Journal of Applied Mathematics
Volume 2012, Article ID 804538, 16 pages
http://dx.doi.org/10.1155/2012/804538
Research Article

Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

1School of Science, University of Phayao, Phayao 56000, Thailand
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 13 February 2012; Accepted 9 March 2012

Academic Editor: Rudong Chen

Copyright © 2012 Kamonrat Nammanee 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

We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

1. Introduction

Let be a real Banach space and a nonempty subset of . Let be the dual space of . We denote the value of at by . Let be a nonlinear mapping. We denote by the fixed points set of , that is, . Let be a set-valued mapping. We denote by the domain of , that is, and also denote by the graph of , that is, . A set-valued mapping is said to be monotone if whenever . It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operators on . It is known that if is maximal monotone, then the set is closed and convex.

The problem of finding a zero point of maximal monotone operators plays an important role in optimizations. This is because it can be reformulated to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in various spaces (see, e.g., ). Initiated by Martinet , in a real Hilbert space , Rockafellar  introduced the following iterative scheme: and where , is the resolvent of defined by for all , and is a maximal monotone operator on . Such an algorithm is called the proximal point algorithm. It was proved that the sequence generated by (1.1) converges weakly to an element in provided that . Recently, Kamimura and Takahashi  introduced the following iteration in a real Hilbert space: and where and . The weak convergence theorems are also established in a real Hilbert space under suitable conditions imposed on and .

In 2004, Kamimura et al.  extended the above iteration process to a much more general setting. In fact, they proposed the following algorithm: and where , , and for all . They proved, in a uniformly smooth and uniformly convex Banach space, a weak convergence theorem.

Let , where is the set of real numbers, be a bifunction. The equilibrium problem is to find such that The solutions set of (1.4) is denoted by .

For solving the equilibrium problem, we assume that(A1) for all ,(A2) is monotone, that is for all ,(A3) for all , ,(A4) for all is convex and lower semi-continuous.

Recently, Takahashi and Zembayashi  introduced the following iterative scheme for a relatively nonexpansive mapping in a uniformly smooth and uniformly convex Banach space: and where and . Such an algorithm is called the shrinking projection method which was introduced by Takahashi et al. . They proved that the sequence converges strongly to an element in under appropriate conditions. The equilibrium problem has been intensively studied by many authors (see, e.g., ).

Motivated by the previous results, we introduce a hybrid-iterative scheme for finding a zero point of maximal monotone operators () which is also a common element in the solutions set of an equilibrium problem for and in the fixed points set of weak relatively nonexpansive mappings   (). Using the projection technique, we also prove that the sequence generated by a constructed algorithm converges strongly to an element in in a uniformly smooth and uniformly convex Banach space. Finally, we apply our results to a system of convex minimization problems.

2. Preliminaries and Lemmas

In this section, we give some useful preliminaries and lemmas which will be used in the sequel.

Let be a real Banach space and let be the unit sphere of . A Banach space is said to be strictly convex if for any , A Banach space is said to be uniformly convex if, for each , there exists such that for any , It is known that a uniformly convex Banach space is reflexive and strictly convex. The function which is called the modulus of convexity of is defined as follows: Then is uniformly convex if and only if for all . A Banach space is said to be smooth if the limit exists for all . It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for . The duality mapping is defined by for all . It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on bounded subsets of (see  for more details).

Let be a smooth Banach space. The function is defined by for all . From the definition of , we see that for all .

Let be a closed and convex subset of , and let be a mapping from into itself. A point in is said to be an asymptotic fixed point of  if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping is said to be relatively nonexpansive [33, 34] if and for all and . A point in is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to such that . The set of strong asymptotic fixed points of will be denoted by . A mapping is said to be weak relatively nonexpansive  if and for all and . It is obvious by definition that the class of weak relatively nonexpansive mappings contains the class of relatively nonexpansive mappings. Indeed, for any mapping , we see that . Therefore, if is a relatively nonexpansive mapping, then .

Nontrivial examples of weak relatively nonexpansive mappings which are not relatively nonexpansive can be found in .

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 , is a mapping , that assigns to an arbitrary point the minimum point of the function , that is, , where is the solution to the minimization problem In a Hilbert space, is coincident with the metric projection denoted by .

Lemma 2.1 (see ). Let be a uniformly convex and smooth Banach space and let be two sequences in . If and either or is bounded, then .

Lemma 2.2 (see [37, 38]). Let be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space , let and let . Then if and only if for all .

Lemma 2.3 (see [37, 38]). Let be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space . Then

Lemma 2.4 (see ). Let be a smooth and strictly convex Banach space, and let be a nonempty, closed, and convex subset of . Let be a mapping from into itself such that is nonempty and for all . Then is closed and convex.

Let be a reflexive, strictly convex, and smooth Banach space. It is known that is maximal monotone if and only if for all , where stands for the range of .

Define the resolvent of by for all . It is known that is a single-valued mapping from to and for all . For each , the Yosida approximation of is defined by for all . We know that for all and .

Lemma 2.5 (see ). Let be a smooth, strictly convex, and reflexive Banach space, let be a maximal monotone operator with , and let for each . Then for all , , and .

