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A Hybrid Proximal Point ThreeStep Algorithm for Nonlinear SetValued QuasiVariational Inclusions System Involving Accretive Mappings
Fixed Point Theory and Applications volumeÂ 2010, ArticleÂ number:Â 635382 (2010)
Abstract
The main purpose of this paper is to introduce and study a new class of generalized nonlinear setvalued quasivariational inclusions system involving accretive mappings in Banach spaces. By using the resolvent operator due to LanChoVerma associated with accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point threestep iterative algorithm for this system of setvalued variational inclusions and an existence theorem of solutions for this kind of the variational inclusions system. The results presented in this paper generalize, improve, and unify some recent results in this field.
1. Introduction
The variational inclusion, which was introduced and studied by Hassouni and Moudafi [1], is a useful and important extension of the variational inequality. It provides us with a unified, natural, novel, innovative, and general technique to study a wide class of problems arising in different branches of mathematical and engineering sciences. Various variational inclusions have been intensively studied in recent years. Ding and Luo [2], Verma [3, 4], Huang [5], Fang et al. [6], Fang and Huang [7], Fang et al. [8], Lan et al. [9], Zhang et al. [10] introduced the concepts of subdifferential operators, maximal monotone operators, monotone operators, monotone operators, monotone operators, accretive mappings, monotone operators, and defined resolvent operators associated with them, respectively. Moreover, by using the resolvent operator technique, many authors constructed some approximation algorithms for some nonlinear variational inclusions in Hilbert spaces or Banach spaces. Verma has developed a hybrid version of the EcksteinBertsekas [11] proximal point algorithm, introduced the algorithm based on the maximal monotonicity framework [12], and studied convergence of the algorithm. For the past few years, many existence results and iterative algorithms for various variational inequalities and variational inclusion problems have been studied. For details, please see [1â€“37] and the references therein.
On the other hand, some new and interesting problems for systems of variational inequalities were introduced and studied. Peng and Zhu [14], Cohen and Chaplais [15], Bianchi [16], and Ansari and Yao [17] considered a system of scalar variational inequalities. Ansari et al. [18] introduced and studied a system of vector equilibrium problems and a system of vector variational inequalities using a fixed point theorem. Allevi et al. [19] considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonotonicity. Kassay and KolumbÃ¡n [20] introduced a system of variational inequalities and proved an existence theorem through the Ky Fan lemma. Kassay et al. [21] studied Minty and Stampacchia variational inequality systems with the help of the KakutaniFanGlicksberg fixed point theorem. J. K. Kim and D. S. Kim [22] introduced a new system of generalized nonlinear quasivariational inequalities and obtained some existence and uniqueness results on solutions for this system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho et al. [23] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. They proved some existence and uniqueness theorems for solutions for the system of nonlinear variational inequalities. As generalizations of a system of variational inequalities, Agarwal et al. [24] introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Kazmi and Bhat [25] introduced a system of nonlinear variationallike inclusions and gave an iterative algorithm for finding its approximate solution. Fang and Huang [26], Fang et al. [8] introduced and studied a new system of variational inclusions involving Hmonotone operators and monotone operators, respectively. Yan et al. [27] introduced and studied a system of setvalued variational inclusions which is more general than the model in [3].
Inspired and motivated by recent research work in this field, in this paper, a general setvalued quasivariational inclusions system with accretive mappings is studied in Banach spaces, which includes many variational inclusions (inequalities) as special cases. By using the resolvent operator associated with accretive operator due to Lan, an existence theorem of solution for this class of variational inclusions is proved, and a new hybrid proximal point algorithm is established and suggested, and the convergence of iterative sequences generated by the algorithm is discussed in uniformly smooth Banach spaces. The results presented in this paper generalize, and unify some recent results in this field.
2. Preliminaries
Let be a real Banach space with dual space , be the dual pair between and , denote the family of all the nonempty subsets of , and denote the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by
where is a constant.
The modulus of smoothness of is the function defined by
A Banach space is called uniformly smooth if
is called uniformly smooth if there exists a constant such that
Remark 2.1.
In particular, is the usual normalized duality mapping. It is known that, for all , is singlevalued if is strictly convex [10], or is uniformly smooth (Hilbert space and space are uniformly Banach space), and if , the Hilbert space, then becomes the identity mapping on . In what follows we always denote the singlevalued generalized duality mapping by in real uniformly smooth Banach space unless otherwise states.
