Open Access

On parametric implicit vector variational inequality problems

Journal of Inequalities and Applications20142014:151

https://doi.org/10.1186/1029-242X-2014-151

Received: 13 August 2013

Accepted: 28 March 2014

Published: 2 May 2014

Abstract

Recently Huang et al. (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.

Keywords

topological vector space equilibrium problem KKM mapping

1 Introduction and preliminaries

Equilibrium problems have been extensively studied in recent years, the origin ofwhich can be traced back to Takahashi [[1], Lemma 1], Blum and Oettli [2], and Noor and Oettli [3]. It is well known that vector equilibrium problems provide a unifiedmodel for several classes of problems, for example, vector variational inequalityproblems, vector complementarity problems, vector optimization problems, and vectorsaddle point problems; see [24] and the references therein. In 2003, Huang et al.[5] considered the implicit vector equilibrium problem (for short IVEP) whichconsists of finding x E such that
f ( g ( x ) , y ) int C ( x ) , y E ,
where f : E × E Y and g : E E , are mappings, X and Y are twoHausdorff topological vector spaces, E is a nonempty closed convex subsetof X and C : E 2 Y be a set-valued mapping such that for any x E , C ( x ) is a closed and convex cone with C ( x ) C ( x ) = { 0 } , that is pointed, with nonempty interior. Theycontinued their research and introduced the parametric implicit vector equilibriumproblem, which consists of finding x K ( λ ) , for each given ( λ , ϵ ) Λ 1 × Λ 2 such that
f ( ϵ , g ( x ) , y ) int C ( x ) , y K ( λ ) ,

where Λ i ( i = 1 , 2 ) are Hausdorff topological vector spaces (theparametric spaces), K : Λ 1 2 X a set-valued mapping such that for any λ Λ 1 , K ( λ ) is a nonempty, closed and convex subset of Xwith K ( Λ 1 ) = λ Λ 1 K ( λ ) E and f : Λ 1 × Λ 2 × E Y . They obtained some existence results for a solutionof PIVEP and further they studied upper and lower semi-continuity of the solution ofPIVEP in locally convex Hausdorff topological vector spaces. This paper is motivatedand inspired by the recent paper [5] and its aim is to extend the results to the setting of Hausdorfftopological vector spaces with mild assumptions and removing the condition of beinglocally non-positive at a point has been applied in Proposition 3.3 of [6] and lower semi-continuity of the parametric mapping used inTheorem 3.2 of [6]. More precisely, we first establish an existence result for a solution ofIVEP and then by using it we will deal with the behavior of the solution set ofPIVEP when the parameters ( λ , ϵ ) start to change. In fact we will show that thesolution set as a mapping S : Λ 1 × Λ 2 2 X is upper semi-continuous and lower semi-continuousunder special conditions. In the rest of this section we recall some definitions andresults that we need in the next section.

A subset P of Y is called a pointed and convex cone if and only if P + P P , t P P , for all t 0 , and P P = { 0 } . The domain of a set-valued mapping W : X 2 Y is defined as D ( W ) = { x X : W ( x ) } and its graph is defined as
Graph ( W ) = { ( x , z ) X × Y : z W ( x ) } .

Also W is said to be closed if its graph, that is, Graph ( W ) , is a closed subset of X × Y . A set-valued mapping T : X 2 Y is called upper semi-continuous (u.s.c.) at x X if for every open set V containing T ( x ) there exists an open set U containingx such that T ( u ) V , for all u U . The mapping T is said to be lowersemi-continuous (l.s.c.) if for every open set V with T ( x ) V there exists an open set U containingx such that T ( u ) V . The mapping T is continuous at xif it is both u.s.c. and l.s.c. at x. Moreover, T is u.s.c.(l.s.c.) on X if T is u.s.c. (l.s.c.) at each point ofX.

We need the following lemma in the sequel.

Lemma 1.1 ([7])

Let X and Y be topological spaces and T : X 2 Y be a mapping. The following statements are true:
  1. (i)

    If for any x X , T ( x ) is compact, then T is u.s.c. at x X if and only if for any net { x i } X such that x i x and for every y i T ( x i ) , there exist y T ( x ) and a subnet { y j } of { y i } such that y j y .

     
  2. (ii)

    T is l.s.c. at x X if and only if for any net { x i } X with x i x and for any y T ( x ) , there exists a net { y i } such that y i T ( x i ) and y i y .

