Strong Convergence of a New Iteration for a Finite Family of Accretive Operators

Abstract

The viscosity approximation methods are employed to establish strong convergence of the modified Mann iteration scheme to a common zero of a finite family of accretive operators on a strictly convex Banach space with uniformly Gâteaux differentiable norm. Our work improves and extends various results existing in the current literature.

1. Introduction

Let be a Banach space with dual space of , and let a nonempty closed convex subset . Let be a positive integer, and let . We denote by the normalized duality map from E to defined by

(1.1)

A mapping is said to be nonexpansive if , for all . A mapping is called -contraction if there exists a constant such that

(1.2)

In the last ten years, many papers have been written on the approximation of fixed point for nonlinear mappings by using some iterative processes (see, e.g.,[1–20]).

An operator is said to be accretive if , for all , and . If is accretive and is identity mapping, then we define, for each , a nonexpansive single-valued mapping by , which is called the resolvent of . we also know that for an accretive operator , , where and . An accretive operator is said to be -accretive, if for all . If is a Hilbert space, then accretive operator is monotone operator. There are many papers throughout literature dealing with the solution of () by utilizing certain iterative sequence (see [1–3, 8–10, 13, 16, 20]).

In 2005, Kim and Xu [10] introduced the following Halpern type iterative sequence for -accretive operator : Let    be a nonempty closed convex subset of  . For any  , the sequence  is generated by

(1.3)

where  and   , for some  , satisfy the following conditions:

(C1),

(C2),

(C3), and

(C4).

They proved that the iterative sequence converges strongly to a zero of .

Recently, Zegeye and Shahzad [20] proved a strong convergence theorem for a finite family of accretive operators by using the Halpern type iteration: Let    be a nonempty closed convex subset of  . For any  , the sequence    is generated by

(1.4)

where     with  , ,  for  , , and     satisfies the conditions: (C1), (C2), (C3), or (). ).

More recently, Hu and Liu [8] proposed a generalized Halpern type iteration: Let  be a nonempty closed convex subset of. For any  , the sequence    is generated by

(1.5)

where    with  , for  ,   and  . Assume , , ,   and    satisfy the following conditions: (C1), (C2),

(1.6)

They proved that the sequence converges strongly to a common zero of .

In this paper, we introduce and study a new iterative sequence: Let    be a nonempty closed convex subset of    and    a  -contraction. For any  , the sequence    is defined by

(1.7)

where    with  ,  for  ,   and  ,   and  . The iterative sequence (1.7) is a natural generalization of all the above mentioned iterative sequences.

(i)In contrast to the iterations (1.3)–(1.5), the convex composition of the iteration (1.7) deals with only instead of and .

(ii)If we take , for all , in (1.7), then (1.7) reduces to Mann iteration. In 2000, Kamimura and Takahashi [9] proved that if is a Hilbert space and and are chosen such that , and , then the Mann iterative sequence,

(1.8)

converges weakly to a zero of . However, the Mann iteration scheme has only weak convergence for nonexpansive mappings even in a Hilbert space (see [4]).

Our main purpose is to prove strong convergence theorems for a finite family of accretive operators on a strictly convex Banach space with uniformly Gteaux differentiable norm by using viscosity approximation methods. Our theorems extend the comparable results in the following three aspects.

(1)In contrast to weak convergence results on a Hilbert Space in [9], strong convergence of the iterative sequence is obtained in the general setup of a Banach space.

(2)The restrictions (C3), (), and (C4) on the results in [10, 20] are dropped.

(3)A single mapping of the results in [3] is replaced by a finite family of mappings.

2. Preliminaries and Lemmas

A Banach space is said to have Gteaux differentiable norm if the limit

(2.1)

exists for each , where . The norm of is uniformly Gteaux differentiable   if for each , the limit is attained uniformly for . The norm of is uniformly Fréchet differentiable ( is also called uniformly smooth) if the limit is attained uniformly for each . It is well known that if is uniformly Gteaux differentiable norm, then the duality mapping is single-valued and uniformly continuous on each bounded subset of .

