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# Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 294 (2013)

## Abstract

In this paper, we introduce and study a new class of generalized nonlinear vector mixed quasi-variational-like inequalities governed by a multi-valued map in Hausdorff topological vector spaces which includes generalized vector mixed general quasi-variational-like inequalities, generalized nonlinear mixed variational-like inequalities, and so on. By using the fixed point theorem, we prove some existence theorems for the proposed inequality.

## 1 Introduction

Variational inequality theory has appeared as an effective and powerful tool to study and investigate a wide class of problems arising in pure and applied sciences including elasticity, optimization, economics, transportation, and structural analysis; see, for instance, [1–4] and the references therein. A vector variational inequality in a finite-dimensional Euclidean space was first introduced by Giannessi [5]. This is a generalization of scalar variational inequality to the vector case by virtue of multi-criterion consideration. In 1966, Browder [6] first introduced and proved the basic existence theorems of solutions to a class of nonlinear variational inequalities. The Browder’s results was extended to more generalized nonlinear variational inequalities by Liu *et al.* [7], Ahmad and Irfan [8], Husain and Gupta [9] and Xiao *et al.* [10], Zhao *et al.* [11].

In this paper, we consider a generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map and establish some existence results in locally convex topological vector spaces by using the fixed point theorem.

Let *Y* be a locally convex Hausdorff topological vector space (l.c.s., in short) and let *K* be a nonempty convex subset of a Hausdorff topological vector space (t.v.s., in short) *E*. We denote by $L(E,Y)$ the space of all continuous linear operators from *E* into *Y*, where $L(E,Y)$ is equipped with a *σ*-topology, and by $\u3008l,x\u3009$ the evaluation of $l\in L(E,Y)$ at $x\in E$. Let $X\subseteq L(E,Y)$. From the corollary of Schaefer [12], $L(E,Y)$ becomes a l.c.s. By Ding and Tarafdar [13], we have the bilinear map $\u3008\cdot ,\cdot \u3009:L(K,Y)\times K\to Y$ is continuous. Let int*A* and $co(A)$ represent the interior and convex hull of a set *A*, respectively. Let $C:K\to {2}^{Y}$ be a set-valued mapping such that $intC(x)\ne \mathrm{\varnothing}$ for each $x\in K$, let $\eta :K\times K\to E$ be a vector-valued mapping.

Let $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to {2}^{L(E,Y)}$ be a set-valued mapping, $H:K\times K\to {2}^{Y}$, $D:K\to {2}^{K}$ and $T,A,M:K\to {2}^{X}$ be set-valued mappings. For each ${\omega}^{\ast}\in L(E,Y)$ and $g:K\to K$ a single-valued mapping, we consider the following class of generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map :

The problem ($\mathcal{P}$) encompasses many models of variational inequality problems. The following problems are the special cases of ($\mathcal{P}$).

(a) If $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to L(E,Y)$ and $H:K\times K\to Y$ are two single-valued mappings, $N(x,y,z)=A(x)$, where $A:L(E,Y)\to L(E,Y)$ and ${\omega}^{\ast}=0$, then the problem ($\mathcal{P}$) reduces to the following generalized vector mixed general quasi-variational-like inequality problem for finding $u\in K$ such that $u\in D(u)$ and for each $v\in D(u)$, there exists $x\in T(u)$ satisfying

The problem (1.2) was studied by Ding and Salahuddin [14]. Some existence results of solutions are established under suitable assumptions without monotonicity and compactness.

(b) If *g* is an identity mapping and ${\omega}^{\ast}=0$, then the problem ($\mathcal{P}$) reduces to the following generalized nonlinear vector quasi-variational-like inequality problem for finding $(u,x,y,z)\in K\times U\times V\times W$ such that $u\in D(u)$ and for each $v\in D(u)$, there exist $x\in T(u)$, $y\in A(u)$ and $z\in M(u)$ satisfying

The problem (1.3) was studied by Husain and Gupta [15].

