Existence and Algorithm for Solving the System of Mixed Variational Inequalities in Banach Spaces
Siwaporn Saewan1and Poom Kumam1
Academic Editor: Hong-Kun Xu
Received22 Dec 2011
Accepted29 Jan 2012
Published09 Apr 2012
Abstract
The purpose of this paper is to study the existence and convergence analysis of the solutions of the system of mixed variational inequalities in Banach spaces by using the generalized f projection operator. The results presented in this paper improve and extend important recent results of Zhang et al. (2011) and Wu and Huang (2007) and some recent results.
1. Introduction
Let be a real Banach space with norm , let be a nonempty closed and convex subset of , and let denote the dual of . Let denote the duality pairing of and . If is a Hilbert space, denotes an inner product on . It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, and so forth (see, e.g., [1, 2] and the references therein). In 1993, Alber [3] introduced and studied the generalized projections and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [1] presented some applications of the generalized projections to approximately solving variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [2] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Later, Wu and Huang [4] introduced a new generalized -projection operator in Banach spaces which extended the definition of the generalized projection operators introduced by Abler [3] and proved some properties of the generalized -projection operator. As an application, they studied the existence of solution for a class of variational inequalities in Banach spaces. In 2007, Wu and Huang [5] proved some properties of the generalized -projection operator and proposed iterative method of approximating solutions for a class of generalized variational inequalities in Banach spaces. In 2009, Fan et al. [6] presented some basic results for the generalized -projection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces. In 2011, Zhang et al. [7] introduced and considered the system of mixed variational inequalities in Banach spaces. Using the generalized -projection operator technique, they introduced some iterative methods for solving the system of mixed variational inequalities and proved the convergence of the proposed iterative methods under suitable conditions in Banach spaces. Recently, many authors studied methods for solving the system of generalized (mixed) variational inequalities and the system of nonlinear variational inequalities problems (see, e.g., [8β17] and references therein).
We first introduce and consider the system of mixed variational inequalities (SMVI) which is to find such that
where for are mappings and is the normalized duality mapping from to .
As special case of the problem (1.1), we have the following.
If for , for all , (1.1) is equivalent to find , and such that
The problem (1.2) is called the system of variational inequalities we denote by (SVI).
If , for all and , then (1.1) is reduced to find such that
which is studied by Zhang et al. [7].
If , for all and , (1.1) is reduced to find such that
This iterative method is studied by Wu and Huang [5].
If , for all , (1.4) is reduced to find such that
which is studied by Alber [1, 18], Li [2], and Fan [19]. If is a Hilbert space, (1.5) holds which is known as the classical variational inequality introduced and studied by Stampacchia [20].
If is a Hilbert space, then (1.1) is reduced to find such that
If for , for all , (1.6) reduces to the following (SVI):
The purpose of this paper is to study the existence and convergence analysis of solutions of the system of mixed variational inequalities in Banach spaces by using the generalized -projection operator. The results presented in this paper improve and extend important recent results in the literature.
2. Preliminaries
A Banach space is said to be strictly convex if for all with and . Let be the unit sphere of . Then, a Banach space is said to be smooth if the limit exists for each . It is also said to be uniformly smooth if the limit exists uniformly in . Let be a Banach space. The modulus of smoothness of is the function defined by . The modulus of convexity of is the function defined by . The normalized duality mapping is defined by . If is a Hilbert space, then , where is the identity mapping.
If is a reflexive smooth and strictly convex Banach space and is the normalized duality mapping on , then , and , where and are the identity mappings on and . If is a uniformly smooth and uniformly convex Banach space, then is uniformly norm-to-norm continuous on bounded subsets of and is also uniformly norm-to-norm continuous on bounded subsets of .
Let and be Banach spaces, , the operator is said to be compact if it is continuous and maps the bounded subsets of onto the relatively compact subsets of ; the operator is said to be weak to norm continuous if it is continuous from the weak topology of to the strong topology of .
We also need the following lemmas for the proof of our main results.
Lemma 2.1 (Xu [21]). Let and be two fixed real numbers. Let be a -uniformly convex Banach space if and only if there exists a continuous strictly increasing and convex function , , such that
for all and , where .
For case , we have
Lemma 2.2 (Chang [22]). Let be a uniformly convex and uniformly smooth Banach space. The following holds:
Next we recall the concept of the generalized -projection operator. Let be a functional defined as follows:
where is positive number and is proper, convex, and lower semicontinuous. From definitions of and , it is easy to see the following properties:(1);(2) is convex and continuous with respect to when is fixed;(3) is convex and lower semicontinuous with respect to when is fixed.
Definition 2.3. Let be a real Banach space with its dual . Let be a nonempty closed convex subset of . It is said that is the generalized -projection operator if
In this paper, we fixed , we have
For the generalized -projection operator, Wu and Hung [5] proved the following basic properties.
