On parametric implicit vector variational inequality problems
© Farajzadeh and Plubteing; licensee Springer. 2014
Received: 13 August 2013
Accepted: 28 March 2014
Published: 2 May 2014
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
where () are Hausdorff topological vector spaces (theparametric spaces), a set-valued mapping such that for any, is a nonempty, closed and convex subset of Xwith and . 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  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  and lower semi-continuity of the parametric mapping used inTheorem 3.2 of . 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 start to change. In fact we will show that thesolution set as a mapping 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.
Also W is said to be closed if its graph, that is, , is a closed subset of . A set-valued mapping is called upper semi-continuous (u.s.c.) at if for every open set V containing there exists an open set U containingx such that , for all . The mapping T is said to be lowersemi-continuous (l.s.c.) if for every open set V with there exists an open set U containingx such that . 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 ()
If for any , is compact, then T is u.s.c. at if and only if for any net such that and for every , there exist and a subnet of such that .
T is l.s.c. at if and only if for any net with and for any , there exists a net such that and .
where coA denotes the convex hull of A.
The following lemma plays a crucial rule in this paper.
Lemma 1.3 ()
Let K be a nonempty subset of a topological vector space X andbe amapping with closed values in K. Assume that there exists a nonempty compact convex subset B of K such thatis compact. Then.
2 Main results
The next result provides an existence result for a solution of IVEP.
the mapping is continuous, for all ,
for each , the set is convex,
the mapping defined by is closed,
there exist subsets M and N of K, compact convex and compact, respectively, such that for all there is such that ,
then the solution set of IVEP is nonempty and compact.
and it is easy to see that the solution set of IVEP is equal to the set and hence is a solution of IVEP and further it is compact (note) and hence the proof is complete. □
which implies condition (c) of Theorem 2.1, while if we take and let K be any nonempty convex and compactsubset of X and define , for all and we let g be an arbitrary mapping, wetake the example , for 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 .
Definition 2.2 ()
where is the boundary of . In the case that g is the identity mapping,the mapping f is called locally non-positive at .
The following corollary is an extension of Proposition 3.3 in  for topological vector spaces. Furthermore, the condition that fis locally non-positive at has been omitted.
the mapping is continuous, for all ,
for each the mapping is -convex,
the mapping defined by is closed,
- (e)there exist a nonempty compact and convex subset D of and such that for all
Moreover, the solution set is a compact subset of.
which is a contradiction. Hence 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  with mild assumptions for mappings which do not need to satisfy thelocally non-positive condition. In fact this condition has been removed.
is a continuous mapping with nonempty convex compact values;
the set is convex and , for each ;
the mapping defined by is closed;
- (i)for each , the solution set
- (ii)the solution set mapping defined by
and hence . Consequently, for each point there is , and so . 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  without using the lower semi-continuity of the mapping; its proof is similar to the proof presented for thesecond part of Theorem 2.4 and so we omit the proof.
is a mapping with nonempty convex and compact values;
mapping is continuous, for each ;
for each the mapping is -convex and , for each ;
the mapping defined by is closed.
, for each is nonempty and compact;
the mapping S is l.s.c. at .
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