(c) If $D(u)=K$, then the problem (1.3) reduces to the problem of finding $u\in K$ such that there exist $x\in T(u)$, $y\in A(u)$ and $z\in M(u)$ satisfying

which is introduced and studied by Xiao *et al.* [5]. When $N:L(E,Y)\times L(E,Y)\times L(E,Y)\to L(E,Y)$ and $H:K\times K\to Y$ are two single-valued mappings, the problem (1.4) includes some generalized variational inequality problems investigated in [8, 11, 16–19] as special cases.

(d) If $T(u)=A(u)=\mathrm{\varnothing}$ for all $u\in K$, and *N* is an identity mapping, the problem (1.3) reduces to the problem of finding $u\in K$ such that $u\in D(u)$ and for all $v\in D(u)$,

which is introduced and studied by Peng and Yang [20].

For suitable and appropriate conditions imposed on the mappings *C*, *N*, *H*, *D*, *T*, *A*, *M*, *η* and *g* and by means of the fixed point theorem, we establish some existence results of solutions for the problem ($\mathcal{P}$). It is clear that the problem ($\mathcal{P}$) is the most general and unifying one, which is also one of the main motivations of this paper.

**Definition 1.1** [21]

Let *A* and *B* be two topological vector spaces and let $T:A\to {2}^{B}$ be a multi-valued mapping, then

(i) *T* is said to be upper semicontinuous if for any ${x}_{0}\in A$ and for each open set *U* in *B* containing $T({x}_{0})$, there is a neighborhood *V* of ${x}_{0}$ in *A* such that $T(x)\subset U$ for all $x\in V$.

(ii) *T* is said to have open lower sections if the set ${T}^{-1}(y)=\{x\in A:y\in T(x)\}$ is open in *X* for each $y\in B$.

(iii) *T* is said to be closed if any net $\{{x}_{\alpha}\}$ in *A* such that ${x}_{\alpha}\to x$ and any $\{{y}_{\alpha}\}$ in *B* such that ${y}_{\alpha}\to y$ and ${y}_{\alpha}\in T({x}_{\alpha})$ for any *α*, we have $y\in T(x)$.

(iv) *T* is said to be lower semicontinuous if for any ${x}_{0}\in A$ and for each open set *U* in *B* containing $T({x}_{0})$, there is a neighborhood *V* of ${x}_{0}$ in *A* such that $T(x)\cap U\ne \mathrm{\varnothing}$ for all $x\in V$.

(v) *T* is said to be continuous if it is both lower and upper semicontinuous.

**Lemma 1.2** [22]

*Let* *A* *and* *B* *be two topological spaces*. *Suppose* $T:A\to {2}^{B}$ *and* $H:A\to {2}^{B}$ *are multi*-*valued mappings having open lower sections*, *then*

(i) $G:A\to {2}^{B}$ *defined by*, *for each* $x\in A$, $G(x)=co(T(x))$ *has open lower sections*;

(ii) $\rho :A\to {2}^{B}$ *defined by*, *for each* $x\in A$, $\rho (x)=T(x)\cap H(x)$ *has open lower sections*.

**Lemma 1.3** [23]

*Let* *A* *and* *B* *be two topological spaces*. *If* $T:A\to {2}^{B}$ *is an upper semicontinuous mapping with closed values*, *then* *T* *is closed*.

**Lemma 1.4** [24]

*Let* *A* *and* *B* *be two topological spaces and let* $T:A\to {2}^{B}$ *be an upper semicontinuous mapping with compact values*. *Suppose* $\{{x}_{\alpha}\}$ *is a net in* *A* *such that* ${x}_{\alpha}\to {x}_{0}$. *If* ${y}_{\alpha}\in T({x}_{\alpha})$ *for each* *α*, *then there is a* ${y}_{0}\in T({x}_{0})$ *and a subset* $\{{y}_{\beta}\}$ *of* $\{{y}_{\alpha}\}$ *such that* ${y}_{\beta}\to {y}_{0}$.

Let *I* be an index set, ${E}_{i}$ be a Hausdorff topological vector space for each $i\in I$. Let ${K}_{i}$ be a family of nonempty compact convex subsets in ${E}_{i}$. Let $K={\prod}_{i\in I}{K}_{i}$ and $E={\prod}_{i\in I}{E}_{i}$.