Let us recall the following results and concepts.
Definition 2.2.
A singlevalued mapping is said to be Lipschitz continuous if there exists a constant such that
Definition 2.3.
A singlevalued mapping is said to be
(i)accretive if
(ii)strictly accretive if is accretive and if and only if for all ;
(iii)strongly accretive if there exists a constant such that
(iv)Lipschitz continuous if there exists a constant such that
Definition 2.4.
A setvalued mapping is said to be
(i)Lipschitz continuous if there exists a constant such that
where is the Hausdorff metric on
(ii)strongly accretiveif there exists a constant such that
(iii)relaxed cocoercive if there exist two constants such that
(iv)strongly accretive with respect to the first argument of the mapping , if there exists a constant such that
where .
Definition 2.5.
Let be a singlevalued mapping and be a setvalued mapping . For , a singlevalued mapping is said to be
(i)Lipschitz continuous if there exist three constants such that
(ii)relaxed cocoercive with respect to in the first argument, if there exist constants such that
In a similar way, we can define Lipschitz continuity and relaxed cocoercive with respect to of in the second, or the three argument.
Definition 2.6.
Let and be singlevalued mappings. A setvalued mapping is said to be

(i)
accretive if
(215)
(ii)accretive if
(iii)relaxed accretive, if there exists a constant such that
(iv)accretive if is accretive and for all ;
(v)accretive if is relaxed accretive and for every .
Based on [9], we can define the resolvent operator as follows.
Definition 2.7 (see [9]).
Let be a singlevalued mapping be a strictly accretive singlevalued mapping and be a ()accretive mapping. The resolvent operator is defined by
where is a constant.
Remark 2.8.
The accretive mappings are more general than monotone mappings, accretive mappings, monotone operators, subdifferential operators, and accretive mappings in Banach space or Hilbert space, and the resolvent operators associated with accretive mappings include as special cases the corresponding resolvent operators associated with monotone operators, accretive mappings, accretive mappings, monotone operators, subdifferential operators [5, 6, 11, 14, 15, 26, 27, 35â€“37].
Lemma 2.9 (see [9]).
Let be Lipschtiz continuous mapping, be an strongly accretive mapping, and be setvalued accretive mapping, respectively. Then the generalized resolvent operator is Lipschitz continuous, that is,
where , , and .
In the study of characteristic inequalities in uniformly smooth Banach spaces , Xu [29] proved the following result.
Lemma 2.10 (see [29]).
Let be a real uniformly smooth Banach space. Then is uniformly smooth if andonly if there exists a constant such that for all
Theorem 2.11.
Let the function where and , then
Proof.
Let , where , , , . Then by the . We can obtain
Let , and , where . It follows that
This completes the proof.
Corollary 2.12.
Let be real, for any real , if , then
Proof.
The proof directly follows from the (i) in the Theorem 2.11.
Definition 2.13 (see [38]).
Let is a real be a real matrix set, then the mappings
is called the norm, and norm, respectively.
Obviously, may be a Banach space on real field , which is called the real matrixBanach space.
Definition 2.14 (see [38]).
Let is a real be a real matrixBanach Space with the matrixnorm (, or ). If
then the matrix is called the limit matrix of matrix sequence , noted by , where is a real sequence, , and , .
Lemma 2.15 (see [38]).
, if and only if
Hence, if , then
In this paper, the matrix norm symbol is noted by .
Definition 2.16.
Let be real numbers, and and be two real vectors, then if and only if .
3. QuasiVariational Inclusions System Problem and Hybrid Proximal Point Algorithm
Let be a real uniformly smooth Banach space with dual space , be the dual pair between and , denote the family of all the nonempty subsets of , and denote the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by
where is a constant. Now, we consider the following generational nonlinear setvalued quasivariational inclusions system problem with accretive mappings (GNSVQVIS) problem.
Let , , and be singlevalued mappings for . Let be a setvalued accretive mapping and be setvalued mappings for .
For any , finding such that , , , , , and
where
Remark 3.1.
Some special cases of problem (3.2) are as follows.

(i)
If , , , , and is a Hilbert space, then the problem (3.2) reduces to the problem associated with the system of variational inclusions with monotone operators, which is finding such that , , , , , , , , , and
(33)
where
Problem (3.3) contains the system of variational inclusions with monotone operators in Peng and Zhu [14], and the system of variational inclusions with monotone operators in [8] as special cases.