Lemma 2.4 (Wu and Hung [4]). Let be a reflexive Banach space with its dual and is a nonempty closed convex subset of . The following statements hold:(1) is nonempty closed convex subset of for all ;(2)if is smooth, then for all , if and only if
(3)if is smooth, then for any , , where is the subdifferential of the proper convex and lower semicontinuous functional .
Lemma 2.5 (Wu and Hung [4]). If for all , then for any ,
Lemma 2.6 (Fan et al. [6]). Let be a reflexive strictly convex Banach space with its dual and is a nonempty closed convex subset of . If is proper, convex, and lower semicontinuous, then(1) is single valued and norm to weak continuous;(2)if has the property (h), that is, for any sequence and , imply that , then is continuous.
Defined the functional by
3. Generalized Projection Algorithms
Proposition 3.1. Let be a nonempty closed and convex subset of a reflexive strictly convex and smooth Banach space . If for is proper, convex, and lower semicontinuous, then is a solution of (SMVI) equivalent to finding such that
Proof. From Lemma 2.4 (2) and is a reflexive strictly convex and smooth Banach space, we known that is single valued and for is well defined and single valued. So, we can conclude that Proposition 3.1 holds.
For solving the system of mixed variational inequality (1.1), we defined some projection algorithms as follow.
Algorithm 3.2. For an initial point , define the sequences , and as follows:
where .
If , for all , then Algorithm 3.2 reduces to the following iterative method for solving the system of variational inequalities (1.2).
Algorithm 3.3. For an initial point , define the sequences , and as follows:
where .
For solving the problem (1.6), we defined the algorithm as follows:
If is a Hilbert space, then Algorithm 3.2 reduces to the following.
Algorithm 3.4. For an initial point , define the sequences , and as follows:
where .
If , for all , then Algorithm 3.4 reduces to the following iterative method for solving the problem (1.7) as follows.
Algorithm 3.5. For an initial point , define the sequences ,
and as follows:
where .
4. Existence and Convergence Analysis
Theorem 4.1. Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping and which is convex lower semicontinuous mappings for satisfying the following conditions:(i), for all for ;(ii) are compact for ;(iii) and , for all and ; then the system of mixed variational inequality (1.1) has a solution and sequences , and defined by Algorithm 3.2 have convergent subsequences , and such that
Proof. Since is a uniformly convex and uniform smooth Banach space, we known that is bijection from to and uniformly continuous on any bounded subsets of . Hence, for is well-defined and single-value implies that , and are well defined. Let , for any and , we have
By (4.2) and Lemma 2.5, we have
From Lemma 2.2, and for all , , so for , we obtain
Again by Lemma 2.2, for all , and for , we have
In similar way, for all , , and , we also have
It follows from (4.5) and (4.6) that
From (4.5) and (4.6), we compute
This implies that the sequences , and are bounded. For a positive number such that , by Lemma 2.1, for , there exists a continuous, strictly increasing, and convex function with such that for , we have
Applying (4.3), (4.4), and (4.7), we have
Summing (4.10), for , we have
taking , we get
This shows that series (4.12) is converge, we obtain that
From for all , thus and (4.13), we have
By property of functional , we have
Since is bounded sequence and is compact on , then sequence has a convergence subsequence such that
By the continuity of the , we have
Again since are bounded and are compact on , then sequences and have convergence subsequences such that
By the continuity of and , we have
Let
By using the triangle inequality, we have
From (4.15) and (4.17), we have
By definition of , we get
It follows by (4.20) and (4.23), we obtain
In the same way, we also have
By the continuity properties of , and for . We conclude that
This completes of proof.
Theorem 4.2. Let be a nonempty compact and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping and which is convex lower semicontinuous mappings for satisfy the following conditions:(i) for all for ;(ii) and , for all for ; then the system of mixed variational inequality (1.1) has a solution and sequences , and defined by Algorithm 3.2 have a convergent subsequences , and such that
Proof. In the same way to the proof in Theorem 4.1, we have
Hence, there exist subsequences and such that
From the compactness of , we have that
where are points in . Also, for a sequence , where is a points in . By the continuity properties of , and , we obtain that
From definition of , we get
By (4.25) and (4.31), we have
This completes of proof.
Corollary 4.3. Let be a nonempty closed and convex subset of a uniformly convex and uniformly smooth Banach space with dual space . If the mapping for satisfy the following conditions:(i), for all for ;(ii) are compact for ; then the system of mixed variational inequality (1.2) has a solution and sequences , and defined by Algorithm 3.3 have convergent subsequences , and such that , and .
If is a Hilbert space, then , so one obtains the following corollary.
Corollary 4.4. Let be a nonempty closed and convex subset of a Hilbert space . If the mapping and which is convex lower semicontinuous mappings for satisfy the following conditions:(i) for ;(ii) and for all for ; then the system of mixed variational inequality (1.6) has a solution and sequences , and defined by Algorithm 3.4 have a convergent subsequences , and such that , and .