**Lemma 1.5** [8]

*For each* $i\in I$, *let* ${T}_{i}:K\to {2}^{{K}_{i}}$ *be a set*-*valued mapping*. *Assume that the following conditions hold*.

(i) *For each* $i\in I$, ${T}_{i}$ *is a convex set*-*valued mapping*;

(ii) $K=\cup \{int{T}_{i}^{-1}({x}_{i}):{x}_{i}\in {K}_{i}\}$.

*Then there exists* $\overline{x}\in K$ *such that* $\overline{x}\in T(\overline{x})={\prod}_{i\in I}{T}_{i}({\overline{x}}_{i})$, *that is*, ${\overline{x}}_{i}\in {T}_{i}({\overline{x}}_{i})$ *for each* $i\in I$, *where* ${\overline{x}}_{i}$ *is the projection of* $\overline{x}$ *onto* ${K}_{i}$.

## 2 Main results

In this section, we shall derive the solvability for the problem ($\mathcal{P}$) under certain conditions.

First, we give the concept of 0-diagonally convex which is useful for establishing the existence theorem for the problem ($\mathcal{P}$).

**Definition 2.1** Let K be a convex subset of a t.v.s. *E* and *Y* be a t.v.s. Let $C:K\to {2}^{Y}$ be a set-valued mapping and $g:K\to K$ be a single-valued mapping. Then the multi-valued mapping $H:K\times K\to {2}^{Y}$ is said to be 0-diagonally convex with respect to *g* in the second variable if for any finite subset $\{{x}_{1},\dots ,{x}_{n}\}$ of *K* and any $x={\sum}_{i=1}^{n}{\alpha}_{i}{x}_{i}$ with ${\alpha}_{i}\ge 0$ for $i=1,\dots ,n$, and ${\sum}_{i=1}^{n}{\alpha}_{i}=1$,

**Remark 2.2**

(i) If *g* is an identity mapping, then the concept in Definition 2.1 reduces to the corresponding concept of 0-diagonal convexity in [25].

(ii) If $H:K\times K\to Y$ is a single-valued mapping, then the concept in Definition 2.1 reduces to the corresponding concept of 0-diagonally convex with respect to *g* in the second variable in [14].

**Theorem 2.3** *Let* *Y* *be a l*.*c*.*s*., *K* *be a nonempty convex subset of a Hausdorff t*.*v*.*s*. *E*, *X* *be a nonempty compact convex subset of* $L(E,Y)$, *which is equipped with a* *σ*-*topology*. *Let* $g:K\to K$, ${\omega}^{\ast}\in L(E,Y)$ *and* $T,A,M:K\to {2}^{X}$ *be upper semicontinuous set*-*valued mappings with nonempty compact values*. *Assume that the following conditions are satisfied*:

(i) $D:K\to {2}^{K}$ *is a nonempty convex set*-*valued mapping and has open lower sections*;

(ii) *for each* $v\in K$, *the mapping*

*is an upper semicontinuous set*-*valued mapping with compact values*;

(iii) $C:K\to {2}^{Y}$ *is a convex set*-*valued mapping with* $intC(u)\ne \mathrm{\varnothing}$ *for all* $u\in K$;

(iv) $\eta :K\times K\to E$ *is affine in the first argument and for all* $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to {2}^{Y}$ *is generalized vector* 0-*diagonally convex with respect to* *g*;

(vi) $g:K\to K$ *is continuous*;

(vii) *for each* $u\in K$, *the set* $\{u\in K:co\mathrm{\Lambda}(u)\cap D(u)\ne \mathrm{\varnothing}\}$ *is closed in* *K*, *where* $\mathrm{\Lambda}(u)$ *is defined as*

*Then the problem* ($\mathcal{P}$) *admits at least one solution*.

