Iterative Schemes for Finite Families of Maximal Monotone Operators Based on Resolvents

Abstract

The purpose of this paper is to present two iterative schemes based on the relative resolvent and the generalized resolvent, respectively. And, it is shown that the iterative schemes converge weakly to common solutions for two finite families of maximal monotone operators in a real smooth and uniformly convex Banach space and one example is demonstrated to explain that some assumptions in the main results are meaningful, which extend the corresponding works by some authors.

1. Introduction and Preliminaries

Let be a real Banach space with norm and let denote the dual space of . We use “” and “” to denote strong and weak convergence either in or , respectively. A Banach space is said to be strictly convex if Also, is said to be uniformly convex if, for each , there exists such that A Banach space is said to be smooth if exists for each . In this case, the norm of is said to be Gâteaux differentiable. The space is said to have a uniformly Gâteaux differentiable norm if, for each , the limit (3) is attained uniformly for . The norm of is said to be Frêchet differentiable if, for each , the limit (3) is attained uniformly for . The norm of is said to be uniformly Frêchet differentiable if the limit (3) is attained uniformly for .

The normalized duality mapping is defined by

We call that is weakly sequentially continuous if is a sequence in which converges weakly to it follows that converges in weak^* to .

We know the following properties of (see [1] for details):(i) for each ;(ii)if is smooth, then is single-valued and strictly monotone;(iii)if is strictly convex, then is one to one; that is, ;(iv)if has a uniformly Gâteaux differentiable norm, then is norm to weak^* uniformly continuous on each bounded subset of ;(v)if is a smooth and uniformly convex Banach space, then is also a duality mapping and is uniformly continuous on each bounded subset of .

An operator is said to be monotone if , for . A monotone operator is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. If is maximal monotone, then the set is closed and convex: moreover, if is a real smooth and uniformly convex Banach space, then is demiclosed; that is, ,  , ,  , , and . If is reflexive and strictly convex, then a monotone operator is maximal if and only if , for each (see [2] for more details).

A mapping is said to be accretive (c.f. [3]) if , for , and . In a Hilbert space , the -accretive mapping is exactly the maximal monotone operator.

The Lyapunov functional is defined as follows: It is obvious from the definition of Lyapunov functional that for each .

We have the following well-known result.

Lemma 1 (see [4]). Let be a real smooth and uniformly convex Banach space, and let and be two sequences in . If either or is bounded and , then ,  .

Definition 2 (see [5]). Let be a real smooth and uniformly convex Banach space and let be a maximal monotone operator. Then , define by , which is called the relative resolvent.

We have the following property of the relative resolvent.

Lemma 3 (see [5]). Let be a real reflexive, strictly convex, and smooth Banach space and let be a maximal monotone operator such that . Then , and , we have .

Definition 4 (see [4]). Let be a real reflexive, strictly convex, and smooth Banach space and let be a nonempty closed and convex subset of . Then , there exists a unique element satisfying . In this case, , define by , and then is called the generalized projection from onto .

Lemma 5 (see [4]). Let be a real reflexive, strictly convex, and smooth Banach space and let be a nonempty closed and convex subset of . Then ,

Lemma 6 (see [4]). Let be a real smooth Banach space and let be a nonempty closed and convex subset of . Let , and . Then if and only if .

Let be a smooth Banach space and let be a nonempty closed and convex subset of . A mapping is said to be generalized nonexpansive (c.f. [5]) if and , for and , where is a set of the fixed points of ; that is, .

Let be a nonempty, closed subset of and let be a mapping of onto . Then is said to be sunny (c.f. [5]) if , for all and . A mapping is said to be a retraction (c.f. [5]) if for every . If is smooth and strictly convex, then a sunny generalized nonexpansive retraction of onto is uniquely decided (c.f. [5]). Then, if is smooth and strictly convex, a sunny generalized nonexpansive retraction of onto is denoted by .

A subset of is said to be a sunny nonexpansive retract of (c.f. [5]) if there exists a sunny nonexpansive retraction of onto and it is called a generalized nonexpansive retract of if there exists a generalized nonexpansive retraction of onto .

Definition 7 (see [5]). Let be a real reflexive, strictly convex, and smooth Banach space and let be a maximal monotone operator. Then , define by , which is called the generalized resolvent.

