Convergence Theorems of Fixed Points for a Finite Family of Nonexpansive Mappings in Banach Spaces

Abstract

We modify the normal Mann iterative process to have strong convergence for a finite family nonexpansive mappings in the framework of Banach spaces without any commutative assumption. Our results improve the results announced by many others.

1. Introduction and Preliminaries

Throughout this paper, we assume that is a real Banach space with the normalized duality mapping from into give by

(1.1)

where denotes the dual space of and denotes the generalized duality pairing. We assume that is a nonempty closed convex subset of and a mapping. A point is a fixed point of provided . Denote by the set of fixed points of , that is, . Recall that is nonexpansive if for all

One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping (see [1, 2]). More precisely, take and define a contraction by

(1.2)

where is a fixed point. Banach's contraction mapping principle guarantees that has a unique fixed point in . It is unclear, in general, what is the behavior of as even if has a fixed point. However, in the case of having a fixed point, Browder [1] proved that if is a Hilbert space, then converges strongly to a fixed point of that is nearest to . Reich [2] extended Broweder's result to the setting of Banach spaces and proved that if is a uniformly smooth Banach space, then converges strongly to a fixed point of and the limit defines the (unique) sunny nonexpansive retraction from onto .

Recall that the normal Mann iterative process was introduced by Mann [3] in 1953. The normal Mann iterative process generates a sequence in the following manner:

(1.3)

where the sequence is in the interval (0,1). If is a nonexpansive mapping with a fixed point and the control sequence is chosen so that then the sequence generated by normal Mann's iterative process (1.3) converges weakly to a fixed point of (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [4]). In an infinite-dimensional Hilbert space, the normal Mann iteration algorithm has only weak convergence, in general, even for nonexpansive mappings. Therefore, many authors try to modify normal Mann's iteration process to have strong convergence for nonexpansive mappings (see, e.g., [5–8] and the references therein).

Recently, Kim and Xu [5] introduced the following iteration process:

(1.4)

where is a nonexpansive mapping of into itself and is a given point. They proved that the sequence defined by (1.4) converges strongly to a fixed point of provided the control sequences and satisfy appropriate conditions.

Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, [9]. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see, e.g., [10]).

In this paper, we consider the mapping defined by

(1.5)

where are sequences in . Such a mapping is called the -mapping generated by and . Nonexpansivity of each ensures the nonexpansivity of . Moreover, in [11], it is shown that

Motivated by Atsushiba and Takahashi [11], Kim and Xu [5], and Shang et al. [7], we study the following iterative algorithm:

(1.6)

where is defined by (1.5) and is given point. We prove, under certain appropriate assumptions on the sequences and , that defined by (1.6) converges to a common fixed point of the finite family nonexpansive mappings without any commutative assumptions.

In order to prove our main results, we need the following definitions and lemmas.

Recall that if and are nonempty subsets of a Banach space such that is nonempty closed convex and , then a map is sunny (see [12, 13]) provided for all and whenever A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. Sunny nonexpansive retractions play an important role in our argument. They are characterized as follows [12, 13]: if is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality for all and

Reich [2] showed that if is uniformly smooth and is the fixed point set of a nonexpansive mapping from into itself, then there is a sunny nonexpansive retraction from onto and it can be constructed as follows.

Lemma 1.1.

Let be a uniformly smooth Banach space and let be a nonexpansive mapping with a fixed point. For each fixed and , the unique fixed point of the contraction converges strongly as to a fixed point of . Define by . Then is the unique sunny nonexpansive retract from onto , that is, satisfies the property for all and

Lemma 1.2 (See [14]).

Let and be bounded sequences in a Banach space and let be a sequence in [0,1] with . Suppose for all integers and Then

Lemma 1.3.

In a Banach space , there holds the inequality for all where .

Lemma 1.4 (See [15]).

Assume that is a sequence of nonnegative real numbers such that where is a sequence in (0,1) and is a sequence such that and or Then

2. Main Results

Theorem 2.1.

Let be a closed convex subset of a uniformly smooth and strictly convex Banach space . Let be a nonexpansive mapping from into itself for . Assume that . Given a point and given sequences and in (0,1), the following conditions are satisfied:

(i)

(ii)

(iii)

Let be the composite process defined by (1.6). Then converges strongly to , where and is the unique sunny nonexpansive retraction from onto .

Proof.

We divide the proof into four parts.

Step 1.

First we observe that sequences and are bounded.

Indeed, take a point and notice that

(2.1)

It follows that

(2.2)

By simple inductions, we have which gives that the sequence is bounded, so is .

Step 2.

In this part, we will claim that as

Put . Now, we compute that is,

(2.3)

Observing that

(2.4)

we have

(2.5)

From the proof of Yao [8], we have

(2.6)

where is an appropriate constant. Substituting (2.6) into (2.5), we have

(2.7)

Observing the conditions (i)–(iii), we get We can obtain easily by Lemma 1.2. Observe that (2.3) yields Therefore, we have

(2.8)

Step 3.

We will prove .

Observing that and the condition (i), we can easily get

(2.9)

On the other hand, we have Combining (2.8) with (2.9), we have

(2.10)

Notice that This implies From the condition (iii) and (2.10), we obtain

(2.11)

Step 4.

Finally, we will show as .

First, we claim that

(2.12)

where with being the fixed point of the contraction Then solves the fixed point equation Thus we have

(2.13)

It follows from Lemma 1.3 that

(2.14)

where

(2.15)

It follows from (2.14) that

(2.16)

Letting in (2.16) and noting (2.15) yield

(2.17)

where is an appropriate constant. Taking in (2.17), we have

(2.18)

On the other hand, we have

(2.19)

It follows that

(2.20)

Noticing that is norm-to-norm uniformly continuous on bounded subsets of and from (2.18), we have It follows that

(2.21)

Hence, (2.12) holds. Now, from Lemma 1.3, we have

(2.22)

Applying Lemma 1.4 to (2.22) we have as

Remark 2.2.

Theorem 2.1 improves the results of Kim and Xu [5] from a single nonexpansive mapping to a finite family of nonexpansive mappings.

Remark 2.3.

If is a contraction map and we replace by in the recursion formula (1.6), we obtain what some authors now call viscosity iteration method. We note that our theorem in this paper carries over trivially to the so-called viscosity process. Therefore, our results also include Yao et al. [16] as a special case.

Remark 2.4.

Our results partially improve Shang et al. [7] from a Hilbert space to a Banach space.

Remark 2.5.

If is a single nonexpansive mapping, then the strict convexity of may not be needed.