A Banach space is called strictly convex if for , , and , we have for , and for . In a strictly convex Banach space , we have that if , for , , and , then .

Lemma 2.1 (The Resolvent Identity).

For and ,

(2.2)

We denote by the set of all natural numbers, and let be a mean on , that is, a continuous linear functional on satisfying . We know that is a mean on if and only if

(2.3)

for each . In general, we use instead of . Let with , and let be a Banach limit on . Then . Further, we know the following result.

Lemma 2.2 (see [15, 16]).

Let be a nonempty closed convex subset of a Banach space with uniformly Gteaux differentiable norm. Assume that is a bounded sequence in . Let , and let a Banach limit. Then if and only if , .

Let be a closed convex and, let a mapping of onto . Then is said to be sunny [12, 13] if for all and . A mapping of onto is said to be retraction if ; If a mapping is a retraction then for any , the range of . A subset of is said to be a sunny nonexpansive retraction of if there exists a sunny nonexpansive retraction of onto , and it is said to be a nonexpansive retraction of if there exists a nonexpansive retraction of onto . In a smooth Banach space , it is known ([5, Page 48]) that is a sunny nonexpansive retraction if and only if the following condition holds: , and .

Lemma 2.3 (see [14]).

Let and be bounded sequences in a Banach space such that

(2.4)

where is a sequence in such that . Assume

(2.5)

Then .

Lemma 2.4.

Let be a real Banach space. Then for all in and , the following inequality holds

(2.6)

Lemma 2.5 (see [18]).

Let is a sequence of nonnegative real number such that

(2.7)

where is a sequence in and is a sequence in satisfying the following conditions:

(i);

(ii) or .

Then  .

Lemma 2.6 ([8]).

Let be a nonempty closed convex subset of a strictly convex Banach space . Suppose that is a finite family of accretive operators such that and satisfies the range conditions:

(2.8)

Let be real numbers in with and , where and . Then is nonexpansive and .

3. Main Results

For the sake of convenience, we list the assumptions to be used in this paper as follows.

(i) is a strictly convex Banach space which has uniformly Gteaux differentiable norm, and is a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings.

(ii)The real sequence satisfies the conditions: (C1). and (C2). .

We will employ the viscosity approximation methods [11, 19] to obtain a strong convergence theorem. The method of proof is closely related to [2, 3, 19].

Theorem 3.1.

Let be a finite family of accretive operators satisfying the following range conditions:

(3.1)

Assume that . Let be a -contraction with . For , the net is generated by

(I)

where with , for , and . If , then the net converges strongly to , as , where is the unique solution of a variational inequality:

(VI)

Proof.

Put , for all and . Then we have

(3.2)

and so is a contraction of into itself. Hence, for each , there exists a unique element such that

(3.3)

Thus the net is well defined.

Lemma 2.6 implies that . Taking , we have for any

(3.4)

Consequently, we get

(3.5)

that is, the net is bounded, and so are and . Rewriting (I) to find

(3.6)

and hence for any , it yields that

(3.7)

Obviously, estimate (I) yields

(3.8)

In view of the Resolvent Identity, we deduce

(3.9)

and so

(3.10)

Combining (3.8) and the above inequality, we obtain

(3.11)

Assume , as . Set and define ( is the set of all real numbers) by

(3.12)

where LIM is a Banach limit on . Let

(3.13)

It is easy to see that is a nonempty closed convex and bounded subset of and is invariant under . Indeed, as , we have for any ,

(3.14)

and so is an element of . Since has the fixed point property for nonexpansive mappings, has a fixed point in . Using Lemma 2.2, we have

(3.15)

Clearly

(3.16)

Consequently, by (3.15), we obtain

(3.17)

, that is,

(3.18)

and there exists a subsequence which is still denoted by such that .

On the other hand, let of be such that . Now (3.7) implies

(3.19)

Thus

(3.20)

Interchange and to get

(3.21)

Addition of (3.20) and (3.21) yields

(3.22)

and so we have

(3.23)

Since , it follows that . Consequently as . Likewise, using (3.7), it implies for all

(3.24)

Letting yields

(3.25)

for all .