*Proof* Let ${\omega}^{\ast}\in L(E,Y)$. Define a set-valued mapping $Q:K\to {2}^{K}$ by

for all $u\in K$. We first prove that $u\notin coQ(u)$ for all $u\in K$. To see this, suppose, by the method of contradiction, that there exists some point $\overline{u}\in K$ such that $\overline{u}\in coQ(\overline{u})$. Then there exists a finite subset $\{{v}_{1},{v}_{2},\dots ,{v}_{n}\}\subset Q(\overline{u})$, for $\overline{u}\in co\{{v}_{1},{v}_{2},\dots ,{v}_{n}\}$, such that

Since $intC(\overline{u})$ is a convex set and *η* is affine in the first argument, for $i=1,2,\dots ,n$, ${\alpha}_{i}\ge 0$ with ${\sum}_{i=1}^{n}{\alpha}_{i}=1$, $\overline{u}={\sum}_{i=1}^{n}{\alpha}_{i}{v}_{i}$, we have

Since $\eta (u,g(u))=0$, for all $u\in K$, we have

which contradicts the condition (v), so that $u\notin coQ(u)$ for all $u\in K$.

We now prove that

is open for all $v\in K$, that is, *Q* has open lower sections.

Consider a set-valued mapping $J:K\to {2}^{K}$ is defined by

We only need to prove that $J(v)$ is closed for all $v\in K$. Let $\{{u}_{\alpha}\}$ be a net in $J(v)$ such that

Since *g* is continuous, we have

Then there exist ${x}_{\alpha}\in T({u}_{\alpha})$, ${y}_{\alpha}\in A({u}_{\alpha})$ and ${z}_{\alpha}\in M({u}_{\alpha})$ such that

Since *T*, *A*, *M* are upper semicontinuous set-valued mappings with compact values, by Lemma 1.4, $\{{x}_{\alpha}\}$, $\{{y}_{\alpha}\}$, $\{{z}_{\alpha}\}$ have convergent subnets with limits, say ${x}^{\ast}$, ${y}^{\ast}$, ${z}^{\ast}$ and ${x}^{\ast}\in T({u}^{\ast})$, ${y}^{\ast}\in A({u}^{\ast})$ and ${z}^{\ast}\in M({u}^{\ast})$. Without loss of generality, we may assume that ${x}_{\alpha}\to {x}^{\ast}$, ${y}_{\alpha}\to {y}^{\ast}$ and ${z}_{\alpha}\to {z}^{\ast}$. Suppose that

Since $\u3008N(\cdot ,\cdot ,\cdot )-{\omega}^{\ast},\eta (v,\cdot )\u3009+H(\cdot ,v)$ is upper semicontinuous with compact values, by Lemma 1.4, there exist ${m}^{\ast}\in \u3008N({x}^{\ast},{y}^{\ast},{z}^{\ast})-{\omega}^{\ast},\eta ({v}^{\ast},g({u}^{\ast}))\u3009+H(g({u}^{\ast}),{v}^{\ast})$ and a subnet $\{{m}_{\beta}\}$ of $\{{m}_{\alpha}\}$ such that ${m}_{\beta}\to {m}^{\ast}$. Hence $J(v)$ is closed in *K*. So that ${Q}^{-}(v)$ is open for each $v\in K$. Therefore *Q* has open lower sections.

Consider a set-valued mapping $G:K\times U\times V\times W\to {2}^{K}$ defined by

Since *D* has open lower sections by hypothesis (i), we may apply Lemma 1.2 to assert that the set-valued mapping *G* has also open lower sections. Let

There are two cases to consider. In the case $Z=\mathrm{\varnothing}$, we have

This implies that for each $u\in K$,

On the other hand, by the condition (i), and the fact that *K* is a compact convex subset of *Y*, we can apply Lemma 1.5, in this case that $I=\{1\}$, to assert the existence of a fixed point ${u}^{\ast}\in D({u}^{\ast})$, we have

This implies $\mathrm{\forall}v\in D({u}^{\ast})$, $v\notin Q({u}^{\ast})$. Hence, in this particular case, the assertion of the theorem holds.

