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On parametric implicit vector variational inequality problems

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.

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 xE such that

f ( g ( x ) , y ) intC(x),yE,

where f:E×EY and g:EE, 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 anyxE, C(x) is a closed and convex cone withC(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 ) intC ( x ) ,yK(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 anyλ 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 ×EY. 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 ifP+PP, tPP, for all t0, and PP={0}. The domain of a set-valued mappingW:X 2 Y is defined as D(W)={xX: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.) atxX if for every open set V containingT(x) there exists an open set U containingx such that T(u)V, for all uU. The mapping T is said to be lowersemi-continuous (l.s.c.) if for every open set V withT(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 andT:X 2 Y be a mapping. The following statements are true:

  1. (i)

    If for any xX, T(x) is compact, then T is u.s.c. at xX 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 yT(x) and a subnet { y j } of { y i } such that y j y.

  2. (ii)

    T is l.s.c. at xX if and only if for any net { x i }X with x i x and for any yT(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:KX 2 X is said to be a KKM mapping, if, for any finite setAK,

coAF(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 andF:K 2 X be aKKMmapping 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 andf:K×KYandg:KKbe two mappings. If the following assumptions are satisfied:

  1. (a)

    f(g(x),x)intC(x), xK,

  2. (b)

    the mapping xf(g(x),y) is continuous, for all yK,

  3. (c)

    for each xK, the set {yK:f(g(x),y)intC(x)} is convex,

  4. (d)

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

  5. (e)

    there exist subsets M and N of K, compact convex and compact, respectively, such that for all xKN there is yM such that f(g(x),y)intC(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 allyK. 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 )intC(z) and so by (c) we deduce that

f ( g ( z ) , z ) intC(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 yY 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 xf(g(x),y) is continuous and so condition (b) holds while thesimple example g(x)=1 if x is rational, andg(x)=0 if x is irrational, andf(x,y)=1, for x rational, and f(x,y)=0 if x is irrational, shows that it is easy tocheck that the mapping xf(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 xK,

tf(x, y 1 )+(1t)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 takeX= and let K be any nonempty convex and compactsubset of X and define f(x,y)= y 2 , for all x,yK and we let g be an arbitrary mapping, wetake the example f(x,y)= y 2 , for x,yK 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×EY is said to be locally non-positive at x 0 E with respect to a mapping g:EE if there exist a neighborhood V( x 0 ) of x 0 and a point z 0 EintV( x 0 ) such that

f ( g ( x ) , z 0 ) C(x),xEV( 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 letf:K×KYandC:KYbe two mappings such that:

  1. (a)

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

  2. (b)

    the mapping xf(g(x),y) is continuous, for all yK,

  3. (c)

    for each xK the mapping yf(g(x),y) is C(x)-convex,

  4. (d)

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

  5. (e)

    there exist a nonempty compact and convex subset D of KV( x 0 ) and y 0 D such that for all x(KV( x 0 ))D

    f ( g ( x ) , y ) intC(x).

Then IVEP has a solution in the neighborhoodV( x 0 )of x 0 , that is, there exists x (KV( x 0 ))such that

f ( g ( x ) , y ) intC ( x ) ,yK.

Moreover, the solution set is a compact subset ofKV( x 0 ).

Proof There is neighborhood U of x 0 such that coUV( x 0 ) (see, for example, [10]). Hence by Theorem 2.1, IVEP has a solution onB=K co ( co U { y 0 } ) ¯ . Then there exists x B such that

f ( g ( x ) , y ) intC ( x ) ,yB.

We claim that

f ( g ( x ) , y ) intC ( x ) ,yK.

Indeed, if the sentence is not true then there is yK so that

f ( g ( x ) , y ) intC ( 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 ) (1t)f ( g ( x ) , y ) +tf ( g ( x ) , x ) intC ( x ) +0=intC ( 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 LetF: Λ 2 ×E×EYandg:EEbe 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)intC(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)=YintC(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 existzK(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 ),yK( λ i j ).
(1)

We claim that zS(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 yK(λ) such that

F ( ϵ , g ( z ) , y ) intC(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 ) YintC(z)=W(z),

which is contradicted by (2) and so zS(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 )intC( 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 getf(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 ofS(λ,ϵ)K(λ). Put Ξ={V:V is a neighborhood of z} (note by the relation VW if and only if VW, 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 ) intC( z i ),y H V , i K( λ i ).
(3)

Now if there exists yK( λ i ) such that

F ( ϵ i , g ( z i ) , y ) intC( 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 ) ) intC( 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 ) intC( z i ),yK( 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 mappingK: 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 LetF: Λ 2 ×E×EYandg:EEbe two mappings. For a given( λ 0 , ϵ 0 )( Λ 1 , Λ 2 )there exist neighborhoodsU( λ 0 )of λ 0 andM( 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 xE the mapping yF(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)=YintC(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 ).

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This work presented have was carried out in collaboration between all authors. Themain idea of this paper was proposed by AF and SP prepared the manuscript initiallyand performed all the steps of the proofs in this research. All authors read andapproved the final manuscript.

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Farajzadeh, A., Plubteing, S. On parametric implicit vector variational inequality problems. J Inequal Appl 2014, 151 (2014). https://doi.org/10.1186/1029-242X-2014-151

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