Lemma 8 (see [5]). Let be a real reflexive and strictly Banach space with a Frêchet differential norm and let be a maximal monotone operator with . Then (i);  (ii)    is closed; (iii) is generalized nonexpansive, for .

Lemma 9 (see [5]). Let be a real reflexive, smooth, and strictly Banach space and let be a maximal monotone operator with . Then

Lemma 10 (see [6]). Let and be two sequences of nonnegative real numbers and for . If , then exists.

Finding zeros of maximal monotone operators is a hot topic in applied mathematics since it has practical background. One classical method for studying the problem , where is a maximal monotone operator, is the following so-called proximal method (c.f. [7]), presented in a Hilbert space: where . It was shown that the sequence generated by (9) converges weakly to a point in under some conditions.

In 2004, Kamimura et al. extended the study on zeros of maximal monotone operators to the following iterative scheme based on the relative resolvent in Banach spaces (c.f. [8]): And, they showed that generated by (10) converges weakly to a point in , where is a maximal monotone operator.

In 2007, Ibaraki and Takahashi [9] studied the following iterative scheme based on the generalized resolvent in Banach spaces: And, they showed that generated by (11) converges weakly to a point in , where is a maximal monotone operator.

In 2010, Shehu and Ezeora, [10] presented the following iterative scheme for a family of -accretive mappings in a real uniformly smooth and uniformly convex Banach space : where with , for .  , for , and . Then converges strongly to the common point in , where .

Can we extend the study on -accretive mappings [10] to maximal monotone operators? Inspired by the work on (10)–(12), in Section 2, we will present the following iterative scheme based on the relative resolvent: where are maximal monotone operators, ,  . Suppose .  , and and , for and .  , and and , for and .   are real numbers in with and .

In Section 3, we will study the following iterative scheme based on generalized resolvent: where are maximal monotone operators, ,  . Suppose .  , and . For ,  . For ,  .   and are real numbers in and ,.  , for , and , for and .

In this paper, some weak convergence theorems are obtained, which can be regarded as the extension and complement of the work done in [7–10], and so forth. At the end of Section 3, one example is demonstrated to show that the assumption that in the discussions of and is meaningful.

2. Weak Convergence Theorems Based on the Relative Resolvent

Theorem 11. Let be a real smooth and uniformly convex Banach space. Let be maximal monotone operators, where . Suppose that both and are weakly sequentially continuous and . Let be generated by the iterative scheme , where , and , for ,  , for ,  .  , where , for ,  , for ,  . Suppose ,  , and are three sequences in and , satisfying the following conditions:(i), as ;(ii); (iii) and ;(iv) and , for and .

Then converges weakly to the unique element which satisfies

Proof. We will split the proof into six steps.
Step 1. is bounded.
For , noticing the definition of the Lyapunov functional and by using Lemma 3 repeatedly, we have
Lemma 10 ensures that exists, which implies that is bounded in view of (6).
Then from iterative scheme , is bounded. Since , for , then is bounded, which ensures that is bounded. For ,  , then we know that is bounded.
Step 2. , where is the set of the weak limit points of all of the weakly convergent subsequences of .
Since is bounded, then . And, there exists a subsequence of ; for simplicity, we still denote it by such that .
For , using Lemma 3 again, we have the following: Then (15) implies that
Since exists and is bounded, then, using Lemma 1, we know that as . Revise (14) in the following way: Then repeating the above process, we have as . Similarly, we have
On the other hand, noticing (14) and using Lemma 3, we have
Similar to the discussion of (20), we have as ,  .
Since both and are weakly sequentially continuous, , and , then , as . Now, (22) implies that . If we set , then from (22) and the fact that is uniformly norm to norm continuous on each bounded subset of , we have , as , for . Since is demiclosed, then .
Now, from , we have , and then , which implies that , as , since . Thus (20) implies that . In the same way as the proof of , we have .
From the fact that and (20), we have , and , as . Then, if we set , we have , which ensures that . By induction, using (20) repeatedly, we know that .
Therefore, , and then .
Step 3. There exists a unique element such that
In fact, let ,  . Then is proper, convex, and lower-semicontinuous and , as . Thus there exists such that . Since is strictly convex, then is unique.
Step 4. exists.
From the definition of , we have .
Using (14), we have
Thus
Then Lemma 10 ensures that exists.
Step 5. , where is the same as that in Step 3.
From Lemma 5, we have . Thus
Therefore, Lemma 1 implies that , as .
Step 6. where is the same as that in Step 3.
From Lemma 6, we know that, Since is weakly sequentially continuous, then from Step 5, we have , as .
Since is bounded, then there exists a subsequence of such that , as . From Step 2, . And, , as . Substituting by in (27) and taking limits on both sides, we have
Letting in (28), then , which implies that , since is strictly monotone.
Suppose there exists another subsequence of such that , as . Then and , as . Repeating the above process, we know that . Therefore, all of the weakly convergent subsequences of converge weakly to the same element , and then which satisfies (13), as .
This completes the proof.