We now consider the case $Z\ne \mathrm{\varnothing}$. Define a set-valued mapping $S:K\to {2}^{K}$ by

Then, for each $u\in K$, $S(u)$ is a convex set and for each $t\in K$,

Since ${D}^{-}(t)$, $co{Q}^{-}(t)$ are open in *K* and $K\setminus Z$ is open in *K* by the condition (vii), we have ${S}^{-}(t)$ is open in *K*. This implies that *S* has open lower sections. Therefore, there exists ${u}^{\ast}\in K$ such that ${u}^{\ast}\in S({u}^{\ast})$. Suppose that ${u}^{\ast}\in Z$, then

so that ${u}^{\ast}\in coQ({u}^{\ast})$. This is a contradiction. Hence, ${u}^{\ast}\notin Z$. Therefore,

Thus

This implies

Consequently, the assertion of the theorem holds in this case. The problem ($\mathcal{P}$) admits at least one solution. □

**Corollary 2.4** *Let* *Y* *be a l*.*c*.*s*., *K* *be a nonempty convex subset of a Hausdorff t*.*v*.*s*. *E*, *X* *be a nonempty compact convex subset of* $L(E,Y)$, *which is equipped with a* *σ*-*topology*. *Assume that* *N* *and* *H* *are single*-*valued mappings and* $T,A,M:K\to {2}^{X}$ *are upper semicontinuous set*-*valued mappings with nonempty compact values*. *Let* ${\omega}^{\ast}\in L(E,Y)$ *and* $g:K\to K$. *Assume that the following conditions are satisfied*:

(i) $D:K\to {2}^{K}$ *is a nonempty convex set*-*valued mapping and has open lower sections*;

(ii) *for each* $v\in K$, *the mapping*

*is continuous*;

(iii) $C:K\to {2}^{Y}$ *is a convex set*-*valued mapping with* $intC(u)\ne \mathrm{\varnothing}$ *for all* $u\in K$;

(iv) $\eta :K\times K\to E$ *is affine in the first argument and for all* $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to {2}^{Y}$ *is vector* 0-*diagonally convex with respect to* *g*;

(vi) $g:K\to K$ *is continuous*;

(vii) *for each* $u\in K$, *the set* $\{u\in K:co\mathrm{\Lambda}(u)\cap D(u)\ne \mathrm{\varnothing}\}$ *is closed in* *K*, *where* $\mathrm{\Lambda}(u)$ *is defined as*

(viii) $Y\setminus \{-intC(u)\}$ *is an upper semicontinuous set*-*valued mapping*.

*Then there exists a point* $\overline{u}\in K$ *such that* $\overline{u}\in D(\overline{u})$ *and for each* $v\in D(\overline{u})$, *there exist* $\overline{x}\in T(\overline{u})$, $\overline{y}\in A(\overline{u})$ *and* $\overline{z}\in M(\overline{u})$ *such that*

*Proof* Define a set-valued mapping $Q:K\to {2}^{K}$ by

for all $u\in K$. We now prove that ${Q}^{-}(v)$ is open for each $v\in K$, that is,

is closed in *K*. Let $\{{u}_{t}\}$ be a net in ${({Q}^{-1}(v))}^{c}$ such that

Then there exist ${x}_{t}\in T({u}_{t})$, ${y}_{t}\in A({u}_{t})$ and ${z}_{t}\in M({u}_{t})$ such that

The upper semicontinuity, compact values of *T*, *A*, *M* and Lemma 1.4 imply that there exist convergent subnets $\{{x}_{{t}_{j}}\}$, $\{{y}_{{t}_{j}}\}$ and $\{{z}_{{t}_{j}}\}$ such that

for some ${x}^{\ast}\in T(u),{y}^{\ast}\in A(u)$ and ${z}^{\ast}\in M(u)$. Since $\u3008N(\cdot ,\cdot ,\cdot )-{\omega}^{\ast},\eta (v,\cdot )\u3009+H(\cdot ,v)$ is continuous, we have

From Lemma 1.3 and upper semicontinuity of $Y\setminus (-intC(u))$, we have

and hence ${u}^{\ast}\in {({Q}^{-1}(v))}^{c}$, which gives that ${({Q}^{-1}(v))}^{c}$ is closed. Therefore *Q* has open lower sections. For the remainder of the proof, we can just follow that of Theorem 2.3. This completes the proof. □

**Theorem 2.5** *Let* *Y* *be a l*.*c*.*s*., *K* *be a nonempty convex subset of a Hausdorff t*.*v*.*s*. *E*, *X* *be a nonempty compact convex subset of* $L(E,Y)$, *which is equipped with a* *σ*-*topology*. *Let* ${\omega}^{\ast}\in L(E,Y)$, $g:K\to K$ *and* $T,A,M:K\to {2}^{X}$ *be upper semicontinuous set*-*valued mappings*. *Assume that the following conditions are satisfied*.

