Open Access

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

Journal of Inequalities and Applications20132013:294

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

Received: 9 October 2012

Accepted: 3 May 2013

Published: 17 June 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.

Keywords

generalized nonlinear vector mixed quasi-variational-like inequalitymulti-valued mapfixed point theoremopen lower section0-diagonally convexlocally convex topological vector space

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, [14] 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 l , x the evaluation of l L ( E , Y ) at x E . Let X 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 , : L ( K , Y ) × K Y is continuous. Let intA and co ( A ) represent the interior and convex hull of a set A, respectively. Let C : K 2 Y be a set-valued mapping such that int C ( x ) for each x K , let η : K × K E be a vector-valued mapping.

Let N : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) 2 L ( E , Y ) be a set-valued mapping, H : K × K 2 Y , D : K 2 K and T , A , M : K 2 X be set-valued mappings. For each ω L ( E , Y ) and g : K 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 :
( P ) { find  u K  such that  u D ( u )  and for each  v D ( u ) , there exist  x T ( u ) , y A ( u )  and  z M ( u )  satisfying N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) .
(1.1)

The problem ( P ) encompasses many models of variational inequality problems. The following problems are the special cases of ( P ).

(a) If N : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) L ( E , Y ) and H : K × K Y are two single-valued mappings, N ( x , y , z ) = A ( x ) , where A : L ( E , Y ) L ( E , Y ) and ω = 0 , then the problem ( P ) reduces to the following generalized vector mixed general quasi-variational-like inequality problem for finding u K such that u D ( u ) and for each v D ( u ) , there exists x T ( u ) satisfying
A ( x ) , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) .
(1.2)

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 ω = 0 , then the problem ( P ) reduces to the following generalized nonlinear vector quasi-variational-like inequality problem for finding ( u , x , y , z ) K × U × V × W such that u D ( u ) and for each v D ( u ) , there exist x T ( u ) , y A ( u ) and z M ( u ) satisfying
N ( x , y , z ) , η ( v , u ) + H ( u , v ) int C ( u ) .
(1.3)

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 K such that there exist x T ( u ) , y A ( u ) and z M ( u ) satisfying
N ( x , y , z ) , η ( v , u ) + H ( u , v ) int C ( u ) , v K ,
(1.4)

which is introduced and studied by Xiao et al. [5]. When N : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) L ( E , Y ) and H : K × K Y are two single-valued mappings, the problem (1.4) includes some generalized variational inequality problems investigated in [8, 11, 1619] as special cases.

(d) If T ( u ) = A ( u ) = for all u K , and N is an identity mapping, the problem (1.3) reduces to the problem of finding u K such that u D ( u ) and for all v D ( u ) ,
T ( u ) , η ( v , u ) + H ( u , v ) int C ( 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 ( P ). It is clear that the problem ( 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 2 B be a multi-valued mapping, then

(i) T is said to be upper semicontinuous if for any x 0 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 ) U for all x V .

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

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

(iv) T is said to be lower semicontinuous if for any x 0 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 ) U for all x 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 2 B and H : A 2 B are multi-valued mappings having open lower sections, then

(i) G : A 2 B defined by, for each x A , G ( x ) = co ( T ( x ) ) has open lower sections;

(ii) ρ : A 2 B defined by, for each x A , ρ ( x ) = T ( x ) H ( x ) has open lower sections.

Lemma 1.3 [23]

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

Let I be an index set, E i be a Hausdorff topological vector space for each i I . Let K i be a family of nonempty compact convex subsets in E i . Let K = i I K i and E = i I E i .

Lemma 1.5 [8]

For each i I , let T i : K 2 K i be a set-valued mapping. Assume that the following conditions hold.

(i) For each i I , T i is a convex set-valued mapping;

(ii) K = { int T i 1 ( x i ) : x i K i } .

Then there exists x ¯ K such that x ¯ T ( x ¯ ) = i I T i ( x ¯ i ) , that is, x ¯ i T i ( x ¯ i ) for each i I , where x ¯ i is the projection of x ¯ onto K i .

2 Main results

In this section, we shall derive the solvability for the problem ( P ) under certain conditions.

First, we give the concept of 0-diagonally convex which is useful for establishing the existence theorem for the problem ( 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 2 Y be a set-valued mapping and g : K K be a single-valued mapping. Then the multi-valued mapping H : K × K 2 Y is said to be 0-diagonally convex with respect to g in the second variable if for any finite subset { x 1 , , x n } of K and any x = i = 1 n α i x i with α i 0 for i = 1 , , n , and i = 1 n α i = 1 ,
i = 1 n α i H ( g ( x ) , x i ) int C ( x ) .