(ii)
If , , (Hilbert space) and, , where is a proper, subdifferentiable functional and denotes the subdifferential operator of , then problem(3.3) changes to the problem associated with the following system of variationallike inequalities, which is finding such that
(34)
where

(iii)
If , , , (Hilbert space) and , where is a proper, convex, lower semicontinuous functional and denotes the subdifferential operator of , then problem (3.3) changes to the problem associated with the following system of variational inequalities, which is finding such that
(35)
where

(iv)
If , (Hilbert space), and , where is a nonempty, closed, and convex subsets and denotes the indicator of , then problem (3.5) reduces to the problem associated with the following system of variational inequalities, which is finding such that
(36)
where

(v)
If , and is a Hilbert space, is a nonempty, closed and convex subset, for all , , where is a mapping on , is a constant, then problem (3.6) changes to the following problem: find such that
(37)
where
Moreover, if , problem (3.7) becomes the problem introduced and studied by Verma [31].
We can see that problem (3.2) includes a number of known classes of system of variational inequalities and variational inclusions as special cases (see, e.g., [2â€“9, 11â€“27, 29, 32â€“37]). It is worth noting that problems (3.2)â€“(3.7) are all new mathematical models.
Theorem 3.2.
Let be a Banach space, be Lipschtiz continuous mapping, be an strongly accretive mapping, and be a setvalued accretive mapping for . Then the following statements are mutually equivalent.
(i)An element is a solution of problem (3.2), .
(ii)For and , , , , the following relations hold:
where is a constant and , respectively.
Proof.
This directly follows from definition of and the problem (3.2) for .
Algorithm 3.3.
Let , and be three nonnegative sequences such that
Step 1.
For arbitrarily chosen initial points , , , , , , Set
where the satisfies
By using [39], we can choose suitable , , and such that
for .
Step 2.
The sequences , and are generated by an iterative procedure:
where
Thus, we can choose suitable , , and such that
for and .
Remark 3.4.
If we choose suitable some operators , and space , then Algorithm 3.3 can be degenerated to a number of known algorithms for solving the system of variational inequalities and variational inclusions (see, e.g., [2â€“9, 11â€“27, 29, 31â€“35, 38, 39]).
4. Existence and Convergence
In this section, we prove the existence of solutions for problem (3.2) and the convergence of iterative sequences generated by Algorithm 3.3.
Theorem 4.1.
Let be a uniformly smooth Banach space, be a Lipschtiz continuous mapping, and be a strongly accretive mapping and Lipschitz continuous. Let be a setvalued mappings of Lipschitz continuous with constants , and be relaxed cocoercive, respectively. Let be Lipschitz continuous with constants and for all , and relaxed cocoercive with respect to in the first, second and third arguments, respectively. Let be Lipschitz continuous with constants and for all . Let , , be some setvalued mappings such that for each given , , , and be an accretive mapping, respectively. Suppose that , and are three nonnegative sequences with
where is the same as in Lemma 2.10, , and . Then the problem (3.2) has a solution .
Proof.
Let
for . Then it follows from (3.13) that
Since is Lipschitz continuous with constants and relaxed cocoercive,
By (3.15), we have
Since and , by Lemma 2.9, we have
Since is Lipschitz continuous with constants , and relaxed cocoercive with respect to in the first arguments, be Lipschitz continuous with constants , respectively, Lemma 2.10, we have
By (3.13), we know that . Since is Lipschitz continuous with constants , and is Lipschitz continuous with constants , respectively, combing (4.4)â€“(4.8) and using Corollary 2.12, we have
and so
For the sequences , we have
Since is Lipschitz continuous with constant and relaxed cocoercive, we have
It follows from (3.15) that
Since and , by using Lemma 2.9, we obtain
Since is Lipschitz continuous with constants , and relaxed cocoercive with respect to in the first arguments, is Lipschitz continuous with constants , respectively, it follows from Lemma 2.10 that
By (3.13), we know that . Since is Lipschitz continuous with constants , and is Lipschitz continuous with constants , respectively, combing (4.11)â€“(4.20) and using Corollary 2.12, we have
and so
Using the same as the method, we can obtain
Let
Letting , then combining (4.10),(4.17)â€“(4.19), we have , where
which is called the iterative matrix for Hybrid proximal point threestep algorithm of nonlinear setvalued quasivariational inclusions system involving Accretive mappings. Using (4.20), , , , we have
where , and
By using [38], we have
Letting
It follows from (4.22) and assumption condition (4.2) that and hence there exists and such that for all . Therefore, by (4.23), we have
Without loss of generality we assume
By the property of the matrix norm [38], for , we have
Hence, for any and , we have
It follows that , as , and so that the is a Cauchy sequence in . Let as . By the Lipschitz continuity of , we can obtain
It follows that , , , and are also Cauchy sequences in . We can assume that , , , and , respectively. Noting that , we have
Hence and therefore . Similarly, we can prove that , , and . By the Lipschitz continuity of , , , and , we have
for , where is a constant. Thus, by Theorem (3.3), we know that is solution of problem (3.2). This completes the proof.
