: 442 (2014)
Abstract
In this paper, we prove the existence of solutions for a variationallike inequality and a generalized variationallike inequality in the relaxed ηα pseudomonotone and strictly ηquasimonotone 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 finitedimensional 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 HartmanStampacchia 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 variationallike inequalities with relaxed ηα 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 finitedimensional 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 infinitedimensional reflexive Banach spaces.
On the other hand, Bai et al. [20] introduced the new concept of relaxed ηα pseudomonotone mappings. By using the KKM technique, they obtain some existence results for variationallike inequalities with relaxed ηα pseudomonotone mappings in reflexive Banach spaces. In 2007, Wu and Huang [21] introduced the two new concepts of relaxed ηα pseudomonotonicity and relaxed ηα 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 variationallike inequalities with relaxed ηα 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 ηα pseudomonotone mappings and prove the existence of solutions for variationallike inequality and generalized variationallike inequality with relaxed ηα pseudomonotone mappings and strictly ηquasimonotone 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
Let 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 variationallike inequality defined by K and Φ, denoted by $GVLI(K,\mathrm{\Phi})$, is the problem of finding a point $x\in K$ such that
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 singlevalued mapping, then the problem $GVLI(K,\mathrm{\Phi})$ is called a variationallike inequality and the abbreviation $VLI(K,F)$ is the problem of finding an $x\in K$ such that
We introduce the definition of relaxed ηα 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 introduce the family
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 ηα 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 $\eta (x,y)=xy$ for all distinct points x, y in K, then (2.3) becomes
$$\u3008F(y),xy\u3009\ge 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),xy\u3009\ge \alpha (xy),$$and F is said to be relaxed α pseudomonotone.

(ii)
Strictly ηquasimonotone 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 $\eta (x,y)=xy$ for all distinct points x, y in K, then (2.4) becomes
$$\u3008F(y),xy\u3009>0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\u3008F(x),xy\u3009>0,$$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 ηα 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 ηquasimonotone 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
where $c>0$, then the mapping F is a relaxed ηα pseudomonotone mapping with
But it is not a relaxed αpseudomonotone mapping. In fact, if we let $x=1$, $y=0$, $\u3008F(y),xy\u3009\ge 0$, but $\u3008F(x),xy\u3009<\alpha (xy)$, 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 ηquasimonotone but fails to be strictly quasimonotone since if $x\in (1,1)$ and $y<1$, then we have $\u3008F(y),xy\u3009\ge 0$ but $\u3008F(x),xy\u3009<0$.
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 ηhemicontinuous if, for any fixed $x,y\in K$, the mapping $f:[0,1]\to (\mathrm{\infty},+\mathrm{\infty})$ defined by $f(t)=\u3008F(x+t(yx)),\eta (y,x)\u3009$ is continuous at 0^{+}.
If $\eta (x,y)=xy$ $\mathrm{\forall}x,y\in K$, 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 variationallike inequality with relaxed ηα pseudomonotone mappings
In this section, we will discuss the existence of solutions for the following variationallike inequality and generalized variationallike inequality with relaxed ηα 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 ηhemicontinuous and relaxed ηα pseudomonotone;

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

(iii)
$\eta (tx+(1t)z,y)=t\eta (x,y)+(1t)\eta (z,y)$ 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 ηα pseudomonotone, we have
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 ${x}_{t}=ty+(1t)x$, $t\in (0,1]$, we have ${x}_{t}\in K$. It follows from (3.1) that
By the conditions of η, we have
It follows from (3.2) and (3.3) that
So, we have
Letting $t\to {0}^{+}$, we get
□
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 ηα pseudomonotone mapping and ηhemicontinuous;

