Existence of Solutions and Algorithm for a System of Variational Inequalities
© Yali Zhao et al 2010
Received: 13 May 2009
Accepted: 24 January 2010
Published: 10 February 2010
We obtain some existence results for a system of variational inequalities (for short, denoted by SVI) by Brouwer fixed point theorem. We also establish the existence and uniqueness theorem using the projection technique for the SVI and suggest an iterative algorithm and analysis convergence of the algorithm.
1. Questions under Consideration in This Paper
2. Existence and Uniqueness of Solutions for SVI
In order to obtain our main results, we recall the following definitions and lemmas.
Lemma 2.3 (see ).
Lemma 2.4 (see ).
In , assume that and are two compact and convex sets, is continuous, and is strictly monotone in . Then, for any given , has a unique solution and for all , there exists an implicit function which is the unique solution to . In addition, the implicit function determined by is continuous on .
For any given , since is compact and convex and is continuous on , then by Lemma 2.3, parametric variational inequality has solutions. In terms of strict monotonicity of the mapping in and Lemma 2.4, we know that has a unique solution. So, for all , the implicit function determined by is well defined.
Thus, by the uniqueness of the solution to the problem , we conclude that . Since, is arbitrary, we can conclude that as , which means that implicit function is continuous at . For is arbitrary, we know that is continuous on .
From Theorem 2.5, we see that in order to ensure that is well defined, the condition that is strictly monotone on is necessary, but the boundedness of is a strong condition. As usual, is unbounded (e.g., inequality constraint set , where , is always unbounded). So, we try to weaken the boundedness of . For this, we introduce the concept uniform coercivity of in .
In , let be nonempty closed and convex set, and let be uniformly coercive on with respect to and strict monotone in . Then for each , has a unique solution, and for all , the implicit function determined by is continuous on .
For given , , satisfying as . By Lemma 2.7, there exists some neighbourhood of and bounded open set , such that , that is, the solution set of denoted by . such that as . Let without generality, then is bounded; the following argument is similar to Theorem 2.5, so it is omitted, and this completes the proof.
In , let be two nonempty compact and convex subsets, be continuous and strict monotone in . Then for each given , has a unique solution, and for all , the implicit function determined by is continuous on .
The conclusion holds directly from Theorem 2.5.
Lemma 2.10 (see [3, (Brouwer fixed point theorem)]).
By the given conditions of Theorems 2.11 and 2.5, we know that there exists continuous implicit function determined by parametric variational inequality with respect to in SVI. Also denoted the range of by . By Corollary 2.9, there exists continuous implicit function determined by parametric variational inequality with respect to in SVI such that for all , is the unique solution to . Let for all Making use of Brouwer fixed point theorem (Lemma 2.10), we have that there exists , such that . Setting , by the definitions of and , we know that is a solution of SVI.
In SVI, let be nonempty compact and convex subset, let be nonempty closed and convex subset, let and be two continuous mappings, and be strict monotone in and , respectively. Let be uniformly coercive on with respect to . Then SVI has solution.
By Theorem 2.8 and similar argument in Theorem 2.11, our conclusion holds.
Now, we give the definition of uniformly strong monotonicity, which is stronger condition than the uniformly coercivity.
In SVI, let be a nonempty closed and convex set, let be a compact and convex set, let be two continuous mapping, let be uniformly strongly monotone in , and let be strictly monotone in . Then SVI has solution.
By Lemma 2.14 and Corollary 2.12, it is easy to see that the conclusion holds.
In SVI, Let and be nonempty closed and convex subsets, let be two continuous mappings, and let be uniformly strongly monotone in , and let be uniformly coercive and strict monotone in . Then SVI has solution.
then SVI has a unique solution.
In order to prove Theorem 2.17, we need the following lemma.
Lemma 2.18 (see ).
The Proof of Theorem 2.17
3. Iterative Algorithm and Convergence
In this section, we will construct an iterative algorithm for approximating the unique solution of SVI and discuss the convergence analysis of the algorithm.
Lemma 3.1 (see, ).
The authors are grateful to the referee for his/her valuable suggestions and comments that help to present the paper in the present form. This work was supported by the Doctoral Initiating Foundation of Liaoning Province (20071097).
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