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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 431717, 5 pages
Connectedness of Solution Sets for Weak Vector Variational Inequalities on Unbounded Closed Convex Sets
Department of Mathematics, Guangxi Normal University, Guilin, Guangxi 541004, China
Received 20 January 2013; Revised 27 March 2013; Accepted 28 March 2013
Academic Editor: Ngai-Ching Wong
Copyright © 2013 Ren-you Zhong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the connectedness of solution set for set-valued weak vector variational inequality in unbounded closed convex subsets of finite dimensional spaces, when the mapping involved is scalar C-pseudomonotone. Moreover, the path connectedness of solution set for set-valued weak vector variational inequality is established, when the mapping involved is strictly scalar C-pseudomonotone. The results presented in this paper generalize some known results by Cheng (2001), Lee et al. (1998), and Lee and Bu (2005).
The concept of vector variational inequality , which was first introduced by Giannessi  in 1980, has wide applications in many problems such as finance and economics, transportation and optimization, operations research, and engineering sciences. Many authors have devoted to the study of and its various extensions. The main topic of these papers is to establish existence theorems of solution for , see, for example, [1–5] and the references therein. Another important and interesting problem for is to study the topological properties of solutions set. Among them, the connectedness property of the solution set is quite of interest as it provides the possibility of continuously moving from one solution to any other solution.
Some authors have discussed the connectedness of solution set for single-valued weak vector variational inequalities under the assumption that the mapping involved is monotone and the constrained set involved is compact. In , Chen discussed the connectedness of solution set for a single-valued in compact subsets of , when the mapping involved is strictly pseudomonotone. Gong  and Gong and Yao  studied the connectedness of solution sets for vector equilibrium problems and generalized system in compact subsets of infinite dimensional spaces, when the mapping involved is monotone. For more related work, we refer the readers to  and references therein.
It is noted that in the works mentioned above, a compactness assumption of the constrained set is necessary. As for the noncompactness case, we observe that only few papers in the literature have dealt with this. In , Lee et al. established the connectedness of the solution set for a single-valued strongly monotone on unbounded closed convex sets. Lee and Bu  discussed the connectedness of solution set for affine vector variational inequalities with noncompact polyhedral constraint sets and positive semidefinite (or monotone) matrices.
Inspired and motivated by the work in [6, 10, 11], we further study the connectedness properties of solution set for set-valued in noncompact subsets of finite dimensional spaces. We establish the connectedness and path-connectedness results of solution set for set-valued when the mapping involved is scalar -pseudomonotone and strictly scalar -pseudomonotone, which is weaker than monotone mapping and strictly monotone (strongly monotone) mapping, respectively. Compared with the previous connectedness results, we establish our results without putting the compactness assumption on the constrained set, and the mapping involved is set-valued. Moreover, we would like to point out that the image space associated with the problem discussed is infinite dimensional. The results presented in this paper generalize the corresponding results in [6, 10, 11].
The paper is organized as follows. In Section 2, we introduce some basic notations and preliminary results. In Section 3, we establish the connectedness and path-connectedness result of the solution set for set-valued .
Let be a finite-dimensional norm space and a normed space with its dual space of . Let be a nonempty closed convex subset of and a set-valued mapping with nonempty values, where denotes the space of all continuous linear mappings from to . Let be a closed convex pointed cone in with . The cone induces a partial ordering in , which was defined by Let be the dual cone of . It is clear that Let be fixed and The dual cone is said to admit a -compact base if and only if there exists a -compact set such that and . From , we know that is a -compact base of .
The recession cone of , denoted by , is defined by The negative polar cone of , denoted by , is defined by Let be a nonempty set. The weak and strong -polar cones of , which were introduced in , are defined, respectively, by
In this paper, we consider the following set-valued weak vector variational inequality, denoted by , which is to find and such that
Let be any given point. It is known that is closely related to the following scalar variational inequality problem, denoted by , which is to find and such that Here, means the composition of and . Hence, , for all .
The solution sets of and are denoted by and , respectively. From , the following relationship between and holds: Although the representation is stated in  for single-valued map, the statement and the proof are also valid when is multivalued without any assumption on the values of .
We now recall some definitions for set-valued monotone and pseudomonotone mapping.
Definition 1. A set-valued mapping is said to be as follows.(i)-monotone (resp. strictly -monotone) on if for all (resp., ), for all , , (ii)-pseudomonotone (resp., strictly -monotone) on if for all (resp., ), for all , ,
Definition 2. A set-valued mapping is said to be scalar -pseudomonotone (resp., strictly scalar -pseudomonotone) on if and only if for any , for any (resp., ), , , we have
Example 3. Let Clearly, is strictly scalar -pseudomonotone on and so scalar -pseudomonotone.
Remark 4. (i) If is -monotone (resp., strictly -monotone) on , then is scalar -pseudomonotone (resp., strictly scalar -pseudomonotone) on .
(ii) The scalar -pseudomonotonicity in Definition 2 is weaker than -pseudomonotonicity in Definition 1(ii). In fact, for any , , if , then we have . Then, it follows from the -pseudomonotonicity of that , which implies that .
Definition 5. The topological space is said to be connected if there do not exist nonempty open subsets of , such that and .