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 × K 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 K , ω L ( E , Y ) and T , A , M : K 2 X be upper semicontinuous set-valued mappings with nonempty compact values. Assume that the following conditions are satisfied:

(i) D : K 2 K is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each v K , the mapping
N ( , , ) ω , η ( v , ) + H ( , v ) : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) × K × K 2 Y

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

(iii) C : K 2 Y is a convex set-valued mapping with int C ( u ) for all u K ;

(iv) η : K × K E is affine in the first argument and for all u K , η ( u , g ( u ) ) = 0 ;

(v) H : K × K 2 Y is generalized vector 0-diagonally convex with respect to g;

(vi) g : K K is continuous;

(vii) for each u K , the set { u K : co Λ ( u ) D ( u ) } is closed in K, where Λ ( u ) is defined as
Λ ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) } .

Then the problem ( P ) admits at least one solution.

Proof Let ω L ( E , Y ) . Define a set-valued mapping Q : K 2 K by
Q ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) }
for all u K . We first prove that u co Q ( u ) for all u K . To see this, suppose, by the method of contradiction, that there exists some point u ¯ K such that u ¯ co Q ( u ¯ ) . Then there exists a finite subset { v 1 , v 2 , , v n } Q ( u ¯ ) , for u ¯ co { v 1 , v 2 , , v n } , such that
N ( x ¯ , y ¯ , z ¯ ) ω , η ( v i , g ( u ¯ ) ) + H ( g ( u ¯ ) , v i ) int C ( u ¯ ) , i = 1 , 2 , , n .
Since int C ( u ¯ ) is a convex set and η is affine in the first argument, for i = 1 , 2 , , n , α i 0 with i = 1 n α i = 1 , u ¯ = i = 1 n α i v i , we have
N ( x ¯ , y ¯ , z ¯ ) ω , η ( i = 1 n α i v i , g ( u ¯ ) ) + i = 1 n α i H ( g ( u ¯ ) , v i ) int C ( u ¯ ) .
Since η ( u , g ( u ) ) = 0 , for all u K , we have
i = 1 n α i H ( g ( u ¯ ) , v i ) int C ( u ¯ ) ,

which contradicts the condition (v), so that u co Q ( u ) for all u K .

We now prove that
Q ( v ) = { u K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) }

is open for all v K , that is, Q has open lower sections.

Consider a set-valued mapping J : K 2 K is defined by
J ( v ) = { u K : x T ( u ) , y A ( u ) , z M ( u )  such that N ( x , y , z , ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) } .
We only need to prove that J ( v ) is closed for all v K . Let { u α } be a net in J ( v ) such that
u α u .
Since g is continuous, we have
g ( u α ) g ( u ) .
Then there exist x α T ( u α ) , y α A ( u α ) and z α M ( u α ) such that
N ( x α , y α , z α , ) ω , η ( v α , g ( u α ) ) + H ( g ( u α ) , v α ) int C ( u α ) .
Since T, A, M are upper semicontinuous set-valued mappings with compact values, by Lemma 1.4, { x α } , { y α } , { z α } have convergent subnets with limits, say x , y , z and x T ( u ) , y A ( u ) and z M ( u ) . Without loss of generality, we may assume that x α x , y α y and z α z . Suppose that
m α { N ( x α , y α , z α , ) ω , η ( v α , g ( u α ) ) + H ( g ( u α ) , v α ) int C ( u α ) } .

Since N ( , , ) ω , η ( v , ) + H ( , v ) is upper semicontinuous with compact values, by Lemma 1.4, there exist m N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) and a subnet { m β } of { m α } such that m β m . Hence J ( v ) is closed in K. So that Q ( v ) is open for each v K . Therefore Q has open lower sections.

Consider a set-valued mapping G : K × U × V × W 2 K defined by
G ( u ) = co Q ( u ) D ( u ) , u K .
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
Z = { u K : G ( u ) } .
There are two cases to consider. In the case Z = , we have
co Q ( u ) D ( u ) = for each  u K .
This implies that for each u K ,
Q ( u ) D ( u ) = .
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 D ( u ) , we have
Q ( u ) D ( u ) = .

This implies v D ( u ) , v Q ( u ) . Hence, in this particular case, the assertion of the theorem holds.

We now consider the case Z . Define a set-valued mapping S : K 2 K by
S ( u ) = { G ( u ) , u Z ; D ( u ) , u K Z .
Then, for each u K , S ( u ) is a convex set and for each t K ,
S ( t ) = G ( t ) ( ( K Z ) ( D ( t ) ) ) .
Since D ( t ) , co Q ( t ) are open in K and K 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 K such that u S ( u ) . Suppose that u Z , then
u co Q ( u ) D ( u ) ,
so that u co Q ( u ) . This is a contradiction. Hence, u Z . Therefore,
u D ( u ) and G ( u ) = .
Thus
u D ( u ) and co Q ( u ) D ( u ) = .
This implies
Q ( u ) D ( u ) = .

