On parametric implicit vector variational inequality problems

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


Introduction and preliminaries
Equilibrium problems have been extensively studied in recent years, the origin of which can be traced back to Takahashi [, Lemma ], Blum and Oettli [], and Noor and Oettli []. It is well known that vector equilibrium problems provide a unified model for several classes of problems, for example, vector variational inequality problems, vector complementarity problems, vector optimization problems, and vector saddle point problems; see [-] and the references therein. In , Huang et al. [] considered the implicit vector equilibrium problem (for short IVEP) which consists 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 two Hausdorff topological vector spaces, E is a nonempty closed convex subset of X and C : E →  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) = {}, that is pointed, with nonempty interior. They continued their research and introduced the parametric implicit vector equilibrium problem, which consists of finding x * ∈ K(λ), for each given (λ, ) ∈  ×  such that f , g x * , y / ∈int C x * , ∀y ∈ K(λ), where i (i = , ) are Hausdorff topological vector spaces (the parametric spaces), K :  →  X a set-valued mapping such that for any λ ∈  , K(λ) is a nonempty, closed and convex subset of X with K(  ) = λ∈  K(λ) ⊆ E and f :  ×  × E → Y . They obtained some existence results for a solution of PIVEP and further they studied upper and ©2014 Farajzadeh and Plubteing; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. http://www.journalofinequalitiesandapplications.com/content/2014/1/151 lower semi-continuity of the solution of PIVEP in locally convex Hausdorff topological vector spaces. This paper is motivated and inspired by the recent paper [] and its aim is to extend the results to the setting of Hausdorff topological vector spaces with mild assumptions and removing the condition of being locally non-positive at a point has been applied in Proposition . of [] and lower semi-continuity of the parametric mapping used in Theorem . of []. More precisely, we first establish an existence result for a solution of IVEP and then by using it we will deal with the behavior of the solution set of PIVEP when the parameters (λ, ) start to change. In fact we will show that the solution set as a mapping S :  ×  →  X is upper semi-continuous and lower semi-continuous under special conditions. In the rest of this section we recall some definitions and results 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, tP ⊆ P, for all t ≥ , and P ∩ -P = {}. The domain of a set-valued mapping W : X →  Y is defined as D(W ) = {x ∈ X : W (x) = ∅} and its graph is defined as Also W is said to be closed if its graph, that is, The mapping T is continuous at x if it is both u.s.c. and l.s.c. at x. Moreover, T is u.s.c.
We need the following lemma in the sequel.

Lemma . ([]
) Let X and Y be topological spaces and T : X →  Y be a mapping. The following statements are true: where co A denotes the convex hull of A.
The following lemma plays a crucial rule in this paper.

Main results
The next result provides an existence result for a solution of IVEP.
Theorem . 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: , then the solution set of IVEP is nonempty and compact.
Proof Define the set-valued mapping F : We show that F satisfies all the assumptions of Lemma .. 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 KKM mapping. Indeed, on the contrary of the assertion if there exist y  , y  , . . . , y n in K and z = n i= λ i y i ∈ co{y  , y  , . . . , y n }\ n i= F(y i ), then f (g(z), y i ) ∈int C(z) and so by (c) we deduce that which is a contradiction (by (a)). Then F is a KKM mapping. Also, it is obvious 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 . there exists x ∈ K such that and it is easy to see that the solution set of IVEP is equal to the set x∈K F(x) and hencē 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 to the first variable then the mapping x → f (g(x), y) is continuous and so condition (b) holds while the simple example g(x) =  if x is rational, and g(x) =  if x is irrational, and f (x, y) = , for x rational, and f (x, y) =  if x is irrational, shows that it is easy to check that the mapping x → f (g(x), y) is continuous; nevertheless, neither g nor f is continuous, which shows that the converse does not hold in general. Moreover, in the example, if we take K = [, ] then f and g satisfy all the assumptions of Theorem . and so the solution set of IVEP is nonempty and compact but the example cannot fulfill all the conditions of Proposition . in []. Hence Theorem . extends Proposition . in []. Also one can easily see the C-convexity of f at the second variable, that is, for each x ∈ K , tf (x, y  ) + (t)f (x, y  )f x, ty  + (t)y  ∈ C(x), ∀t ∈ [, ], http://www.journalofinequalitiesandapplications.com/content/2014/1/151 which implies condition (c) of Theorem ., while if we take X = and let K be any nonempty convex and compact subset of X and define f (x, y) = -y  , for all x, y ∈ K and we let g be an arbitrary mapping, we take the example f (x, y) = -y  , for x, y ∈ K and g an arbitrary mapping, then this example fulfills condition (c) (note that it satisfies all the assumptions of Theorem .) but f is not convex at the second variable and hence condition (c) improves condition () in Proposition . of [].
where ∂V (x  ) is the boundary of V (x  ). In the case that g is the identity mapping, the mapping f is called locally non-positive at x  ∈ E.
The following corollary is an extension of Proposition . in [] for topological vector spaces. Furthermore, the condition that f is locally non-positive at x  ∈ K has been omitted.