Corollary 4.5. Let be a nonempty closed and convex subset of a Hilbert space . If the mapping for satisfy the conditions: for ; then the system of mixed variational inequality (1.7) has a solution and sequences , and defined by Algorithm 3.5 have a convergent subsequences , and such that , and .
Remark 4.6. Theorems 4.1 and 4.2 and Corollary 4.3 extend and improve the results of Zhang et al. [7] and Wu and Huang [5].
Acknowledgments
This research was supported by grant from under the Program Strategic Scholarships for Frontier Research Network for the Joint Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. Furthermore, we would like to thank the King Mongkuts Diamond Scholarship for Ph.D. Program at King Mongkuts University of Technology Thonburi (KMUTT) and the National Research University Project of Thailandβs Office of the Higher Education Commission (under CSEC project no.54000267) for their financial support during the preparation of this paper. P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund (Grant no. MRG5380044).
References
Y. I. Alber, βMetric and generalized projection operators in Banach spaces: properties and applications,β in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178, pp. 15β50, Dekker, New York, NY, USA, 1996.
J. Li, βThe generalized projection operator on reflexive Banach spaces and its applications,β Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55β71, 2005.
Y. I. Alber, βGeneralized projection operators in Banach spaces: properties and applications,β in Proceedings of the Israel Seminar on Functional Differential Equations, vol. 1, pp. 1β21, The College of Judea & Samaria, Ariel, Israel, 1993.
K.-q. Wu and N.-j. Huang, βThe generalised f-projection operator with an application,β Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 307β317, 2006.
K.-Q. Wu and N.-J. Huang, βProperties of the generalized f-projection operator and its applications in Banach spaces,β Computers & Mathematics with Applications, vol. 54, no. 3, pp. 399β406, 2007.
J. Fan, X. Liu, and J. Li, βIterative schemes for approximating solutions of generalized variational inequalities in Banach spaces,β Nonlinear Analysis, vol. 70, no. 11, pp. 3997β4007, 2009.
Q.-B. Zhang, R. Deng, and L. Liu, βProjection algorithms for the system of mixed variational inequalities in Banach spaces,β Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1692β1699, 2011.
R. P. Agarwal and Y. J. C. N. Petrot, βSystems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces,β Fixed Point Theory and Applications, vol. 2011, no. 31, 2011.
Y. J. Cho and N. Petrot, βRegularization and iterative method for general variational inequality problem in Hilbert spaces,β Journal of Inequalities and Applications, vol. 2011, no. 21, p. 11, 2011.
P. Kumam, N. Petrot, and R. Wangkeeree, βExistence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces,β Applied Mathematics and Computation, vol. 217, no. 18, pp. 7496β7503, 2011.
S. Suantai and N. Petrot, βExistence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems,β Applied Mathematics Letters, vol. 24, no. 3, pp. 308β313, 2011.
I. Inchan and N. Petrot, βSystem of general variational inequalities involving different nonlinear operators related to fixed point problems and its applications,β Fixed Point Theory and Applications, Article ID 689478, 17 pages, 2011.
Y. J. Cho, N. Petrot, and S. Suantai, βFixed point theorems for nonexpansive mappings with applications to generalized equilibrium and system of nonlinear variational inequalities problems,β Journal of Nonlinear Analysis and Optimization, vol. 1, no. 1, pp. 45β53, 2010.
N. Onjai-uea and P. Kumam, βAlgorithms of common solutions to generalized mixed equilibrium problems and a system of quasivariational inclusions for two difference nonlinear operators in Banach spaces,β Fixed Point Theory and Applications, vol. 2011, Article ID 601910, 23 pages, 2011.
N. Onjai-Uea and P. Kumam, βExistence and Convergence Theorems for the new system of generalized mixed variational inequalities in Banach spaces,β Journal of Inequalities and Applications, vol. 2012, 9 pages, 2012.
N. Petrot, βA resolvent operator technique for approximate solving of generalized system mixed variational inequality and fixed point problems,β Applied Mathematics Letters, vol. 23, no. 4, pp. 440β445, 2010.
Y. J. Cho and N. Petrot, βOn the system of nonlinear mixed implicit equilibrium problems in Hilbert spaces,β Journal of Inequalities and Applications, Article ID 437976, 12 pages, 2010.
Y. I. Alber, βMetric and generalized projection operators in Banach spaces: properties and applications,β in Proceedings of the Israel Seminar on Functional Differential Equations, vol. 1, pp. 1β21, The College of Judea & Samaria, Ariel, Israel, 1994.
J. Fan, βA Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces,β Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1041β1047, 2008.
S.-S. Chang, βOn Chidume's open questions and approximate solutions of multivalued strongly accretive mapping equations in Banach spaces,β Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 94β111, 1997.