     

Definition 1.2 ([8, 9])

Let X be a topological vector space. A mapping F : K X 2 X is said to be a KKM mapping, if, for any finite set A K ,
co A F ( A ) = x A F ( x ) ,

where coA denotes the convex hull of A.

The following lemma plays a crucial rule in this paper.

Lemma 1.3 ([8])

Let K be a nonempty subset of a topological vector space X and F : K 2 X be a K K M mapping with closed values in K. Assume that there exists a nonempty compact convex subset B of K such that x B F ( x ) is compact. Then x K F ( x ) .

2 Main results

The next result provides an existence result for a solution of IVEP.

Theorem 2.1 Let K be closed convex subset of a t.v.s. X and f : K × K Y and g : K K be two mappings. If the following assumptions are satisfied:
  1. (a)

    f ( g ( x ) , x ) int C ( x ) , x K ,

     
  2. (b)

    the mapping x f ( g ( x ) , y ) is continuous, for all y K ,

     
  3. (c)

    for each x K , the set { y K : f ( g ( x ) , y ) int C ( x ) } is convex,

     
  4. (d)

    the mapping W : K 2 Y defined by W ( x ) = Y ( int C ( x ) ) is closed,

     
  5. (e)

    there exist subsets M and N of K, compact convex and compact, respectively, such that for all x K N there is y M such that f ( g ( x ) , y ) int C ( x ) ,

     

then the solution set of IVEP is nonempty and compact.

Proof Define the set-valued mapping F : K 2 K by
F ( y ) = { x K : f ( g ( x ) , y ) int C ( x ) } .
We show that F satisfies all the assumptions of Lemma 1.3. By (b) and(d), F ( y ) is a closed subset of K for all y K . It follows from (c) and (a) that F is a KKMmapping. Indeed, on the contrary of the assertion if there exist y 1 , y 2 , , y n in K and z = i = 1 n λ i y i co { y 1 , y 2 , , y n } i = 1 n F ( y i ) , then f ( g ( z ) , y i ) int C ( z ) and so by (c) we deduce that
f ( g ( z ) , z ) int C ( z ) ,
which is a contradiction (by (a)). Then F is a KKM mapping. Also, it isobvious from (e) that y M F ( y ) N and so y M F ( y ) is compact (note that F ( y ) is closed for each y Y and M is compact). Hence by Lemma 1.3there exists x ¯ K such that
x ¯ x K F ( x )

and it is easy to see that the solution set of IVEP is equal to the set x K F ( x ) and hence x ¯ is a solution of IVEP and further it is compact (note x K F ( x ) x M F ( x ) N ) and hence the proof is complete. □

We note that if g is continuous and f is continuous with respect tothe first variable then the mapping x f ( g ( x ) , y ) is continuous and so condition (b) holds while thesimple example g ( x ) = 1 if x is rational, and g ( x ) = 0 if x is irrational, and f ( x , y ) = 1 , for x rational, and f ( x , y ) = 0 if x is irrational, shows that it is easy tocheck that the mapping x f ( g ( x ) , y ) is continuous; nevertheless, neither g norf is continuous, which shows that the converse does not hold ingeneral. Moreover, in the example, if we take K = [ 0 , 1 ] then f and g satisfy all theassumptions of Theorem 2.1 and so the solution set of IVEP is nonempty andcompact but the example cannot fulfill all the conditions of Proposition 3.1 in [6]. Hence Theorem 2.1 extends Proposition 3.1 in [6]. Also one can easily see the C-convexity of f at thesecond variable, that is, for each x K ,
t f ( x , y 1 ) + ( 1 t ) f ( x , y 2 ) f ( x , t y 1 + ( 1 t ) y 2 ) C ( x ) , t [ 0 , 1 ] ,

which implies condition (c) of Theorem 2.1, while if we take X = and let K be any nonempty convex and compactsubset of X and define f ( x , y ) = y 2 , for all x , y K and we let g be an arbitrary mapping, wetake the example f ( x , y ) = y 2 , for x , y K and g an arbitrary mapping, then thisexample fulfills condition (c) (note that it satisfies all the assumptions ofTheorem 2.1) but f is not convex at the second variable and hencecondition (c) improves condition (3) in Proposition 3.1 of [6].

Definition 2.2 ([6])

A mapping f : E × E Y is said to be locally non-positive at x 0 E with respect to a mapping g : E E if there exist a neighborhood V ( x 0 ) of x 0 and a point z 0 E int V ( x 0 ) such that
f ( g ( x ) , z 0 ) C ( x ) , x E V ( x 0 ) ,

where V ( x 0 ) is the boundary of V ( x 0 ) . In the case that g is the identity mapping,the mapping f is called locally non-positive at x 0 E .

