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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 185104, 17 pages
http://dx.doi.org/10.1155/2012/185104
Research Article

Convergence of an Iterative Algorithm for Common Solutions for Zeros of Maximal Accretive Operator with Applications

1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2Department of Mathematics Education and RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea

Received 18 November 2011; Accepted 30 December 2011

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Uamporn Witthayarat et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is to introduce an iterative algorithm for finding a common solution of the sets (0) and (0), where M is a maximal accretive operator in a Banach space and, by using the proposed algorithm, to establish some strong convergence theorems for common solutions of the two sets above in a uniformly convex and 2-uniformly smooth Banach space. The results obtained in this paper extend and improve the corresponding results of Qin et al. 2011 from Hilbert spaces to Banach spaces and Petrot et al. 2011. Moreover, we also apply our results to some applications for solving convex feasibility problems.

1. Introduction

Let be a real Banach space with norm with the dual space and let denote the pairing between and . Let be a nonempty closed convex subset of . We define the generalized duality mapping by for all . In the special case, for , we called the mapping as the normalized duality mapping and as usual we write . The following is the well-known properties of the generalized duality mapping :(1) for all with ;(2) for all and ;(3) for all .

It is well known that if is smooth, then is single valued, which is denoted by . Recall that the duality mapping is said to be weakly sequentially continuous if, for each sequence with weakly, we have weakly*. We know that, if admits a weakly sequentially continuous duality mapping, then is smooth. For the details, see [13].

Let . A Banach space is said to be(1)uniformly convex if there exists such that, for any and, for any , implies .

We can see that every uniformly convex Banach space is also reflexive and strictly convex.(2)Smooth if exists for all .(3)Uniformly smooth if the limit is attained uniformly for . The modulus of smoothness of is defined by where is a function. In the other way, is uniformly smooth if and only if .(4)-uniformly smooth if there exists a constant such that for all where is a fixed real number with . (see, for instance, [1, 4]).

We note that is a uniformly smooth Banach space if and only if is single valued and uniformly continuous on any bounded subset of . Examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is min-uniformly smooth for any . Note also that no Banach space is -uniformly smooth for (see [1, 5] for more details).

Let be a nonlinear mapping. The mapping is said to be(1)accretive if (2)-strongly accretive if there exists a constant such that (3)-inverse-strongly accretive if there exists a constant such that

Definition 1.1. Let be a multivalued maximal accretive mapping. The single-valued mapping defined by is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Let be a mapping from into itself. We use to denote the set of fixed points of the mapping . Recall that the mapping is said to be nonexpansive if

A mapping is said to be contractive if there exists a constant such that

Recently, Aoyama et al. [4] considered the following generalized variational inequality problem in a smooth Banach space: Find a point such that where is an accretive operator of into . This problem is related to the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator, and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Agarwal et al. [6], Cho et al. [7, 8], Kamimura and Takahashi [9, 10], Qin et al. [11], Song et al. [12], and Wei and Cho [13]. In order to find a solution of the variational inequality (1.9), Aoyama et al. [4] studied the weak convergence theorem for accretive operators in Banach spaces, which is a generalization of the result by Iiduka et al. [14] from the class of Hilbert spaces.

Theorem AIT (see [4], Aoyama et al. Theorem  3.1). Let be a uniformly convex and 2-uniformly smooth Banach space and let be a nonempty closed convex subset of . Let be a sunny nonexpansive retraction from onto , and be an -inverse strongly accretive operator of into with , where If and are chosen such that for some and for some with , then the sequence defined by the following manners: and converges weakly to an element of , where is the 2-uniformly smoothness constant of and is a sunny nonexpansive retraction.

In 2011, Katchang and Kumam [15] presented an iterative algorithm for finding a common solution of fixed point problems and a general system of variational inequality problems for two accretive operators as shown in the following: for all , They proved that the sequence generated by the above algorithm converges strongly to a point . Moreover, they apply their theorem to find zeros of accretive operators and the class of -strictly pseudocontractive mappings.