Corollary . Let K be a nonempty closed and convex subset of a Hausdorff topological vector space X and let f : K × K → Y and C : K → Y be two mappings such that:
( there exist a nonempty compact and convex subset D of K ∩ V (x  ) and y  ∈ D such that for all x ∈ (K ∩ V (x  ))\D f g(x), y ∈int C(x).

Moreover, the solution set is a compact subset of K ∩ V (x  ).
Proof There is neighborhood U of x  such that co U ⊆ V (x  ) (see, for example, []). Hence by Theorem ., IVEP has a solution on B = K ∩ co(co U ∪ {y  }). 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. http://www.journalofinequalitiesandapplications.com/content/2014/1/151 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(yx * ), for t > . It is clear that y t ∈ B, for t that is small enough. Then by condition (c) we have which is a contradiction. Hence x * is a solution of IVEP. The second part follows from condition (e). This completes the proof.
The next theorem is an extension of Theorems ., . and Corollary . in [] with mild assumptions for mappings which do not need to satisfy the locally non-positive condition. In fact this condition has been removed.
Theorem . Let F :  × E × E → Y and g : E → E be two mappings. If the following assumptions hold: (i) K :  →  E is a continuous mapping with nonempty convex compact values; is nonempty and compact, (ii) the solution set mapping S :  ×  →  X defined by is continuous.
Proof The first part, that is, (i), follows from Theorem . by taking, for each (λ, ε) ∈  ×  , 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 , i )} i∈I ⊆  ×  be a net with (λ i , i ) → (λ, ) and z i ∈ S(λ i , i ) ⊂ K(λ i ). Since K is u.s.c., λ i → λ and z i ∈ K(λ i ), using Lemma .(i), there exist z ∈ K(λ) and a subnet {z i j } of {z i } which converges to z. So We claim that z ∈ S(λ, ) (note that if we show the claim then according to Lemma . the mapping S will be an u.s.c.). If the claim is not true then there is y ∈ K(λ) such that F , g(z), y ∈int C(z).
() http://www.journalofinequalitiesandapplications.com/content/2014/1/151 So, since K is l.s.c., there exists net w j ∈ K(λ j ) such that w j → y. Then it follows from () that and so by (ii) and (iv) we get which is contradicted by () and so z ∈ S(λ, ). It follows from Now if there exists y ∈ K(λ i ) such that it follows from F( i , g(z i ), z i ) =  (see condition (i)) and (iii) that which is a contradiction, for t ∈ [, ] small enough, by () (note H V ,i is an open set and z i ∈ H V ,i ). So and hence z i ∈ S( i , λ i ). Consequently, for each point (V , i) ∈ × I there is z i ∈ S( i , λ i ), and so z i → z. Hence it follows from Lemma .(ii) that S is l.s.c. and the proof is completed.
Inspired by the proof of the second part of the previous theorem we can deduce the lower semi-continuity of the solution set mapping. Indeed the next theorem is an improvement of Theorem . in [] without using the lower semi-continuity of the mapping K :  →  E ; its proof is similar to the proof presented for the second part of Theorem . and so we omit the proof.