(i) $D:K\to {2}^{K}$ *is a nonempty convex set*-*valued mapping and has open lower sections*;

(ii) *for each* $y\in K$, *the mapping*

*is upper semicontinuous*;

(iii) $C:K\to {2}^{Y}$ *is a convex set*-*valued mapping with* $intC(u)\ne \mathrm{\varnothing}$ *for all* $u\in K$;

(iv) $\eta :K\times K\to E$ *is affine in the first argument and for all* $x\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to {2}^{Y}$ *is generalized vector* 0-*diagonally convex with respect to* *g*;

(vi) $g:K\to K$ *is continuous*;

(vii) *For each* $u\in K$, *the set* $\{u\in K:co\mathrm{\Lambda}(u)\cap D(u)\ne \mathrm{\varnothing}\}$ *is closed in* *K*, *where* $\mathrm{\Lambda}(u)$ *is defined as*

(viii) *for a given* $u\in K$, *and a neighborhood* *O* *of* *u*, *for all* $t\in O$, $intC(u)=intC(t)$.

*Then the problem* ($\mathcal{P}$) *admits at least one solution*.

*Proof* Define a set-valued mapping $Q:K\to {2}^{K}$ by

for all $u\in K$. We now prove that for each $v\in K$,

is open. That is, *Q* has open lower sections in *K*. Indeed, let $\overline{u}\in {Q}^{-}(v)$, that is,

Since $\u3008N(\cdot ,\cdot ,\cdot )-{\omega}^{\ast},\eta (y,g(\cdot ))\u3009+H(g(\cdot ),y)$ is upper semicontinuous, there exists a neighborhood *O* of $\overline{u}$ such that

By (vii),

Hence, $O\subset {Q}^{-}(v)$. This implies ${Q}^{-}(v)$ is open for each $v\in K$, and so *Q* has open lower sections. For the remainder of the proof, we can just follow that of Theorem 2.3. This completes the proof. □

**Corollary 2.6** *Let* *Y* *be a l*.*c*.*s*., *K* *be a nonempty convex subset of a Hausdorff t*.*v*.*s*. *E*, *X* *be a nonempty compact convex subset of* $L(E,Y)$, *which is equipped with a* *σ*-*topology*. *Let* ${\omega}^{\ast}\in L(E,Y)$, $g:K\to K$ *and* $T,A,M:K\to {2}^{X}$ *be upper semicontinuous set*-*valued mappings*. *Assume that the following conditions are satisfied*.

(i) $D:K\to {2}^{K}$ *is a nonempty convex set*-*valued mapping and has open lower sections*;

(ii) *for each* $y\in K$, *the mapping*

*is upper semicontinuous*;

(iii) $C:K\to {2}^{Y}$ *is a convex set*-*valued mapping such that for each* $u\in K$, $C(u)=C$ *is a convex cone with* $intC(u)\ne \mathrm{\varnothing}$ *for all* $u\in K$;

(iv) $\eta :K\times K\to E$ *is affine in the first argument and for all* $u\in K$, $\eta (u,g(u))=0$;

(v) $H:K\times K\to {2}^{Y}$ *is generalized vector* 0-*diagonally convex with respect to* *g*;

(vi) $g:K\to K$ *is continuous*;

(vii) *for each* $u\in K$, *the set* $\{u\in K:co\mathrm{\Lambda}(u)\cap D(u)\ne \mathrm{\varnothing}\}$ *is closed in* *K*, *where* $\mathrm{\Lambda}(u)$ *is defined as*

*Then the problem* ($\mathcal{P}$) *admits at least one solution*.

*Proof* By hypothesis (iii), the condition (vii) in Theorem 2.5 is satisfied. Hence, all the conditions in Theorem 2.5 are satisfied. □

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## Acknowledgements

The authors were partially supported by the Thailand Research Fund and Naresuan university.

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### Keywords

- generalized nonlinear vector mixed quasi-variational-like inequality
- multi-valued map
- fixed point theorem
- open lower section
- 0-diagonally convex
- locally convex topological vector space