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The existence of solutions for systems of generalized set-valued nonlinearquasi-variational inequalities
Fixed Point Theory and Applications volume 2013, Article number: 4 (2013)
Abstract
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.
1 Introduction
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.
Let H be a real Hilbert space whose inner product and norm are denoted by and respectively. Let K be a nonempty closed convex set inH and be the family of all nonempty compact subsets of H.Given two nonlinear mappings and two set-valued mappings , , we consider the following problem of finding such that , and
which is called the system of generalized set-valued nonlinear quasi-variationalinequalities.
It is worth mentioning that in many important problems, the closed convex set Kalso depends upon the solutions explicitly or implicitly. Given two point-to-set mappings and , which associate two closed convex sets and with any element x, y of H, weconsider the problem of finding such that , and
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).
If the closed convex sets and are of the form and , where and are two nonempty closed convex sets and , are two point-to-point mappings, then the problem (1.2) isequivalent to finding such that , and
If , , , then the problem (1.2) is equivalent to finding such that , and
which is due to Noor [1].
If , , then the problem (1.1) is equivalent to finding such that and
2 Preliminaries
We need the following known concepts and results.
Let H be a Hilbert space, be a nonlinear mapping. A is said to be
-
(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 [7])
Let be a set-valued mapping. B is said to be-H-Lipschitz continuous if there exist constants, such that
where is the Hausdorff metric on .
Let H be a Hilbert space, K be a nonempty closed convex set in H. Then, for a given, satisfies the inequality
if and only if
whereis the projection of H into K. Furthermore, the operatoris nonexpansive, i.e.,
Assumption 2.4 Let H be a real Hilbert space, and be two nonempty closed convex sets. For all, the operators and satisfy the relations
where , are two constants.
Remark 2.5 We remark that Assumption 2.4 is also true for the case, , when the point-to-point mappings , are -Lipschitz continuous respectively. For all, it is well known that
and
which shows that (2.1) holds for . Similarly, (2.2) holds for .
Lemma 2.6 The system of generalized set-valued nonlinear implicitquasi-variational inequalities (1.2) has solutionsif and only ifsatisfy the relations
where, , , are two constants. , are two projection operators.
Proof The conclusion follows directly from Lemma 2.3 . □
Let E be a complete metric space, be the family of all the convex bounded subsets of E, be a set-valued mapping. Then for any givenand any given, , there existssuch that
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.
Algorithm 3.1 Let H be a real Hilbert space, and be two nonempty closed convex sets in H, be two nonlinear mappings, be two set-valued mappings, . For any given such that , , and
Since , , and by Lemma 2.7, there exist , such that
Let
since , , and by Lemma 2.7, there exist , such that
By induction, we can define iterative sequences , , and satisfying
where .
If , we obtain the following algorithm from Algorithm 3.1.
Algorithm 3.2 We define iterative sequences , , and satisfying
where .
Theorem 3.3 Let H be a real Hilbert space, andbe two nonempty closed convex sets in H. For, let nonlinear mappingsbe-Lipschitz continuous and-strongly monotone with respect to the ith argument, be-H-Lipschitz continuous if Assumption 2.4holds, and there exist constants, such that
then the problem (1.2) admits solutionsand sequences, , andwhich are generated by Algorithm 3.1 converge to x, y, andrespectively.
Proof By Lemma 2.3, (2.1) and (3.1), we have
Since is -strongly monotone with respect to the first argument andLipschitz continuous, we have
and
By (3.3) and -H-Lipschitz continuity of , we have
Combining (3.6), (3.7), (3.8) and (3.9), we obtain
Similarly, we can have
Adding (3.10) to (3.11), we have
where
Let
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 .
Now, we prove that and . In fact, since and are both Cauchy sequences and by (3.9), we know that is Cauchy sequences. Similarly, is also Cauchy sequences. Therefore, there exist and such that and . Further,
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.
Theorem 3.4 Let H be a real Hilbert space, K be a nonempty closed convex set in H. For, let nonlinear mappingsbe-Lipschitz continuous and-strongly monotone with respect to the ith argument, be-H-Lipschitz continuous, if there existconstants, such that
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.
To be more precise, let , , where I is the identity operator, and are two projection operators, and are two convex sets. We consider the following problem offinding , such that , and
where , are constants. (4.1) is called the system of generalizedimplicit Wiener-Hopf equations.
If , we obtain the following system of generalized Wiener-Hopfequations from (4.1), which is of finding such that , and
where , are constants.
If , , we obtain the following Wiener-Hopf equation from (4.2), whichis of finding such that and
where is a constant.
Lemma 4.1 The system of generalized set-valued nonlinear implicitquasi-variational inequalities (1.2) has solutionssuch that, if and only if the system of generalized implicit Wiener-Hopf equations(4.1) has solutionsandsuch that, , where
and, are constants.
Proof Let such that , be a solution of (1.2), then by Lemma 2.6, we know that satisfy (2.3).
Let , , then by (2.3), we have , , which is just (4.4). And we have
Using the fact and , we obtain (4.1). That is to say, and such that , is also the solution of (4.1).
