A Hybrid Proximal Point Three-Step Algorithm for Nonlinear Set-Valued Quasi-Variational Inclusions System Involving -Accretive Mappings
© Hong-Gang Li et al. 2010
Received: 29 December 2009
Accepted: 7 April 2010
Published: 16 May 2010
The main purpose of this paper is to introduce and study a new class of generalized nonlinear set-valued quasi-variational inclusions system involving -accretive mappings in Banach spaces. By using the resolvent operator due to Lan-Cho-Verma associated with -accretive mappings and the matrix analysis method, we prove the convergence of a new hybrid proximal point three-step iterative algorithm for this system of set-valued 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.
The variational inclusion, which was introduced and studied by Hassouni and Moudafi , 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 , Verma [3, 4], Huang , Fang et al. , Fang and Huang , Fang et al. , Lan et al. , Zhang et al.  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 Eckstein-Bertsekas  proximal point algorithm, introduced the algorithm based on the -maximal monotonicity framework , 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 [1–37] 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 , Cohen and Chaplais , Bianchi , and Ansari and Yao  considered a system of scalar variational inequalities. Ansari et al.  introduced and studied a system of vector equilibrium problems and a system of vector variational inequalities using a fixed point theorem. Allevi et al.  considered a system of generalized vector variational inequalities and established some existence results with relative pseudomonotonicity. Kassay and Kolumbán  introduced a system of variational inequalities and proved an existence theorem through the Ky Fan lemma. Kassay et al.  studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. J. K. Kim and D. S. Kim  introduced a new system of generalized nonlinear quasi-variational inequalities and obtained some existence and uniqueness results on solutions for this system of generalized nonlinear quasi-variational inequalities in Hilbert spaces. Cho et al.  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.  introduced a system of generalized nonlinear mixed quasi-variational inclusions and investigated the sensitivity analysis of solutions for this system of generalized nonlinear mixed quasi-variational inclusions in Hilbert spaces. Kazmi and Bhat  introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. Fang and Huang , Fang et al.  introduced and studied a new system of variational inclusions involving H-monotone operators and -monotone operators, respectively. Yan et al.  introduced and studied a system of set-valued variational inclusions which is more general than the model in .
Inspired and motivated by recent research work in this field, in this paper, a general set-valued quasi-variational 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.
where is a constant.
In particular, is the usual normalized duality mapping. It is known that, for all , is single-valued if is strictly convex , 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 single-valued generalized duality mapping by in real uniformly smooth Banach space unless otherwise states.
Let us recall the following results and concepts.
A single-valued mapping is said to be
(ii)strictly accretive if is accretive and if and only if for all ;
A set-valued mapping is said to be
where is the Hausdorff metric on
Let be a single-valued mapping and be a set-valued mapping . For , a single-valued mapping is said to be
In a similar way, we can define Lipschitz continuity and -relaxed cocoercive with respect to of in the second, or the three argument.
- (i)accretive if(215)
(iv) -accretive if is accretive and for all ;
(v) -accretive if is -relaxed -accretive and for every .
Based on , we can define the resolvent operator as follows.
Definition 2.7 (see ).
where is a constant.
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, 35–37].
Lemma 2.9 (see ).
where , , and .
In the study of characteristic inequalities in -uniformly smooth Banach spaces , Xu  proved the following result.
Lemma 2.10 (see ).
This completes the proof.
The proof directly follows from the (i) in the Theorem 2.11.
Definition 2.13 (see ).
is called the -norm, and -norm, respectively.
Obviously, may be a Banach space on real field , which is called the real matrix-Banach space.
Definition 2.14 (see ).
then the matrix is called the limit matrix of matrix sequence , noted by , where is a real sequence, , and , .
Lemma 2.15 (see ).
Hence, if , then
In this paper, the matrix norm symbol is noted by .
Let be real numbers, and and be two real vectors, then if and only if .
3. Quasi-Variational Inclusions System Problem and Hybrid Proximal Point Algorithm
where is a constant. Now, we consider the following generational nonlinear set-valued quasi-variational inclusions system problem with -accretive mappings (GNSVQVIS) problem.
Let , , and be single-valued mappings for . Let be a set-valued -accretive mapping and be set-valued mappings for .
- (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)
- (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 variational-like inequalities, which is finding such that(34)
- (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)
- (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)
- (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)
Moreover, if , problem (3.7) becomes the problem introduced and studied by Verma .
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., [2–9, 11–27, 29, 32–37]). It is worth noting that problems (3.2)–(3.7) are all new mathematical models.
Let be a Banach space, be -Lipschtiz continuous mapping, be an -strongly -accretive mapping, and be a set-valued -accretive mapping for . Then the following statements are mutually equivalent.
(i)An element is a solution of problem (3.2), .
where is a constant and , respectively.
This directly follows from definition of and the problem (3.2) for .
for and .
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., [2–9, 11–27, 29, 31–35, 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.
where is the same as in Lemma 2.10, , and . Then the problem (3.2) has a solution .
for , where is a constant. Thus, by Theorem (3.3), we know that is solution of problem (3.2). This completes the proof.
where is the same as in Lemma 2.10, , and . Then problem (3.2) has a solution .
The authors acknowledge the support of the Educational Science Foundation of Chongqing, Chongqing (KJ091315).
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