The existence of solutions for systems of generalized set-valued nonlinearquasi-variational inequalities
© Qiu and Li; licensee Springer. 2013
Received: 24 August 2012
Accepted: 14 December 2012
Published: 8 January 2013
The Erratum to this article has been published in Fixed Point Theory and Applications 2014 2014:2
In this paper, we introduce and study a class of new systems of generalized set-valuednonlinear quasi-variational inequalities in a Hilbert space. By using the projectionoperator technique and the system of Wiener-Hopf equations technique, we suggest severalnew iterative algorithms to find the approximate solutions to these problems and prove theconvergence of the different types of iterative sequences respectively. It is the firsttime that the system of Wiener-Hopf equations technique has been used to solve the systemof variational inequalities problems, and the technique is more general than theprojection operator technique. Our results improve and extend some known results in theliterature.
Variational inequality problems are among the most interesting and intensively studiedclasses of mathematics problems and have wide applications in the fields of optimization andcontrol, economics and transportation equilibrium and engineering science. And there havebeen a substantial number of numerical methods including fixed point, projection operator,Wiener-Hopf equations, auxiliary principle, KKM technique, linear approximation,decomposition methods, penalty function, splitting method, inertial proximal, dynamicalsystem and well-posedness for solving the variational inequalities and related problems inrecent years (see [1–13] and the references therein).
One of the most common methods for solving the variational problem is to transfer thevariational inequality into an operator equation, and then transfer the operator equationinto the fixed point problems. In the present paper, we introduce and study a class of newsystems of generalized set-valued nonlinear quasi-variational inequalities in a Hilbertspace. We prove that the system of generalized set-valued nonlinear quasi-variationalinequalities is equivalent to the fixed point problem and the system of Wiener-Hopfequations. By using the projection operator technique and the system of Wiener-Hopfequations technique, we suggest several new iterative algorithms to find the approximatesolutions to the problems and prove the convergence of the different types of iterativesequences. It is the first time that the system of Wiener-Hopf equations technique has beenused to solve the system of variational inequalities problems, and the technique is moregeneral than the projection operator technique. Our results improve and extend some knownresults in the literature.
which is called the system of generalized set-valued nonlinear quasi-variationalinequalities.
which is called the system of generalized set-valued nonlinear implicit quasi-variationalinequalities. We remark that if , a nonempty closed convex set in H, then the problem(1.2) is exactly the problem (1.1).
which is due to Noor .
We need the following known concepts and results.
- (i)γ-strongly monotone with respect to the first argument, if there exists a constant such that
Similarly, we can define A is strongly monotone with respect to the second argument.
- (ii)-relaxed co-coercive, if there exist constants , such that
- (iii)-Lipschitz continuous, if there exist constants , such that
Definition 2.2 (see )
where is the Hausdorff metric on .
where , are two constants.
which shows that (2.1) holds for . Similarly, (2.2) holds for .
where, , , are two constants. , are two projection operators.
Proof The conclusion follows directly from Lemma 2.3 . □
whereis the Hausdorff metric on.
3 Projection operator technique
Using the projection operator technique, Lemma 2.6 and Lemma 2.7, we constructthe following iterative algorithms.
If , we obtain the following algorithm from Algorithm 3.1.
then the problem (1.2) admits solutionsand sequences, , andwhich are generated by Algorithm 3.1 converge to x, y, andrespectively.
then as . By (3.5), we know that . So, (3.12) implies that and are both Cauchy sequences. Thus, there exist and such that , as .
Since is compact, we have . Similarly, we have .
By the continuity of , , , , , and Algorithm 3.1, we know that satisfy the relations (2.3). By Lemma 2.6, we claim that is a solution of the problem (1.2). This completes theproof. □
If , we do not need Assumption 2.4 and can obtain thefollowing theorem from Theorem 3.3.
then the problem (1.1) admits solutionsand sequences, , andwhich are generated by Algorithm 3.2 converge to x, y, andrespectively.
4 System of Wiener-Hopf equations technique
Related to the system of generalized set-valued nonlinear implicit quasi-variationalinequalities (1.2), we now consider a new system of generalized implicit Wiener-Hopfequations (4.1). And we will establish the equivalence between them. This equivalence isthen used to suggest a number of new iterative algorithms for solving the given systems ofvariational inequalities.
where , are constants. (4.1) is called the system of generalizedimplicit Wiener-Hopf equations.
where , are constants.
where is a constant.
and, are constants.
Proof Let such that , be a solution of (1.2), then by Lemma 2.6, we know that satisfy (2.3).
Using the fact and , we obtain (4.1). That is to say, and such that , is also the solution of (4.1).
is a solution of (1.2). □
If , we obtain the following lemma from Lemma 4.1.
and, are constants.
Using the system of Wiener-Hopf equations technique, Lemma 4.1 and Lemma 2.7, weconstruct the following iterative algorithms.
If , we have the following iterative algorithm from Algorithm4.3.
then there existsatisfying the system of generalized implicit Wiener-Hopf equations (4.1).So, the problem (1.2) admits solutionsand sequences, , , , andwhich are generated by Algorithm 4.3 converge to x, y, , , andrespectively.
then as . By (4.12), we know that . So, (4.21) implies that and are both Cauchy sequences. By (4.19) and (4.20), we know that and are both Cauchy sequences respectively. So, there exist and such that , , and as . In a similar way as in Theorem 3.3, we know and are also Cauchy sequences and there exist and such that and .
where , are constants. That is just (4.4). By Lemma 4.1, we knowthat satisfy the generalized implicit Wiener-Hopf equations (4.1).So, we claim that is a solution of the problem (1.2). This completes theproof. □
If , we do not need Assumption 2.4 and we can obtain thefollowing theorem from Theorem 4.5.
then there existsatisfying (4.5). So, the generalized Wiener-Hopfequations (4.2) and the problem (1.1) admit the same solutionsand sequences, , , , andwhich are generated by Algorithm 4.4 converge to x, y, , , andrespectively.
Remark 4.7 It is the first time that the system of generalized Wiener-Hopf equationstechnique has been used to solve the system of generalized variational inequalities problem.And for a suitable and appropriate choice of the mappings , and , Theorem 3.3 and Theorem 4.5 include many importantknown results of variational inequality as special cases.
Remark 4.8 It is easy to see that a γ-strongly monotone mapping mustbe a -relaxed co-coercive mapping, whenever , . Therefore, the class of the -relaxed co-coercive mappings is a more general one. Hence, theresults presented in the paper include many known results as special cases.
The work is supported by the National Natural Science Foundation of China (61064006), theNatural Science Foundation of Jiangxi Province (No.2009GZS0009) and the EducationalResearch Project of Jiangxi Province (GJJ09249).
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