Approximation of Common Fixed Points of a Countable Family of Relatively Nonexpansive Mappings
© Boonchari and S. Saejung. 2010
Received: 22 June 2009
Accepted: 21 November 2009
Published: 7 December 2009
We introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.
1. Introduction and Preliminaries
Let be a nonempty subset of a Banach space , and let be a mapping from into itself. When is a sequence in , we denote strong convergence of to by and weak convergence by . We also denote the weak convergence of a sequence to in the dual by . A point is an asymptotic fixed point of if there exists in such that and . We denote and by the set of fixed points and of asymptotic fixed points of , respectively. A Banach space is said to be strictly convex if for and . It is also said to be uniformly convex if for each , there exists such that for and . The space is said to be smooth if the limit
Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced by Matsushita and Takahashi . This type of mappings is closely related to the resolvent of maximal monotone operators (see [2–4]).
We know that if is smooth, strictly convex, and reflexive, then the duality mapping is single-valued, one-to-one, and onto. The duality mapping is said to be weakly sequentially continuous if implies that (see  for more details).
Following Matsushita and Takahashi , a mapping is said to be relatively nonexpansive if the following conditions are satisfied:
If satisfies (R1) and (R2), then is called relatively quasi-nonexpansive . Obviously, relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.
In , Alber introduced the generalized projection from onto as follows:
In 2004, Masushita and Takahashi [1, 6] also proved weak and strong convergence theorems for finding a fixed point of a single relatively nonexpansive mapping. Several iterative methods, as a generalization of [1, 6], for finding a common fixed point of the family of relatively nonexpansive mappings have been further studied in [7, 9–14].
Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied by Takahashi and Zembayashi in [15, 16]. The purpose of this paper is to introduce a new iterative scheme which unifies several ones studied by many authors and to deduce the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions as were the case in other papers are dropped away.
First, we start with some preliminaries which will be used throughout the paper.
Lemma 1.1 (see [7, Lemma ]).
Lemma 1.2 (see [17, Proposition ]).
Lemma 1.3 (see ).
Lemma 1.4 (see [17, Proposition ]).
Lemma 1.5 (see ).
Lemma 1.6 (see ).
We next prove the following three lemmas which are very useful for our main results.
The proof of this lemma can be extracted from that of Lemma 1.8; so it is omitted.
2. Weak Convergence Theorem
We first establish weak convergence theorem for finding a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of mappings with , we say that satisfies the NST-condition  if for each bounded sequence in ,
We next apply our result for finding a common element of a fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem. This problem is extensively studied in [11, 14–16]. Let be a subset of a Banach space and let be a bifunction. The equilibrium problem for a bifunction is to find such that for all . The set of solutions above is denoted by , that is
The following lemma gives a characterization of a solution of an equilibrium problem.
Takahashi and Zembayashi proved the following important result.
Lemma 2.4 (see [15, Lemma ]).
We now deduce Takahashi and Zembayashi's recent result from Theorem 2.2.
Corollary 2.5 (see [15, Theorem ]).
Using the same proof as above, we have the following result.
Corollary 2.6 (see [11, Theorem ]).
The following result also follows from Theorem 2.2.
Corollary 2.7 (see [9, Theorem ]).
Then the following hold:
Thus the conclusions of this corollary follow.
3. Strong Convergence Theorem
In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.
Lemma 3.2 (see [21, Lemma ]).
3.1. The CQ-Method
The proof is broken into 4 steps.
We apply Theorem 3.3 and the proof of Corollary 2.5 and then obtain the following result.
3.2. The Monotone CQ-Method
Let be a closed subset of a Banach space . Recall that a mapping is closed if for each in , if and , then . A family of mappings with is said to satisfy the -condition if for each bounded sequence in ,
This step is almost the same as Step 1 of the proof of Theorem 3.3, so it is omitted.
Corollary 3.8 (see [12, Theorem ]).
Corollary 3.9 (see [13, Theorem ]).
3.3. The Shrinking Projection Method
The proof is almost the same as the proofs of Theorems 3.3 and 3.7; so it is omitted.
In particular, applying Theorem 3.11 gives the following result.
Corollary 3.14 (see [11, Theorem ]).
The conclusion of Corollary 3.14 remains true under the more general assumption; that is, we can replace (b) by the following one:
We also deduce the following result.
Corollary 3.16 (see [14, Theorem ]).
The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replace (b) and (c) by the following one:
Corollary 3.18 (see [10, Theorem ]).
The authors would like to thank the referee for their comments on the manuscript. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand, and the second author is supported by the Thailand Research Fund under Grant MRG5180146.
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