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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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Wangkeeree, R., Yimmuang, P. Generalized nonlinear vector mixed quasi-variational-like inequality governed by a multi-valued map.
*J Inequal Appl* **2013**, 294 (2013). https://doi.org/10.1186/1029-242X-2013-294

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DOI: https://doi.org/10.1186/1029-242X-2013-294

### 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