Consequently, the assertion of the theorem holds in this case. The problem ( 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 2 X are upper semicontinuous set-valued mappings with nonempty compact values. Let ω L ( E , Y ) and g : K K . Assume that the following conditions are satisfied:

(i) D : K 2 K is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each v K , the mapping
N ( , , ) ω , η ( v , ) + H ( , v ) : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) × K × K 2 Y

is continuous;

(iii) C : K 2 Y is a convex set-valued mapping with int C ( u ) for all u K ;

(iv) η : K × K E is affine in the first argument and for all u K , η ( u , g ( u ) ) = 0 ;

(v) H : K × K 2 Y is vector 0-diagonally convex with respect to g;

(vi) g : K K is continuous;

(vii) for each u K , the set { u K : co Λ ( u ) D ( u ) } is closed in K, where Λ ( u ) is defined as
Λ ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) } ;

(viii) Y { int C ( u ) } is an upper semicontinuous set-valued mapping.

Then there exists a point u ¯ K such that u ¯ D ( u ¯ ) and for each v D ( u ¯ ) , there exist x ¯ T ( u ¯ ) , y ¯ A ( u ¯ ) and z ¯ M ( u ¯ ) such that
N ( x ¯ , y ¯ , z ¯ ) ω , η ( v , g ( u ¯ ) ) + H ( g ( u ¯ ) , v ) int C ( u ¯ ) .
Proof Define a set-valued mapping Q : K 2 K by
Q ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) }
for all u K . We now prove that Q ( v ) is open for each v K , that is,
( Q 1 ( v ) ) c = { u K : x T ( u ) , y A ( u ) , z M ( u )  such that N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) Y { int C ( u ) } }
is closed in K. Let { u t } be a net in ( Q 1 ( v ) ) c such that
g ( u t ) g ( u ) K .
Then there exist x t T ( u t ) , y t A ( u t ) and z t M ( u t ) such that
N ( x t , y t , z t ) ω , η ( v , g ( u t ) ) + H ( g ( u t ) , v ) Y { int C ( u t ) } .
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
x t j x , y t j y and z t j z
for some x T ( u ) , y A ( u ) and z M ( u ) . Since N ( , , ) ω , η ( v , ) + H ( , v ) is continuous, we have
N ( x t j , y t j , z t j ) ω , η ( v , g ( u t j ) ) + H ( g ( u t j ) , v ) N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) .
From Lemma 1.3 and upper semicontinuity of Y ( int C ( u ) ) , we have
N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) Y ( int C ( u ) ) ,

and hence u ( 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 ω L ( E , Y ) , g : K K and T , A , M : K 2 X be upper semicontinuous set-valued mappings. Assume that the following conditions are satisfied.

(i) D : K 2 K is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each y K , the mapping
N ( , , ) ω , η ( v , ) + H ( , v ) : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) × K × K 2 Y

is upper semicontinuous;

(iii) C : K 2 Y is a convex set-valued mapping with int C ( u ) for all u K ;

(iv) η : K × K E is affine in the first argument and for all x K , η ( u , g ( u ) ) = 0 ;

(v) H : K × K 2 Y is generalized vector 0-diagonally convex with respect to g;

(vi) g : K K is continuous;

(vii) For each u K , the set { u K : co Λ ( u ) D ( u ) } is closed in K, where Λ ( u ) is defined as
Λ ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) } ;

(viii) for a given u K , and a neighborhood O of u, for all t O , int C ( u ) = int C ( t ) .

Then the problem ( P ) admits at least one solution.

Proof Define a set-valued mapping Q : K 2 K by
Q ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) }
for all u K . We now prove that for each v K ,
Q 1 ( v ) = { u K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) }
is open. That is, Q has open lower sections in K. Indeed, let u ¯ Q ( v ) , that is,
N ( x , y , z ) ω , η ( v , g ( u ¯ ) ) + H ( g ( u ¯ ) , v ) int C ( u ¯ ) .
Since N ( , , ) ω , η ( y , g ( ) ) + H ( g ( ) , y ) is upper semicontinuous, there exists a neighborhood O of u ¯ such that
N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , u O .
By (vii),
N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ¯ ) , u O .

Hence, O Q ( v ) . This implies Q ( v ) is open for each v 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 ω L ( E , Y ) , g : K K and T , A , M : K 2 X be upper semicontinuous set-valued mappings. Assume that the following conditions are satisfied.

(i) D : K 2 K is a nonempty convex set-valued mapping and has open lower sections;

(ii) for each y K , the mapping
N ( , , ) ω , η ( v , g ( ) ) + H ( g ( ) , v ) : L ( E , Y ) × L ( E , Y ) × L ( E , Y ) × K × K 2 Y

is upper semicontinuous;

(iii) C : K 2 Y is a convex set-valued mapping such that for each u K , C ( u ) = C is a convex cone with int C ( u ) for all u K ;

(iv) η : K × K E is affine in the first argument and for all u K , η ( u , g ( u ) ) = 0 ;

(v) H : K × K 2 Y is generalized vector 0-diagonally convex with respect to g;

(vi) g : K K is continuous;

(vii) for each u K , the set { u K : co Λ ( u ) D ( u ) } is closed in K, where Λ ( u ) is defined as
Λ ( u ) = { v K : N ( x , y , z ) ω , η ( v , g ( u ) ) + H ( g ( u ) , v ) int C ( u ) , x T ( u ) , y A ( u ) , z M ( u ) } .

Then the problem ( 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. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, Thailand

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