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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 232170, 7 pages
http://dx.doi.org/10.1155/2013/232170
Research Article

Proximal Point Algorithms for Finding a Zero of a Finite Sum of Monotone Mappings in Banach Spaces

1Departement of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 15 February 2013; Revised 30 March 2013; Accepted 31 March 2013

Academic Editor: Yisheng Song

Copyright © 2013 H. Zegeye and N. Shahzad. 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

We introduce an iterative process which converges strongly to a zero of a finite sum of monotone mappings under certain conditions. Applications to a convex minimization problem are included. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.

1. Introduction

Let be a nonempty subset of a real Banach space with dual . A mapping is said to be monotone if for each , the following inequality holds: A monotone mapping is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. We know that if is maximal monotone mapping, then is closed and convex (see [1] for more details).

Monotone mappings were introduced by Zarantonello [2], Minty [3], and Kačurovskiĭ [4]. The notion of monotone in the context of variational methods for nonlinear operator equations was also used by Vaĭnberg and Kačurovskiĭ [5]. The central problem is to iteratively find a zero of a finite sum of monotone mappings in a Banach space , namely, a solution to the inclusion problem It is known that many physically significant problems can be formulated as problems of the type (2). For instance, a stationary solution to the initial value problem of the evolution equation can be formulated as (2) when the governing maximal monotone is of the form (see, e.g., [6]). In addition, optimization problems often need [7] to solve a minimization problem of the form where , are proper lower semicontinuous convex functions from to the extended real line . If in (2), we assume that , for , where is the subdifferential operator of in the sense of convex analysis, then (4) is equivalent to (2). Consequently, considerable research efforts have been devoted to methods of finding approximate solutions (when they exist) of equations of the form (2) for a sum of a finite number of monotone mappings (see, e.g., [6, 812]).

A well-known method for solving the equation in a Hilbert space is the proximal point algorithm: and where and for all . This algorithm was first introduced by Martinet [10]. In 1976, Rockafellar [11] proved that if and , then the sequence defined by (5) converges weakly to an element of . Later, many researchers have studied the convergence of the sequence defined by (5) in Hilbert spaces; see, for instance, [8, 1218] and the references therein.

In 2000, Kamimura and Takahashi [9] proved that for a maximal monotone mapping in a Hilbert spaces and for all , the sequence defined by where and satisfy certain conditions, called Halpern type, converges strongly to a point in .

In a reflexive Banach space and for a maximal monotone mapping , Reich and Sabach [19] proved that the sequence defined by where and is the Bergman projection of on to a closed and convex subset induced by a well-chosen convex function , converges strongly to a point in .

Furthermore, many authors (see, e.g., [12, 2025]) have studied strong convergence of an iterative process of Halpern type or proximal type to a common zero of a finite family of maximal monotone mappings in Hilbert spaces (or in Banach spaces).

Regarding iterative solution of a zero of sum of two maximal monotone mappings, Lions and Mercier [6] introduced the nonlinear Douglas-Rachford splitting iterative algorithm which generates a sequence by the recursion where denotes the resolvent of a monotone mapping ; that is, . They proved that the nonlinear Douglas-Rachford algorithm (8) converges weakly to a point , a solution of the inclusion, for maximal monotone mappings in Hilbert spaces.

A natural question arises whether we can obtain an iterative scheme which converges strongly to a zero of sum of a finite number of monotone mappings in Banach spaces or not?

Motivated and inspired by the work mentioned above, it is our purpose in this paper to introduce an iterative scheme (see (21)) which converges strongly to a zero of a finite sum of monotone mappings under certain conditions. Applications to a convex minimization problem are included. Our theorems improve the results of Lions and Mercier [6] and most of the results that have been proved in this direction.

2. Preliminaries

Let be a Banach space and let . Then, a Banach space is said to be smooth provided that the limit exists for each . The norm of is said to be uniformly smooth if the limit (10) is attained uniformly for in (see [1]).

The modulus of convexity of is the function defined by is called uniformly convex if and only if , for every (see [26]).

Lemma 1 (see [27]). Let be a smooth, strictly convex, and reflexive Banach space. Let be a nonempty closed convex subset of , and let be a monotone mapping. Then, is maximal if and only if , for all , where is the normalized duality mapping from into defined, for each , by where denotes the generalized duality pairing between members of and . We recall that is smooth if and only if is single valued (see [1]). If , a Hilbert space, then the duality mapping becomes the identity map on .

Lemma 2 (see [27]). Let be a reflexive with as its dual. Let , and let be maximal monotone mappings. Suppose that . Then, is a maximal monotone mapping.

Lemma 3 (see [28]). Let be a reflexive with as its dual. Let be maximal monotone mapping, and let be monotone mappings such that , is hemicontinuous (i.e., continuous from the segments in to the weak star topology in ) and carries bounded sets into bounded sets. Then, is maximal monotone mapping.

Let be a smooth Banach space with dual . Let the Lyapunov function , introduced by Alber [29], be defined by where is the normalized duality mapping from into . If , a Hilbert space, then (13) reduces to , for .

Let be a reflexive, strictly convex, and smooth Banach space, and let be a nonempty closed and convex subset of . The generalized projection mapping, introduced by Alber [29], is a mapping that assigns an arbitrary point to the minimizer, , of over ; that is, , where is the solution to the minimization problem We know the following lemmas.