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

(iii)
$\eta (tx+(1t)z,y)=t\eta (x,y)+(1t)\eta (z,y)$ 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 variationallike 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
We combine this with the obvious property $\partial \mathrm{\Omega}\cap {L}_{<}(F,{x}^{\mathrm{ref}})=\mathrm{\varnothing}$. Thus $\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009\ge \alpha (\eta (x,{x}^{\mathrm{ref}}))$ $\mathrm{\forall}x\in K\cap \partial \mathrm{\Omega}$. Define the setvalued mappings $T,S:K\to {2}^{X}$, for any $x\in K$, by
and
We claim 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
Since ${\sum}_{i=1}^{n}{t}_{i}=1$ for ${t}_{i}\ge 0$ ($i=1,2,\dots ,n$), it follows that
On the other hand, we note that
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 ηα 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.
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
It follows that there exists $z\in K\cap \overline{\mathrm{\Omega}}$ such that
Hence $z\in SOL(K,F)$.
Assume that (b) holds. We take any ${x}^{\mathrm{ref}}\in SOL(K,F)$. That is,
By the relaxed ηα pseudomonotonicity of F, we have
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
and
Substituting $y={x}^{\mathrm{ref}}$ into the inequality in (3.4), we have
This implies that $\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009\le 0$. From (3.5) and (3.6), we have $\alpha (\eta (x,{x}^{\mathrm{ref}}))<0$. By (3.6) and F is relaxed ηα pseudomonotone, we obtain
It implies that $\u3008F({x}^{\mathrm{ref}}),\eta (x,{x}^{\mathrm{ref}})\u3009\le \alpha (\eta (x,{x}^{\mathrm{ref}}))$. By Theorem 3.1 and (3.5), we get $\u3008F({x}^{\mathrm{ref}}),\eta (x,{x}^{\mathrm{ref}})\u3009\ge 0$. Hence
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 ηα pseudomonotone mapping;

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

(iv)
$\eta (tx+(1t)z,y)=t\eta (x,y)+(1t)\eta (z,y)$ 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 variationallike 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
We combine this with the obvious property $\partial \mathrm{\Omega}\cap {L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}})=\mathrm{\varnothing}$. Thus, we have
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
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
Because Φ is a relaxed ηα pseudomonotone, we obtain
It follows that
Hence ${L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}})=\mathrm{\varnothing}$ and (a) is valid. □
4 Generalized variationallike inequality with strictly ηquasimonotone mappings
In this section, we will discuss the existence of solutions for the following variationallike inequality and generalized variationallike inequality with strictly ηquasimonotone 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 ηhemicontinuous and strictly ηquasimonotone;

(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 ηquasimonotone, 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 $\u3008F(y),\eta (y,x)\u3009\ge 0$ $\mathrm{\forall}y\in K$.
Conversely, suppose that $x\in K$ is a solution of (4.1) and $y\in K$ is arbitrary. Letting ${x}_{t}=ty+(1t)x$, $t\in (0,1]$, we have ${x}_{t}\in K$. It follows from (4.1) that
By assumption, we have
It follows from (4.2) and (4.3) that
Since F is ηhemicontinuous 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 ηquasimonotone mapping and ηhemicontinuous;