Conversely, let and such that , be a solution of (4.1). Then we have
Now, by invoking Lemma 2.3 and the above relations, we have
Thus , where
is a solution of (1.2). □
If , we obtain the following lemma from Lemma 4.1.
Lemma 4.2 The system of generalized set-valued nonlinear quasi-variationalinequalities (1.1) has solutionssuch that, if and only if the system of generalized Wiener-Hopf equations (4.2)has solutionsandsuch that, , where
and, are constants.
Using the system of Wiener-Hopf equations technique, Lemma 4.1 and Lemma 2.7, weconstruct the following iterative algorithms.
Algorithm 4.3 Let H be a real Hilbert space, and be two nonempty closed convex sets in H, be two nonlinear mappings, be two set-valued mappings, . For any given , such that , , , . We compute , , , , and by the following iterative schemes:
where .
If , we have the following iterative algorithm from Algorithm4.3.
Algorithm 4.4 For any given , such that , , , , we compute , , , , and by the following iterative schemes:
where .
Theorem 4.5 Let H be a real Hilbert space, andbe two nonempty closed convex sets in H. For, let nonlinear mappingsbe-Lipschitz continuous and-relaxed co-coercive with respect to the ith argument, be-H-Lipschitz continuous, ifAssumption 2.4 holds and there exist constantssuch that
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.
Proof By (4.8), we have
Since is -relaxed co-coercive with respect to the first argument andLipschitz continuous, we have
and
From -H-Lipschitz continuity of and (4.10), we have
Combining (4.13), (4.14), (4.15) and (4.16), we obtain
Similarly, we can have
By (4.6), Lemma 2.3 and Assumption 2.4,
which implies that
Similarly, we can obtain
By (4.17)-(4.20), we have
where
Let
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 .
By the continuity of the mappings , , , , , and Algorithm 4.3, as , we have
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.
Theorem 4.6 Let H be a real Hilbert space, K be a nonempty closed convex set in H. For, let nonlinear mappingsbe-Lipschitz continuous and-relaxed co-coercive with respect to the ith argument, be-H-Lipschitz continuous if there existconstantssuch that
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.
References
Noor MA: Generalized set-valued mixed nonlinear quasi variational inequalities. Korean J. Comput. Appl. Math. 1998, 5(1):73–89.
Noor MA: Some developments in general variational inequalities. Appl. Math. Comput. 2004, 152(1):199–277. 10.1016/S0096-3003(03)00558-7
Glowinski R, Lions JJ, Tremolieres R: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam; 1981.
Huang NJ: Generalized nonlinear variational inclusions with noncompact valued mappings. Appl. Math. Lett. 1996, 9(3):25–29. 10.1016/0893-9659(96)00026-2
Zhong RY, Huang NJ: Strict feasibility for generalized mixed variational inequality in reflexive Banachspaces. J. Optim. Theory Appl. 2012, 152: 696–709. 10.1007/s10957-011-9914-3
Qiu YQ, Liu LW: A new system of generalized quasi-variational-like inclusion in Hilbert spaces. Comput. Math. Appl. 2010, 59(1):1–8. 10.1016/j.camwa.2009.06.053
Ding XP: Existence and algorithm of solutions for generalized mixed implicit quasi-variationalinequalities. Appl. Math. Comput. 2000, 113(1):67–80. 10.1016/S0096-3003(99)00068-5
Verma RU: Projection methods, algorithms, and a new system of nonlinear variationalinequalities. Comput. Math. Appl. 2001, 41(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9
Xia FQ, Huang NJ: A projection-proximal point algorithm for solving generalized variationalinequalities. J. Optim. Theory Appl. 2011, 150: 98–117. 10.1007/s10957-011-9825-3
Chang SS: Set-valued variational inclusions in Banach spaces. J. Math. Anal. Appl. 2000, 248(2):438–454. 10.1006/jmaa.2000.6919
Wang SH, Huang NJ, ORegan D: Well-posedness for generalized quasi-variational inclusion problems and foroptimization problems with constraints. J. Glob. Optim. 2012. 10.1007/s10898-012-9980-6
Noor MA: Projection iterative methods for extended general variational inequalities. J. Appl. Math. Comput. 2010, 32(1):83–95. 10.1007/s12190-009-0234-9
Petrot N:Existence and algorithm of solutions for general set-valued Noor variationalinequalities with relaxed -cocoercive operators in Hilbert spaces. J. Appl. Math. Comput. 2010, 32(2):393–404. 10.1007/s12190-009-0258-1
Acknowledgements
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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An erratum to this article can be found online at 10.1186/1687-1812-2014-2.
An erratum to this article is available at http://dx.doi.org/10.1186/1687-1812-2014-2.
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Qiu, Y., Li, X. The existence of solutions for systems of generalized set-valued nonlinearquasi-variational inequalities. Fixed Point Theory Appl 2013, 4 (2013). https://doi.org/10.1186/1687-1812-2013-4
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DOI: https://doi.org/10.1186/1687-1812-2013-4