Recently, Petrot et al. [16] considered the problem so-called quasivariational inclusion problem, that is, determine an element such that where is a single-valued nonlinear mapping and is a multivalued mapping. The set of solutions of the above problem is denoted by . Therefore, they presented a new iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inclusion problem with a multivalued maximal monotone mapping and an -inverse-strongly monotone mapping by using the iterative sequence defined as follows: and, under appropriated conditions, they proved the the sequence generated by (1.14) converges strongly to a point , which is the unique solution in to the following variational inequality:

Very recently, Qin et al. [17] introduced an iterative scheme for a general variational inequality () and proved the strong convergence theorems of common solutions of two variational inequalities in a uniformly convex and 2-uniformly smooth Banach space by using the following iterative sequence : They proved that the sequence generated by the above algorithm converges strongly to a point , where is the unique sunny nonexpansive retraction from onto VI.

Motivated and inspired by the above recent works, in this paper, we introduce an iterative scheme for finding zeros of maximal accretive operators. Furthermore, we prove some strong convergence theorems and also propose applications for solving the convex feasibility problems. Our results improve and extend the corresponding results of Qin et al. [17] and Katchang and Kumam [15], Petrot et al. [16], and many others.

2. Preliminaries

Note that, if and are nonempty subsets of a Banach space such that is a subset of a closed convex subset and . Then is said to be sunny if whenever for any and . A subset of is said to be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto . A mapping is called a retraction if . If a mapping is a retraction, then for all is in the range of (see [4, 18] for more details).

The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 2.1 (see [19]). Let be a smooth Banach space and let be a nonempty subset of . Let be a retraction and let be the normalized duality mapping on . Then the following are equivalent:(1) is sunny and nonexpansive;(2) for all ;(3) for all and .

Proposition 2.2 (see [20]). Let be a nonempty closed convex subset of a uniformly convex and let uniformly smooth Banach space and be a nonexpansive mapping of into itself with . Then the set is a sunny nonexpansive retract of .

We need the following lemmas in order to prove our main results.

Lemma 2.3 (see [5]). Let be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:

Lemma 2.4 (see [21]). Let and be bounded sequences in a Banach space and be a sequence in with Suppose that for all and Then .

Lemma 2.5 (see [22]). Assume that is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that(1);(2) or . Then .

Lemma 2.6 (see [23]). Let be a closed convex subset of a strictly convex Banach space . Let be a nonexpansive mappings for each , where is some integer. Suppose that is nonempty. Let be a sequence of positive numbers with . Then the mapping defined by is well defined, nonexpansive, and holds.

Lemma 2.7 (see [24]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and be nonexpansive mapping of into itself. If is a sequence in such that weakly and strongly, then is a fixed point of .

Lemma 2.8 (see [3, 4]). Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space . Let a mapping be -inverse-strongly accretive. Then one has
If , then is nonexpansive.

Proof. For any , it follows from Lemma 2.3 that If , then is nonexpansive. This completes the proof.

Lemma 2.9. Let be a nonempty subset of a Banach space . Let be a mapping of into , be a maximal accretive operator on and be the resolvent of for any . Then for all .

Proof. Let be fixed. Then we have This completes the proof.

Lemma 2.10. Let be a Banach space. Then for all ,

3. Main Results

In this section, we prove strong convergence theorems for a -inverse-strongly accretive mapping and a -inverse-strongly accretive in a real 2-uniformly smooth Banach space .

In order to prove our main results, we need the following lemma.

Lemma 3.1. Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space with the best smooth constant . Let be a resolvent operator associated with , where is a multivalued maximal accretive mapping. Let the mappings be -inverse-strongly accretive and -inverse-strongly accretive, respectively. Let be a mapping defined by If and , then is nonexpansive.

Proof. Since and are nonexpansive, for any , it follows from Lemma 2.8 that Therefore, is nonexpansive. This completes the proof.

Next, we state the main result of this work.

Theorem 3.2. Let be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and be a nonempty closed convex subset of . Let be -inverse-strongly accretive and -inverse-strongly accretive, respectively, and be the best smooth constant. Let be a contraction of into itself with coefficient . Suppose that and is a mapping defined by Lemma 3.1. Let , be any positive real numbers such that and . For arbitrary , define the iterative sequence as follows: where the sequences , , and in satisfy the following conditions:(C1);(C2) and ;(C3);(C4). Then the sequence generated by (3.3) converges strongly to a point , where is a sunny nonexpansive retraction on .