The following corollary is an extension of Proposition 3.3 in [6] for topological vector spaces. Furthermore, the condition that fis locally non-positive at x 0 K has been omitted.

Corollary 2.3 Let K be a nonempty closed and convex subset of a Hausdorff topological vectorspace X and let f : K × K Y and C : K Y be two mappings such that:
  1. (a)

    f ( g ( x ) , x ) = 0 , x K ,

     
  2. (b)

    the mapping x f ( g ( x ) , y ) is continuous, for all y K ,

     
  3. (c)

    for each x K the mapping y f ( g ( x ) , y ) is C ( x ) -convex,

     
  4. (d)

    the mapping W : K 2 Y defined by W ( x ) = Y ( int C ( x ) ) is closed,

     
  5. (e)
    there exist a nonempty compact and convex subset D of K V ( x 0 ) and y 0 D such that for all x ( K V ( x 0 ) ) D
    f ( g ( x ) , y ) int C ( x ) .
     
Then IVEP has a solution in the neighborhood V ( x 0 ) of x 0 , that is, there exists x ( K V ( x 0 ) ) such that
f ( g ( x ) , y ) int C ( x ) , y K .

Moreover, the solution set is a compact subset of K V ( x 0 ) .

Proof There is neighborhood U of x 0 such that co U V ( x 0 ) (see, for example, [10]). Hence by Theorem 2.1, IVEP has a solution on B = K co ( co U { y 0 } ) ¯ . Then there exists x B such that
f ( g ( x ) , y ) int C ( x ) , y B .
We claim that
f ( g ( x ) , y ) int C ( x ) , y K .
Indeed, if the sentence is not true then there is y K so that
f ( g ( x ) , y ) int C ( x ) .
Put y t = x + t ( y x ) , for t > 0 . It is clear that y t B , for t that is small enough. Then bycondition (c) we have
f ( g ( x ) , y t ) ( 1 t ) f ( g ( x ) , y ) + t f ( g ( x ) , x ) int C ( x ) + 0 = int C ( x ) ,

which is a contradiction. Hence x is a solution of IVEP. The second part follows fromcondition (e). This completes the proof. □

The next theorem is an extension of Theorems 3.1, 3.3 and Corollary 3.3 in [6] with mild assumptions for mappings which do not need to satisfy thelocally non-positive condition. In fact this condition has been removed.

Theorem 2.4 Let F : Λ 2 × E × E Y and g : E E be two mappings. If the following assumptions hold:
  1. (i)

    K : Λ 1 2 E is a continuous mapping with nonempty convex compact values;

     
  2. (ii)

    ( ε , x , y ) F ( ε , g ( x ) , y ) is continuous;

     
  3. (iii)

    the set { y : F ( ε , g ( x ) , y ) int C ( x ) } is convex and F ( ε , g ( x ) , x ) = 0 , for each ( ϵ , x ) Λ 1 × E ;

     
  4. (iv)

    the mapping W : E 2 Y defined by W ( x ) = Y int C ( x ) is closed;

     
then
  1. (i)
    for each ( λ , ε ) Λ 1 × Λ 2 , the solution set
    S ( λ , ε ) = { x K ( λ ) : F ( ε , g ( x ) , y ) int C ( x ) , y K ( λ ) } ,
     
is nonempty and compact,
  1. (ii)
    the solution set mapping S : Λ 1 × Λ 2 2 X defined by
    ( λ , ε ) S ( λ , ε )
     

is continuous.