Moreover, is said to be path connected if for each pair of points and in , there exists a continuous mapping such that and .
Definition 6. Let , be two topological spaces. A set-valued mapping is said to be (i)closed if is closed in ,(ii)upper semicontinuous, if for every and every open set satisfying , there exists a neighborhood of such that .
Lemma 7. Let be a closed convex subset of and scalar pseudomonotone and upper semicontinuous with nonempty compact convex values. If for any , is nonempty and compact, then is nonempty and compact.
Remark 8. It is pointed out in  that if , then is nonempty and compact for any , and so is nonempty and compact.
From Theorem 2 of , we have the following lemma.
Lemma 9. Let be a closed convex subset of and scalar pseudomonotone and upper semicontinuous with nonempty compact convex values. Then, for any , is nonempty and compact if and only if .
Lemma 10 (see ). Let , be two topological spaces. If the set-valued mapping is closed and is compact, then is upper semicontinuous.
Lemma 11 (see ). Let , be two topological spaces. Assume that is connected and the set-valued mapping is upper semicontinuous. If for every , the set is nonempty and connected, then the set is connected.
Lemma 11 follows immediately from the definition of path connectedness.
Lemma 12. Let , be two topological spaces. Assume that is path connected and the mapping is continuous. Then, the set is path connected.
Lemma 13. Let be a normed space with its dual space of . Let a net in and be a net in . Suppose that converges to in the norm topology of and -converges to . Then, .
Proof. Indeed, we have Since -converges , it holds that . Moreover, by the triangle inequality, we have . Note that is bounded and converges to in the norm topology of , this yields that . Consequently, we obtain that . This completes the proof.
3. Connectedness of Solution Sets for WVVI
In this section, we establish the connectedness of solution set for the set-valued , when the mapping is scalar -pseudomonotone. Furthermore, when the mapping is strictly scalar -pseudomonotone, we establish the path-connectedness result for the set-valued .
Theorem 14. Let be a closed convex subset of and scalar -pseudomonotone and upper semicontinuous with nonempty compact convex values, where is equipped with the norm topology. Suppose that for any . Then, is connected.
Proof. From Lemma 9, we know that is nonempty and compact for any . Then, it follows from Lemma 7 that is nonempty and compact.
Now, we show that is connected. First, we claim that is convex. Indeed, by the scalar -pseudomonotonicity of , it follows from Proposition 1 of  that Clearly, the set is convex, and so is convex.
Setting , then is compact. Define a set-valued mapping as follows:
We now show that is closed, where is equipped with the topology. Take with and . The fact implies that there exists such that Since is compact and is upper semicontinuous, we have that is compact. Moreover, since and , there exists a subnet of such that . Clearly, . From (18), we have Setting and , then it follows from Lemma 13 that This implies that and so is closed. Then, from Lemma 10, we know that is upper semicontinuous. This together with Lemma 11 yields that is connected. This completes the proof.
Remark 15. (i) If , and , where and , , then reduces to an affine vector variational inequality. Lee and Bu  obtained the corresponding result of Theorem 14 for affine vector variational inequalities, see Theorems 2.1 and 2.2 of .
(ii) If is compact, and , are single-valued and continuous; Cheng  obtained the corresponding result of Theorem 17, see Theorem 1 of . Similar results can be founded in Gong  and Gong and Yao  for the connectedness of solution sets for vector equilibrium problems. Compared with these results, we do not need to assume that is compact but only a unbounded close convex set.
Example 16. Consider problem , where Clearly, , are upper semicontinuous, and is scalar -pseudomonotone on , . Then, all the assumptions of Theorem 14 are satisfied. By a simple computation, we obtain that and so is connected.
When is strictly scalar -pseudomonotone, we further obtain the following path-connectedness result for set-valued .
Theorem 17. Let be a closed convex subset of and strictly scalar -pseudomonotone and upper semicontinuous with nonempty compact convex values, where is equipped with the norm topology. Suppose that for any . Then, is path connected.
Proof. The nonemptiness of is obvious. We now claim that for every , is unique. Let with be the solutions of . Since is a solution of , then there exists such that
Taking in the above inequality, we have
By the strictly scalar -pseudomonotonicity of , it follows that
This contradicts the fact that is a solution of . Thus, for every , has a unique solution in .
Then, from the proof of Theorem 14, we know that is a single-valued and upper semicontinuous mapping and so continuous. Since is path connected, from Lemma 12, the solution set is also path connected. This completes the proof.
Remark 18. Note that a strongly monotone mapping is strictly monotone and hence strictly pseudomonotone; Theorem 17 generalized Theorem 4.2 of  from single-valued mappings to set-valued mappings under a weaker monotonicity assumption.
Example 19. Consider problems and , where Clearly, , are upper semicontinuous, and is strictly -pseudomonotone on , . Then, all the assumptions of Theorem 17 are satisfied. By a simple computation, we obtain that and so is path connected.
This work was supported by the National Natural Science Foundation of China (11061006 and 11226224), the Program for Excellent Talents in Guangxi Higher Education Institutions, the Guangxi Natural Science Foundation (2012GXNSFBA053008), and the Initial Scientific Research Foundation for PhD of Guangxi Normal University.
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