Remark 3.2.

In addition,  if is a uniformly smooth Banach space in Theorem 3.1 and we define  ,   then we obtain from Theorem 3.1 and [19, Theorem 4.1] that the net  converges strongly to  ,   as ,   where    and   is a sunny nonexpansive retraction of    onto  .

Theorem 3.3.

Let be a finite family of accretive operators satisfying the following range conditions:

(3.26)

Assume that . Let be a -contraction with . For any , the sequence is generated by (1.7). Suppose further that sequences in the iterative sequence (1.7) satisfy the conditions:

(3.27)

Then the sequence converges strongly to , where is the unique solution of a variational inequality .

Proof.

Lemma 2.6 implies that . Rewrite (1.7) as follows:

(3.28)

where

(3.29)

Taking , we obtain

(3.30)

Therefore, the sequence is bounded, and so are the sequences , , , and, . We estimate from (3.29)

(3.31)

In view of the Resolvent Identity, we get

(3.32)

where

(3.33)

Since , we have

(3.34)

 and   imply

(3.35)

Consequently, by Lemma 2.3, we obtain

(3.36)

From (3.29), we get

(3.37)

and so it follows from (3.36) and (3.37) that

(3.38)

Using the Resolvent Identity and , we discover

(3.39)

Hence, we have

(3.40)

It follows from Theorem 3.1 that generated by converges strongly to , as , where is the unique solution of a variational inequality . Furthermore,

(3.41)

In view of Lemma 2.4, we find

(3.42)

and hence

(3.43)

Since the sequences , , and are bounded and  , we obtain

(3.44)

where . We also know that

(3.45)

From the facts that , as , is bounded and the duality mapping is uniformly continuous on bounded subset of , it follows that

(3.46)

Combining (3.44), (3.45), and the two results mentioned above, we get

(3.47)

Similarly, from (3.29) and the duality mapping is uniformly continuous on bounded subset of , it follows that

(3.48)

Write

(3.49)

and apply Lemma 2.4 to find

(3.50)

where

(3.51)

From (3.47), (3.48), (C1), (C2), and , it follows that and . Consequently applying Lemma 2.5 to (3.50), we conclude that .

If we take , for all , in the iteration (1.7), then, from Theorem 3.3, we have what follows

Corollary 3.4.

Let , , , and be as in Theorem 3.3. For any , the sequence is generated by

(3.52)

where with ,  for , and . Then the sequence converges strongly to .

Remark 3.5.

Theorem 3.3 and Corollary 3.4 prove strong convergence results of the new iterative sequences which are different from the iterative sequences (1.4) and (1.5). In contrast to [20], the restriction: (C3).  or   is removed.

If we consider the case of an accretive operator , then as a direct consequence of Theorem 3.1 and Theorem 3.3, we have the following corollaries.

Corollary 3.6 ([3, Theorem 3.1]).

Let (not strictly convex) be an accretive operator satisfying the following range condition:

(3.53)

Assume that . Let be a -contraction with . For , the net is given by:

(3.54)

where . If ,  for some , then converges strongly to , as , where is the unique solution of a variational inequality:

Corollary 3.7.

Let (not strictly convex) be an accretive operator satisfying the following range condition:

(3.55)

Assume that . Let be a -contraction with . Suppose that and are real sequences in and is a sequence in , satisfying the conditions: and , for some . For any , the sequence is generated by

(3.56)

where . Then the sequence converges strongly to , where is the unique solution of a variational inequality .

Remark 3.8.

(i)Corollary 3.7 describes strong convergence result in Banach spaces for a modification of Mann iteration scheme in contrast to the weak convergence result on Hilbert spaces given in [9, Theorem 3].

(ii)In contrast to the result [10, Theorem 4.2], the iterative sequence in Corollary 3.7 is different from the iteration (1.3), and the conditions and are not required.