(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 variationallike 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
We combine this with the obvious property $\partial \mathrm{\Omega}\cap {L}_{<}(F,{x}^{\mathrm{ref}})=\mathrm{\varnothing}$. Thus $\u3008F(x),\eta (x,{x}^{\mathrm{ref}})\u3009\ge \alpha (\eta (x,{x}^{\mathrm{ref}}))$ $\mathrm{\forall}x\in K\cap \partial \mathrm{\Omega}$. Defined the setvalued mappings $T,S:K\to {2}^{X}$, for any $x\in K$, by
and
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 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 ηquasimonotone, 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
It follows that there exists $z\in K\cap \overline{\mathrm{\Omega}}$ such that
Hence $z\in SOL(K,F)$.
Assume that (b) holds. We take any ${x}^{\mathrm{ref}}\in SOL(K,F)$, that is,
By the strict ηquasimonotonicity of F, we have
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
Substituting $y={x}^{\mathrm{ref}}$ into the inequality in (4.4), we have
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 ηquasimonotone 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 variationallike 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
We combine this with the obvious property $\partial \mathrm{\Omega}\cap {L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}})=\mathrm{\varnothing}$. Then we have
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
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
Because Φ is strictly ηquasimonotone and Theorem 4.1, we obtain
It follows that
Hence ${L}_{<}(\mathrm{\Phi},{x}^{\mathrm{ref}})=\mathrm{\varnothing}$ and (a) is valid. □
References
 1.
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.
 2.
Hartmann P, Stampacchia G: On some nonlinear elliptic differential functional equations. Acta Math. 1966, 115: 153–188.
 3.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York; 1980.
 4.
Aussel D, Hadjisavvas N: On quasimonotone variational inequalities. J. Optim. Theory Appl. 2004, 121: 445–450.
 5.
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.
 6.
Crouzeix JP: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 1997, 78: 305–314.
 7.
Daniilidis A, Hadjisavvas N: Coercivity conditions and variational inequalities. Math. Program. 1999, 86: 433–438. 10.1007/s101070050097
 8.
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.
 9.
Yao JC: Variational inequalities with generalized monotone operators. Math. Oper. Res. 1994, 19: 691–705. 10.1287/moor.19.3.691
 10.
Yao JC: Multivalued variational inequalities with K pseudomonotone operators. J. Optim. Theory Appl. 1994, 80: 63–74. 10.1007/BF02196593
 11.
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.
 12.
Ricceri B: Basic existence theorems for generalized variational and quasivariational inequalities. In Variational Inequalities and Network Equilibrium Problems. Edited by: Giannessi F, Maugeri A. Plenum, New York; 1995:251–255.
 13.
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.
 14.
Chang SS, Lee BS, Chen YQ: Variational inequalities for monotone operators in nonreflexive Banach spaces. Appl. Math. Lett. 1995, 8: 29–34.
 15.
Verma RU: On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators. J. Math. Anal. Appl. 1997, 213: 387–392. 10.1006/jmaa.1997.5556
 16.
Fang YP, Huang NJ: Existence results for systems of strong implicit vector variational inequalities. Acta Math. Hung. 2004, 103: 265–277.
 17.
Facchinei F, Pang JS I. In FiniteDimensional Variational Inequalities and Complementarity Problems. Springer, Berlin; 2003.
 18.
Facchinei F, Pang JS II. In FiniteDimensional Variational Inequalities and Complementarity Problems. Springer, Berlin; 2003.
 19.
Kien BT, Yao JC, Yen ND: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 2008, 41: 135–145. 10.1007/s1089800792510
 20.
Bai MR, Zhoua SZ, Ni GY: Variationallike inequalities with relaxed η  α pseudomonotone mappings in Banach spaces. Appl. Math. Lett. 2006, 19: 547–554. 10.1016/j.aml.2005.07.010
 21.
Wu KQ, Huang NJ: Vector variationallike inequalities with relaxed η  α pseudomonotone mappings in Banach spaces. J. Math. Inequal. 2007, 1: 281–290.
 22.
Pourbarat K, Abbasi M: On the vector variationallike inequalities with relaxed η  α pseudomonotone mappings. Iran. J. Math. Sci. Inform. 2009,4(1):37–42.
 23.
Goeleven D, Motreanu D: Eigenvalue and dynamic problems for variational and hemivariational inequalities. Commun. Appl. Nonlinear Anal. 1996, 3: 1–21.
 24.
Siddiqi AH, Ansari QH, Kazmi KR: On nonlinear variational inequalities. Indian J. Pure Appl. Math. 1994, 25: 969–973.
 25.
Verma RU: On monotone nonlinear variational inequalities problems. Comment. Math. Univ. Carol. 1998, 39: 91–98.
 26.
Cottle RW, Yao JC: Pseudomonotone complementarity problems in Hilbert spaces. J. Optim. Theory Appl. 1992, 78: 281–295.
 27.
Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 1976, 18: 445–454. 10.1007/BF00932654
 28.
Ceng LC, Lee GM, Yao JC: Generalized variationallike inequalities with compositely monotone multifunctions. J. Korean Math. Soc. 2008,45(3):841–858. 10.4134/JKMS.2008.45.3.841
 29.
Yuan GXZ: KKM Theory and Applications in Nonlinear Analysis. Dekker, New York; 1999.
 30.
Michael EA: Continuous selections, I. Ann. Math. 1956, 63: 361–382. 10.2307/1969615
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.
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The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between all authors. SP, CFW and AA defined the research theme. SP and CFW designed theorems and methods of proof and interpreted the results. AA proved the theorems, interpreted the results and wrote the paper. All authors read and approved the final manuscript.
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Keywords
 variationallike inequality
 generalized variationallike inequality
 relaxed ηα pseudomonotone operator
 strictly ηquasimonotone operator
 solution existence