Proof The first part, that is, (i), follows from Theorem 2.1 by taking,for each ( λ , ε ) Λ 1 × Λ 2 , M = N = K ( λ ) and defining f ( x , y ) = F ( ε , x , y ) , for all ( x , y ) K ( λ ) × K ( λ ) . To prove (ii), let { ( λ i , ϵ i ) } i I Λ 1 × Λ 2 be a net with ( λ i , ϵ i ) ( λ , ϵ ) and z i S ( λ i , ϵ i ) K ( λ i ) . Since K is u.s.c., λ i λ and z i K ( λ i ) , using Lemma 1.1(i), there exist z K ( λ ) and a subnet { z i j } of { z i } which converges to z. So
F ( ϵ i j , g ( z i j ) , y ) Y ( int C ( z i j ) ) = W ( z i j ) , y K ( λ i j ) .
(1)
We claim that z S ( λ , ϵ ) (note that if we show the claim then according toLemma 1.1 the mapping S will be an u.s.c.). If the claim is not truethen there is y K ( λ ) such that
F ( ϵ , g ( z ) , y ) int C ( z ) .
(2)
So, since K is l.s.c., there exists net w j K ( λ j ) such that w j y . Then it follows from (1) that
F ( ϵ i j , g ( z i j ) , w j ) W ( z i j ) ,
and so by (ii) and (iv) we get
F ( ϵ , g ( z ) , y ) Y int C ( z ) = W ( z ) ,
which is contradicted by (2) and so z S ( λ , ϵ ) . It follows from x i j S ( λ i j , ϵ i j ) that f ( ϵ i j , g ( x i j ) , ϵ i j ) int C ( x i j ) and so f ( ϵ i j , g ( x i j ) , y i j ) W ( x i j ) and by the closedness of W we get f ( ϵ , g ( x ) , y ) W ( x ) , which is a contradiction, and then the solution setmapping S is u.s.c. Now we show that S is l.s.c. Let { ( λ i , ϵ i ) } i I Λ 1 × Λ 2 be a net with ( λ i , ϵ i ) ( λ , ϵ ) and z an arbitrary element of S ( λ , ϵ ) K ( λ ) . Put Ξ = { V : V  is a neighborhood of  z } (note by the relation V W if and only if V W , the set Ξ is a directed set). Then for each ( V , i ) Ξ × I , there is a closed and convex neighborhood H V , i of z such that H V , i V (see [10]) and so it follows from Theorem 2.1 that there is z i H V , i K ( λ i ) such that
F ( ϵ i , g ( z i ) , y ) int C ( z i ) , y H V , i K ( λ i ) .
(3)
Now if there exists y K ( λ i ) such that
F ( ϵ i , g ( z i ) , y ) int C ( z i ) ,
it follows from F ( ϵ i , g ( z i ) , z i ) = 0 (see condition (i)) and (iii) that
F ( ϵ i , g ( z i ) , z i + t ( y z i ) ) int C ( z i ) , t [ 0 , 1 ] ,
which is a contradiction, for t [ 0 , 1 ] small enough, by (3) (note H V , i is an open set and z i H V , i ). So
F ( ϵ i , g ( z i ) , y ) int C ( z i ) , y K ( λ i ) ,

and hence z i S ( ϵ i , λ i ) . Consequently, for each point ( V , i ) Ξ × I there is z i S ( ϵ i , λ i ) , and so z i z . Hence it follows from Lemma 1.1(ii) thatS is l.s.c. and the proof is completed. □

Inspired by the proof of the second part of the previous theorem we can deduce thelower semi-continuity of the solution set mapping. Indeed the next theorem is animprovement of Theorem 3.2 in [6] without using the lower semi-continuity of the mapping K : Λ 1 2 E ; its proof is similar to the proof presented for thesecond part of Theorem 2.4 and so we omit the proof.

Theorem 2.5 Let F : Λ 2 × E × E Y and g : E E be two mappings. For a given ( λ 0 , ϵ 0 ) ( Λ 1 , Λ 2 ) there exist neighborhoods U ( λ 0 ) of λ 0 and M ( ϵ 0 ) of ϵ 0 such that the following assumptions are satisfied:
  1. (i)

    K : U ( λ 0 ) 2 X is a mapping with nonempty convex and compact values;

     
  2. (ii)

    mapping ( ε , x , y ) F ( ε , g ( x ) , y ) is continuous, for each ( ε , x , y ) M ( ϵ 0 ) × E × E ;

     
  3. (iii)

    for each x E the mapping y F ( ε , g ( x ) , y ) is C ( x ) -convex and F ( ε , g ( x ) , x ) = 0 , for each ( ϵ , x ) M ( ϵ 0 ) × E ;

     
  4. (iv)

    the mapping W : E 2 Y defined by W ( x ) = Y int C ( x ) is closed.

     
Then
  1. (i)

    S ( λ , ε ) , for each ( λ , ε ) U ( λ 0 ) × M ( ε 0 ) is nonempty and compact;

     
  2. (ii)

    the mapping S is l.s.c. at ( λ 0 , ε 0 ) .

     

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Razi University
(2)
Department of Mathematics, Faculty of Science, Naresuan University

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Copyright

© Farajzadeh and Plubteing; licensee Springer. 2014

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