Proof. First, we prove that and are nonexpansive mappings. Consider the following: Thus, it follows that is nonexpansive and so is .Step 1. We show that is bounded. For any , we have It follows by induction that Thus the sequence is bounded and so is .Step 2. We show that . Let and for each . Then we have and so where ia an appropriate constant such that .
Next, let for all . Then we have for all . Now, we compute and so Substituting (3.8) into (3.10), we get that is, From the conditions (C2) and (C3), it follows that Thus, from Lemma 2.4, it follows that From the definition of in this step, we observe that . Then we have
Step 3. We show that , where . Define a mapping by Lemma 3.1 Then, it follows that is a nonexpansive mapping such that Consider the following: From the condition (C4), we have Next, we consider Therefore, we have From the conditions (C2), (C3), (3.15), (3.18), and the inequality above, we obtain Thus, since is a contraction, there exists a unique fixed point. We denote that is the unique fixed point to the mapping which means that .
Since is bounded, there exists a subsequence of such that , it follows from (3.21) that Since is nonexpansive, it follows from Lemma  2.7 that we obtain that . By (3.22), we have .
Furthermore, with the reason that is bounded, we can choose the sequence of which such that Now, from (3.23) and Proposition 2.1(3) and the weakly sequential continuity of the duality mapping , we have From (3.15), it follows that
Step 4. We show that converges strongly to a point . In fact, observe that Thus it follows that Therefore, from Condition (C2), (3.25), and Lemma 2.5, we get as . This completes the proof.

Corollary 3.3. Let be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let be a nonempty closed convex subset of . Let be -inverse-strongly accretive and -inverse-strongly accretive, respectively, and be the best smooth constant. Suppose that , where is a mapping defined by Lemma 3.1. Let , be any positive real numbers such that and . For arbitrary , define the iterative sequence by where the sequences , , and in satisfy the following conditions:(C1);(C2) and ;(C3);(C4). Then the sequence generated by (3.28) converges strongly to a point , where is a sunny nonexpansive retraction on .

Proof. Take for all for any fixed in (3.3). Then, by Theorem 3.2, we can conclude the desired conclusion easily.

4. Applications

4.1. Application to Convex Feasibility Problems

In this part, we consider the following convex feasibility problem (CFP): find , where and denotes the set of zeros of a maximal accretive operator.

The following result can be obtained from Theorem 3.2.

Theorem 4.1. Let be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and let be a nonempty closed convex subset of . Let be an -inverse-strongly accretive and be the best smooth constant. Let be a contraction of into itself with coefficient . Suppose that , where is a mapping defined by Lemma 3.1. Let be any positive real numbers such that , . For arbitrary , define the iterative sequence by where the sequences , , and in satisfy the following conditions:(C1) and ;(C2) and ;(C3);(C4). Then the sequence generated by (4.1) converges strongly to a point , where is a sunny nonexpansive retraction on .

4.2. Application to Hilbert Spaces

Assume that is a real Hilbert space with inner product and norm . Let be a single-valued nonlinear mapping and be a multivalued mapping. The problem of finding such that is called the quasivariational inclusion problem, and we denote the set of solutions of the above variational inclusion by .

If , where is a nonempty closed convex subset of and is the indicator function of , that is, Then the variational inclusion problem (4.2) is equivalent to the problem of finding such that which is the well-known Hartman-Stampacchia variational inequality problem [25].

Theorem 4.2. Let be a closed convex subset of a real Hilbert space . Let be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be a contraction of into itself with coefficient . Suppose that , where and are the sets of solutions of variational inequality (4.4). For arbitrary , define the iterative sequence by where the sequences , , and in satisfy the following conditions:(C1);(C2) and ;(C3);(C4). Then the sequence generated by (4.5) converges strongly to a point .

Proof. Take , where is the indicator function of . Let . Then we get Thus the conclusion can be obtained from Theorem 3.2 immediately.

Acknowledgments

This paper was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the project NRU-CSEC no. 54000267) for financial support during the preparation of this paper. Furthermore, the first author would like to thank the Office of the Higher Education Commission, Thailand, for the financial support of the Ph.D. program at KMUTT. This research was partially finished at Department of Mathematics Education, Gyeongsang National University, Republic of Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (Grant no. 2011-0021821).

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