Lemma 4 (see [23]). Let be a real smooth and uniformly convex Banach space, and let and be two sequences of . If either or is bounded and , as , then , as .

Lemma 5 (see [29]). Let be a convex subset of a real smooth Banach space , and let . Then if and only if

We make use of the function defined by studied by Alber [29]. That is, , for all and .

In the sequel, we will make use of the following lemmas.

Lemma 6 (see [29]). Let be a reflexive strictly convex and smooth Banach space with as its dual. Then, for all and .

Lemma 7 (see [30]). Let be a smooth and strictly convex Banach space, be a nonempty closed convex subset of , and be a maximal monotone mapping. Let be the resolvent of defined by , for and a sequence of such that . If is a bounded sequence of such that , then .

Lemma 8 (see [31]). Let be a smooth and strictly convex Banach space, be a nonempty closed convex subset of , and be a maximal monotone mapping, and is nonempty. Let be the resolvent of defined by , for . Then, for each for all and .

Lemma 9 (see [32]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfying the following conditions: , , and . Then, .

Lemma 10 (see [33]). Let be the sequences of real numbers such that there exists a subsequence of such that , for all . Then, there exists a nondecreasing sequence such that , and the following properties are satisfied by all (sufficiently large) numbers : In fact, .

3. Main Result

Theorem 11. Let and be nonempty, closed and convex subsets of a smooth and uniformly convex real Banach space with as its dual. Assume that . Let and be maximal monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

Proof. Observe that by Lemma 2, we have that is maximal monotone. In addition, since , the same lemma implies that is maximal monotone. Now, let , and let . Then, we have that , and since , from Lemma 8, we get that Now from (21), property of , and (22) we get that Thus, by induction, which implies that is bounded. In addition, using Lemma 6 and property of , we obtain that Furthermore, using property of and the fact that , as , imply that which implies from Lemma 4 that Now, following the method of proof of Lemma of Maing’e [33], we consider two cases.
Case 1. Suppose that there exists such that is nonincreasing for all . In this situation, is convergent. Since is bounded and is reflexive, we choose a subsequence of such that and . Then, from (27), we get that Thus, by Lemma 7, we get that , and hence . Therefore, by Lemma 5, we immediately obtain that . It follows from Lemma 9 and (25) that , as . Consequently, .
Case 2. Suppose that there exists a subsequence of such that for all . Then, by Lemma 10, there exist a nondecreasing sequence such that , satisfying Thus, following the method of proof of Case 1, we obtain that Then, from (25), we have that
Now, inequalities (30) and (32) imply that In particular, since , we get Then, from (31), we obtain , as . This together with (32) gives , as . But , for all ; thus, we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to , and the proof is complete.

Theorem 12. Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be maximal monotone mapping, and let be bounded and hemicontinuous monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and is a sequence of satisfying: , , and . Then, converges strongly to .

Proof. By Lemma 3, we have that is maximal monotone, and hence following the method of proof of Theorem 11, we obtain the required assertion.

If in Theorem 12, we assume that , for , are continuous monotone mappings, then are hemicontinuous, and hence we get the following corollary.

Corollary 13. Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be a maximal monotone mapping, and let be bounded and continuous monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

If in Theorem 12, we assume that , for , are uniformly continuous monotone mapping, then are bounded and hemicontinuous, and hence we get the following corollary.

Corollary 14. Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space with as its dual. Let be a maximal monotone mapping, and let be monotone uniformly continuous mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

If in Theorem 12 we assume that , for , then we get the following corollary.

Corollary 15. Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space . Let be a maximal monotone mapping. Assume that is nonempty. Let be a sequence generated by where and a sequence of satisfying: , , and . Then, converges strongly to .

If , a real Hilbert space, then is smooth and uniformly convex real Banach space. In this case, , identity map on and , projection mapping from onto . Thus, the following corollaries follow from Theorems 11 and 12.

Corollary 16. Let and be nonempty, closed, and convex subsets of a real Hilbert space . Assume that . Let , and let be maximal monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

Corollary 17. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a maximal monotone mapping, and let be bounded, hemicontinuous, and monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

Corollary 18. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let be a maximal monotone mapping, and let be uniformly continuous monotone mappings. Assume that is nonempty. Let be a sequence generated by where , and a sequence of satisfying: , , and . Then, converges strongly to .

4. Application

In this section, we study the problem of finding a minimizer of a continuously Fréchet differentiable convex functional in Banach spaces. The followings are deduced from Theorems 11 and 12.

Theorem 19. Let and be a nonempty, closed, and convex subsets of a smooth and uniformly convex real Banach space . Let . Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Assume that . Let be a sequence generated by where and a sequence of satisfying: , , and . Then, converges strongly to an element of .

Theorem 20. Let be a nonempty, closed, and convex subset of a smooth and uniformly convex real Banach space. Let be a continuously Fréchet differentiable convex functional, and let be maximal monotone on . Let be a continuously Fréchet differentiable convex functional, and let be bounded, hemicontinuous, and monotone on with . Let be a sequence generated by where and a sequence of satisfying: , , and . Then, converges strongly to an element of .

Remark 21. Our results provide strong convergence theorems for finding a zero of a finite sum of monotone mappings in Banach spaces and hence extend the results of Rockafellar [11], Kamimura and Takahashi [9], and Lions and Mercier [6].

Acknowledgments

The authors thank the referee for his comments that considerably improved the paper. The research of N. Shahzad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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