If, in Theorem 11, the Banach space reduces to the Hilbert space , then we have the following theorem.

Theorem 12. Let be a Hilbert space and let be the same as that in Theorem 11. Let be m-accretive mappings, where . Let and be the same as those in Theorem 11. Let ,   and satisfy some conditions presented in Theorem 11.

Let be generated by the following scheme: where and . Then converges weakly to the unique element , where and is the metric projection from onto .

Remark 13. Compared to the work in [10], we may find that Theorem 11 is not a simple extension from the case of -accretive mappings to maximal monotone operators. In , different and have different coefficients while in (12), different have the same coefficients.

3. Weak Convergence Theorems Based on the Generalized Resolvent

Theorem 14. Let be a real smooth and uniformly convex Banach space. Let be maximal monotone operators, where . Suppose that both and are weakly sequentially continuous and . Let be generated by the iterative scheme , where , and , for ,  , for ,  .  , where ,  for ,  , for ,  . Suppose ,, and are three sequences in and , satisfy the following conditions:(i), as ;(ii); (iii)   ;(iv) and , for and .
Then converges weakly to the unique element , where

Proof. We will split the proof into four steps.
Step 1. .
Since , then we may choose , which implies that and , for ; . Thus and , for ; . And then and , for ; . Therefore, which implies that .
Step 2. is bounded.
For , noticing the definition of the Lyapunov functional and by using Lemma 9 repeatedly, we have Lemma 10 ensures that exists, which ensures that is bounded.
Step 3. , where is the set of weak limit points of all of the weakly convergent subsequences of .
Since is bounded, then . So there exists a subsequence of ; for simplicity, we still denote it by such that .
Using Lemma 9 again, we have for
Then (31) implies that
Similar to the discussion of (17) in Step 2 in Theorem 11, we have as .
Then, similar to the discussions of (19) and (20), we have
On the other hand, noticing (30) and using Lemma 9, we have which implies that as ,  .
Since and , then , as . Now, (36) implies that . Let , then Since is weakly sequentially continuous, then, letting , (37) ensures that Since is maximal monotone, then , which implies that .
From , we have , as . Since , then , as . Using (34), we have , as . By induction, similar to the proof of , we know that .
Therefore, , and then .
Step 4. , as , where is the unique element in .
From Steps 2 and 3, we know that there exists a subsequence of such that , as . If there exists another subsequence of such that , as , then from Step 1, we know that Similarly,
From (39) and (40), we have , which implies that .
This completes the proof.

Remark 15. If, in Theorem 14, the Banach space reduces to the Hilbert space , then the result of Theorem 12 is still true. That is, Theorems 14 and 11 are the same in the frame of Hilbert spaces.

Remark 16. Next, we will present an example to show that the assumptions that in Theorem 11 and in Theorem 14 are meaningful.
Consider the following Laplacian Dirichlet boundary value problem: where is a bounded conical domain of the Euclidean space with its boundary ,  ,  , and are given functions, where is the trace space of ,  is the trace operator, .

Similar to the discussion in [11], we have the following results.

Proposition 17 (see [11]). Define the mapping by for . Define the mapping by for .
Then both and are maximal monotone, for each .

Proposition 18 (see [11]). For and , nonlinear boundary problem has a unique solution , for each . Moreover, if and only if is the solution of , which ensures that , for each .

We can easily get the following result.

Proposition 19. , for each , which ensures that . And, , which means the following nonlinear Laplacian elliptic systems with Dirichlet boundary share the same solution:

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 11071053), the Natural Science Foundation of Hebei Province (no. A2014207010), the Key Project of Science and Research of Hebei Educational Department (ZH2012080), and the Key Project of Science and Research of Hebei University of Economics and Business (2013KYZ01).