Corollary 4.2.
Let be a uniformly smooth Banach space, be a Lipschtiz continuous mapping, and be an strongly accretive mapping and Lipschitz continuous. Let be the same as in Theorem 4.1. If
where is the same as in Lemma 2.10, , and . Then problem (3.2) has a solution .
Remark 4.3.
For a suitable choice of the mappings , we can obtain several known results in [2â€“5, 9, 11â€“27, 29, 32â€“37] as special cases of Theorem 4.1 and Corollary 4.2.
References
Hassouni A, Moudafi A: A perturbed algorithm for variational inclusions. Journal of Mathematical Analysis and Applications 1994,185(3):706â€“712. 10.1006/jmaa.1994.1277
Ding XP, Luo CL: Perturbed proximal point algorithms for general quasivariationallike inclusions. Journal of Computational and Applied Mathematics 2000,113(1â€“2):153â€“165. 10.1016/S03770427(99)002502
Verma RU: monotonicity and applications to nonlinear variational inclusion problems. Journal of Applied Mathematics and Stochastic Analysis 2004, (2):193â€“195.
Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities. Computers & Mathematics with Applications 2001,41(7â€“8):1025â€“1031. 10.1016/S08981221(00)003369
Huang NJ: Nonlinear implicit quasivariational inclusions involving generalized accretive mappings. Archives of Inequalities and Applications 2004,2(4):413â€“425.
Fang YP, Cho YJ, Kin JK: accretive operators and approximating solutions for systems of variational inclusions in Banach spaces. to appear in Applied Mathematics Letters
Fang YP, Huang NJ: accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces. Applied Mathematics Letters 2004,17(6):647â€“653. 10.1016/S08939659(04)900997
Fang YP, Huang NJ, Thompson HB: A new system of variational inclusions with monotone operators in Hilbert spaces. Computers & Mathematics with Applications 2005,49(2â€“3):365â€“374. 10.1016/j.camwa.2004.04.037
Lan HY, Cho YJ, Verma RU: Nonlinear relaxed cocoercive variational inclusions involving accretive mappings in Banach spaces. Computers & Mathematics with Applications 2006,51(9â€“10):1529â€“1538. 10.1016/j.camwa.2005.11.036
Zhang QB, Ding XP, Cheng CZ: Resolvent operator technique for generalized implicit variationallike inclusion in Banach space. Journal of Mathematical Analysis and Applications 2010,361(2):283â€“292. 10.1016/j.jmaa.2006.01.090
Eckstein J, Bertsekas DP: On the DouglasRachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming 1992,55(1â€“3):293â€“318.
Verma RU: A hybrid proximal point algorithm based on the maximal monotonicity framework. Applied Mathematics Letters 2008,21(2):142â€“147. 10.1016/j.aml.2007.02.017
Shim SH, Kang SM, Huang NJ, Cho YJ: Perturbed iterative algorithms with errors for completely generalized strongly nonlinear implicit quasivariational inclusions. Journal of Inequalities and Applications 2000,5(4):381â€“395. 10.1155/S1025583400000205
Peng JW, Zhu DL: Threestep iterative algorithm for a system of setvalued variational inclusions with monotone operators. Nonlinear Analysis: Theory, Methods & Applications 2008,68(1):139â€“153. 10.1016/j.na.2006.10.037
Cohen G, Chaplais F: Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms. Journal of Optimization Theory and Applications 1988,59(3):369â€“390. 10.1007/BF00940305
Bianchi M: Pseudo Pmonotone operators and variational inequalities. Istituto di Econometria e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan, Italy; 1993.
