Weak and strong convergence of a scheme with errors for two nonexpansive mappings

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Abstract

In this paper, we are concerned with the study of an iterative scheme with errors involving two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme in a uniformly convex Banach space under a condition weaker than compactness.

Introduction

Let C be a nonempty convex subset of a normed space E and S,T:CC be two mappings. Xu [11] introduced the following iterative schemes known as Mann iterative scheme with errors and Ishikawa iterative scheme with errors:

(1) The sequence {xn} defined by x1=xC,xn+1=anTxn+bnxn+cnun,n1,where {an},{bn},{cn} are sequences in [0,1] such that an+bn+cn=1 and {un} is a bounded sequence in C, is known as Mann iterative scheme with errors. This scheme reduces to Mann iterative scheme if cn=0.

(2) The sequence {xn} defined by x1=xC,xn+1=anTyn+bnxn+cnun,yn=anTxn+bnxn+cnvn,n1,where {an},{bn},{cn}{an},{bn},{cn} are sequences in [0,1] satisfying an+bn+cn=1=an+bn+cn and {un},{vn} are bounded sequences in C, is known as Ishikawa iterative scheme with errors. This scheme becomes Ishikawa iterative schemes if cn=0=cn. Chidume and Moore [2] studied the above schemes in 1999.

A generalization of Mann and Ishikawa iterative schemes was given by Das and Debata [3] and Takahashi and Tamura [9]. This scheme dealt with two mappings: x1=xC,xn+1=anSyn+(1-an)xn,yn=bnTxn+(1-bn)xn,n1.

We further generalize this scheme to the one with errors as follows.

(3) The sequence {xn}, in this case, is defined byx1=xC,xn+1=anSyn+bnxn+cnun,yn=anTxn+bnxn+cnvn,n1,where {an},{bn},{cn},{an},{bn},{cn} are sequences in [0,1] with 0<δan, an1-δ<1, an+bn+cn=1=an+bn+cn and {un},{vn} are bounded sequences in C.

Nonexpansive mappings since their introduction have been extensively studied by many authors in different frames of work. One is the convergence of iteration schemes constructed through nonexpansive mappings. In this paper, we study the iterative scheme given in (1.1) for weak and strong convergence for a pair of nonexpansive mappings in a uniformly convex Banach space. To proceed in this direction, we first recall the following definitions.

A Banach space E is said to satisfy Opial's condition [6] if for any sequence {xn} in E, xnx implies that limsupnxn-x<limsupnxn-y for all yE with yx.

A mapping T:CE is called demiclosed with respect to yE if for each sequence {xn} in C and each xE, xnx and Txny imply that xC and Tx=y.

Next we state the following useful lemmas.

Lemma 1 Schu [7]

Suppose that E is a uniformly convex Banach space and 0<ptnq<1 for all positive integers n. Also suppose that {xn} and {yn} are two sequences of E such that limsupnxnr, limsupnynr and limntnxn+(1-tn)yn=r hold for some r0. Then limnxn-yn=0.

Lemma 2 Tan and Xu [10]

Let {sn},{tn} be two nonnegative sequences satisfying sn+1sn+tnforalln1.If n=1tn< then limnsn exists.

Lemma 3 Browder [1]

Let E be a uniformly convex Banach space satisfying Opials condition and let C be a nonempty closed convex subset of E. Let T be a nonexpansive mapping of C into itself. Then I-T is demiclosed with respect to zero.

Section snippets

Convergence of the iteration scheme

In this section, we shall prove the weak and strong convergence of the iteration scheme (1.1) to a common fixed point of the nonexpansive mappings S and T. Let F(T) denote the set of all fixed points of T. We first prove the following lemmas.

Lemma 4

Let E be a normed space and C its nonempty bounded convex subset. Let S,T:CC be nonexpansive mappings. Let {xn} be the sequence as defined in (1.1) with n=1cn<, n=1cn<. If F(S)F(T), then limnxn-x* exists for all x*F(S)F(T).

Proof

Since C is

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