Lemma 2.6 (see). Let be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), and let and . Then, there exists such that

Lemma 2.7 (see ). Let be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space , and let be a bifunction from to satisfying (A1)–(A4). For all and , define the mapping as follows: Then, the following holds:(1) is single-valued;(2) is a firmly nonexpansive-type mapping , that is, for all , (3);(4) is closed and convex.

Lemma 2.8 (see ). Let be a closed and convex subset of a smooth, strictly, and reflexive Banach space , let be a bifunction from to satisfying (A1)–(A4), let . Then for all and .

3. Strong Convergence Theorems

In this section, we are now ready to prove our main theorem.

Theorem 3.1. Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty, closed and convex subset of . Let () be maximal monotone operators, let be a bifunction, and let   () be weak relatively nonexpansive mappings such that . Let be the sequence such that . Define the sequence in as follows: If for each and , then the sequence converges strongly to .

Proof. We split the proof into several steps as follows.
Step  1. for all .
From Lemma 2.4, we know that is closed and convex. From Lemma 2.7(4), we also know that is closed and convex. On the other hand, since () are maximal monotone, are closed and convex for each ; consequently, is closed and convex. Hence is a nonempty, closed, and convex subset of .
We next show that is closed and convex for all . Obviously, is closed and convex. Now suppose that is closed and convex for some . Then, for each and , we see that is equivalent to By the construction of the set , we see that Hence, is closed and convex. This shows, by induction, that is closed and convex for all . It is obvious that . Now, suppose that for some . For any , by Lemmas 2.5 and 2.8, we have This shows that . By induction, we can conclude that for all .
Step  2. exists.
From and , we have From Lemma 2.3, for any , we have Combining (3.5) and (3.6), we conclude that exists.
Step  3. .
Since for , by Lemma 2.3, it follows that Letting , we have . By Lemma 2.1, it follows that as . Therefore, is a Cauchy sequence. By the completeness of the space and the closedness of , we can assume that as . In particular, we obtain that Since , we have Since , for each , Since is uniformly smooth, is uniformly norm-to-norm continuous on bounded sets. It follows from (3.9) and by the boundedness of that for all . So from Lemma 2.1, we have and, since , therefore for all . Since is uniformly norm-to-norm continuous on bounded subsets of , for all .
Step  4. for all .
Denote that for each and for each . We note that for each .
To this end, we will show that for all .
For any , by (3.4), we see that Since , by Lemma 2.5 and (3.16), it follows that From (3.13) and (3.14), we get that . So we obtain that Again, since , From (3.13) and (3.14), we get that It also follows that Continuing in this process, we can show that So, we now conclude that for each . By the uniform norm-to-norm continuity of , we also have for each . Using (3.23), it is easily seen that From , by Lemma 2.8, it follows that This implies that and hence Combining (3.13), (3.25), and (3.27), we obtain that for all .
Step  5. .
Since and , . So from (3.25) and (3.27), we have . Note that () are weak relatively nonexpansive. Using (3.28), we can conclude that for all . Hence .
Step  6. .
Noting that for each , we obtain that From (3.24) and , we have We note that for each . If for each , then it follows from the monotonicity of that We see that for each . Thus, from (3.30) and (3.31), we have By the maximality of , it follows that for each . Therefore, .
Step  7. .
From , we have By (A2), we have Note that since . From (A4) and , we get for all . For and , define that . Then , which implies that . From (A1), we obtain that . Thus, . From (A3), we have for all . Hence, . From Steps 5, 6, and 7, we now can conclude that .
Step  8. .
From , we have Since , we also have Letting in (3.36), we obtain that This shows that by Lemma 2.2. We thus complete the proof.

As a direct consequence of Theorem 3.1, we can also apply to a system of convex minimization problems.

Theorem 3.2. Let be a uniformly smooth and uniformly convex Banach space, and let be a nonempty, closed, and convex subset of . Let () be proper lower semicontinuous convex functions, let be a bifunction, and let () be weak relatively nonexpansive mappings such that . Let be the sequence such that . Define the sequence in as follows: If for each and , then the sequence converges strongly to .

Proof. By Rockafellar's theorem [43, 44], are maximal monotone operators for each . Let for each . Then, if and only if which is equivalent to Using Theorem 3.1, we thus complete the proof.

If is a real Hilbert space, we then obtain the following results.

Corollary 3.3. Let be a nonempty, closed and convex subset of a real Hilbert space . Let () be maximal monotone operators, let be a bifunction, and let () be weak relatively nonexpansive mappings such that . Let be the sequence such that . Define the sequence in as follows: If for each and , then the sequence converges strongly to .

Corollary 3.4. Let be a nonempty, closed, and convex subset of a real Hilbert space . Let () be proper lower semi-continuous convex functions, let be a bifunction, and let () be weak relatively nonexpansive mappings such that . Let be the sequence such that . Define the sequence in as follows: If for each and , then the sequence converges strongly to .

Remark 3.5. Using the shrinking projection method, we can construct a hybrid-proximal point algorithm for solving a system of the zero-finding problems, the equilibrium problems, and the fixed point problems of weak relatively nonexpansive mappings.

Remark 3.6. Since every relatively nonexpansive mapping is weak relatively nonexpansive, our results also hold if () are relatively nonexpansive mappings.

Acknowledgments

The authors thank the editor and the referee(s) for valuable suggestions. The first author was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. The second and the third authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

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