Ansari QH, Yao JC: A fixed point theorem and its applications to a system of variational inequalities. Bulletin of the Australian Mathematical Society 1999,59(3):433â€“442. 10.1017/S0004972700033116
Ansari QH, Schaible S, Yao JC: System of vector equilibrium problems and its applications. Journal of Optimization Theory and Applications 2000,107(3):547â€“557. 10.1023/A:1026495115191
Allevi E, Gnudi A, Konnov IV: Generalized vector variational inequalities over product sets. Nonlinear Analysis: Theory, Methods & Applications 2001,47(1):573â€“582. 10.1016/S0362546X(01)002024
Kassay G, KolumbÃ¡n J: System of multivalued variational inequalities. Publicationes Mathematicae Debrecen 2000,56(1â€“2):185â€“195.
Kassay G, KolumbÃ¡n J, PÃ¡les Z: Factorization of Minty and Stampacchia variational inequality systems. European Journal of Operational Research 2002,143(2):377â€“389. 10.1016/S03772217(02)002904
Kim JK, Kim DS: A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces. Journal of Convex Analysis 2004,11(1):235â€“243.
Cho YJ, Fang YP, Huang NJ, Hwang HJ: Algorithms for systems of nonlinear variational inequalities. Journal of the Korean Mathematical Society 2004,41(3):489â€“499.
Agarwal RP, Cho YJ, Huang NJ: Sensitivity analysis for strongly nonlinear quasivariational inclusions. Applied Mathematics Letters 2000,13(6):19â€“24. 10.1016/S08939659(00)000483
Kazmi KR, Bhat MI: Iterative algorithm for a system of nonlinear variationallike inclusions. Computers& Mathematics with Applications 2004,48(12):1929â€“1935. 10.1016/j.camwa.2004.02.009
Fang YP, Huang NJ: monotone operators and system of variational inclusions. Communications on Applied Nonlinear Analysis 2004,11(1):93â€“101.
Yan WY, Fang YP, Huang NJ: A new system of setvalued variational inclusions with monotone operators. Mathematical Inequalities & Applications 2005,8(3):537â€“546.
Zou YZ, Huang NJ: accretive operator with an application for solving variational inclusions in Banach spaces. Applied Mathematics and Computation 2008,204(2):809â€“816. 10.1016/j.amc.2008.07.024
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1127â€“1138. 10.1016/0362546X(91)90200K
Zou YZ, Huang NJ: A new system of variational inclusions involving accretive operator in Banach spaces. Applied Mathematics and Computation 2009,212(1):135â€“144. 10.1016/j.amc.2009.02.007
Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. Journal of Optimization Theory and Applications 2004,121(1):203â€“210.
Chang SS, Cho YJ, Zhou H: Iterative Methods for Nonlinear Operator Equations in Banach Spaces. Nova Science, Huntington, NY, USA; 2002:xiv+459.
Weng X: Fixed point iteration for local strictly pseudocontractive mapping. Proceedings of the American Mathematical Society 1991,113(3):727â€“731. 10.1090/S00029939199110863458
Agarwal RP, Huang NJ, Tan MY: Sensitivity analysis for a new system of generalized nonlinear mixed quasivariational inclusions. Applied Mathematics Letters 2004,17(3):345â€“352. 10.1016/S08939659(04)900730
Huang NJ, Fang YP: A new class of general variational inclusions involving maximal monotone mappings. Publicationes Mathematicae Debrecen 2003,62(1â€“2):83â€“98.
Jin MM: Perturbed algorithm and stability for strongly nonlinear quasivariational inclusion involving accretive operators. Mathematical Inequalities & Applications 2006,9(4):771â€“779.
Peng J, Yang X: On existence of a solution for the system of generalized vector quasiequilibrium problems with upper semicontinuous setvalued maps. International Journal of Mathematics and Mathematical Sciences 2005, (15):2409â€“2420.
Horn RA, Johnson CR: Matrix Analysis. Cambridge University Press, Cambridge, UK; 1985:xiii+561.
Nadler SB Jr.: Multivalued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475â€“488.
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The authors acknowledge the support of the Educational Science Foundation of Chongqing, Chongqing (KJ091315).
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Li, HG., Xu, A. & Jin, M. A Hybrid Proximal Point ThreeStep Algorithm for Nonlinear SetValued QuasiVariational Inclusions System Involving Accretive Mappings. Fixed Point Theory Appl 2010, 635382 (2010). https://doi.org/10.1155/2010/635382
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DOI: https://doi.org/10.1155/2010/635382