Open Access

Construction of Fixed Points by Some Iterative Schemes

Fixed Point Theory and Applications20092009:612491

https://doi.org/10.1155/2009/612491

Received: 23 October 2008

Accepted: 23 February 2009

Published: 8 March 2009

Abstract

We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechét differentiable norm.

1. Introduction

Let be a real Banach space and let be a nonempty closed convex subset of . A self-mapping is said to be nonexpansive if for all A point is a fixed point of provided . Denote by the set of fixed points of that is, It is assumed throughout this paper that is a nonexpansive mapping such that Construction of fixed points of nonexpansive mappings is an important subject in the theory of nonexpansive mappings and its applications in a number of applied areas, in particular, in image recovery and signal processing (see [13]). One way to overcome this difficulty is to use Mann's iteration method that produces a sequence via the recursive sequence manner:
(1.1)

Reich [4] proved that if is a uniformly convex Banach space with a Frechét differentiable norm and if is chosen such that then the sequence defined by (1.1) converges weakly to a fixed point of However, this scheme has only weak convergence even in a Hilbert space (see [5]). Some attempts to modify Mann's iteration method (1.1) so that strong convergence is guaranteed have recently been made.

The following modification of Mann's iteration method (1.1) in a Hilbert space is given by Nakajo and Takahashi [6]:
(1.2)

where denotes the metric projection from onto a closed convex subset of . They proved that if the sequence is bounded from one, then defined by (1.2) converges strongly to Their argument does not work outside the Hilbert space setting. Also, at each iteration step, an additional projection is needed to calculate.

Let be a closed convex subset of a Banach space and is a nonexpansive mapping such that Define in the following way:
(1.3)

where is an arbitrary (but fixed) element in , and and are two sequences in It is proved, under certain appropriate assumptions on the sequences and that defined by (1.3) converges to a fixed point of (see [7]).

The second modification of Mann's iteration method (1.1) is an adaption to (1.3) for finding a zero of an -accretive operator , for which we assume that the zero set

The iteration process is given by
(1.4)

where for each is the resolvent of . In [7], it is proved, in a uniformly smooth Banach space and under certain appropriate assumptions on the sequences and , that defined by (1.4) converges strongly to a zero of

2. Preliminaries

Let be a real Banach space. Recall that the (normalized) duality map from into the dual space of is given by
(2.1)

Now, we define Opial's condition in the sense of doubly sequence.

Definition 2.1.

A Banach space is said to satisfy Opial's condition if for any sequence in implies that
(2.2)

where denotes that converges weakly to

We are going to work in uniformly smooth Banach spaces that can be characterized by duality mappings as follows (see [8] for more details).

Lemma 2.2 (see [8]).

A Banach space is uniformly smooth if and only if the duality map is single-valued and norm-to-norm uniformly continuous on bounded sets of

Lemma 2.3 (see [8]).

In a Banach space there holds the inequality
(2.3)

where

If and are nonempty subsets of a Banach space such that is a nonempty closed convex subset and then the map is called a retraction from onto provided for all A retraction is sunny [1, 4] provided for all and whenever A sunny nonexpansive retraction is a sunny retraction, which is also nonexpansive. A sunny nonexpansive retraction plays an important role in our argument.

If is a smooth Banach space, then is a sunny nonexpansive retraction if and only if there holds the inequality
(2.4)

Lemma 2.4 (see [9]).

Let be a uniformly smooth Banach space and let be a nonexpansive mapping with a fixed point. For each fixed and every , 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
(2.5)

Lemma 2.5 (see [10, 11]).

Let be a sequence of nonnegative real numbers satisfying the property
(2.6)

where and are such that

(i)

(ii)

Then, converges to zero.

Lemma 2.6 (see [8]).

Assume that has a weakly continuous duality map with gauge . Then, is demiclosed in the sense that is closed in the product space , where is equipped with the norm topology and with the weak topology. That is, if then

Lemma 2.7 (see [12]).

Let be a Banach space and Then,

(i) is uniformly convex if and only if, for any positive number r, there is a strictly increasing continuous function such that
(2.7)

where the closed ball of centered at the origin with radius r, and

(ii) is -uniformly convex if and only if there holds the inequality
(2.8)

where is a constant.

Lemma 2.8 (see [4]).

Let be a closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm, and let be a sequence of nonexpansive self mapping of with a nonempty common fixed point set If and for then exists for all In particular, where and are weak limit points of

Lemma 2.9 (the demiclosedness principle of nonexpansive mappings [13]).

Let be a nonexpansive selfmapping of a closed convex subset of of a uniformly convex Banach space. Suppose that has a fixed point. Then is demiclosed. This means that
(2.9)

In 2005, Kim and Xu [7], proved the following theorem.

Theorem 2.

Let be a closed convex subset of a uniformly smooth Banach space , and let be a nonexpansive mapping such that Given a point and given sequences and in the following conditions are satisfied.

(i) ,

(ii) ,

(iii)

Define a sequence in by
(2.10)

Then is strongly converges to a fixed point of .

Recently, the study of fixed points by doubly Mann iteration process began by Moore (see [14]). In [15, 16], we introduced the concept of Mann-type doubly sequence iteration with errors, then we obtained some fixed point theorems for some different classes of mappings. In this paper, we will continue our study in the doubly sequence setting. We propose two modifications of the doubly Mann iteration process with errors in uniformly smooth Banach spaces: one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified doubly Mann iterations are proved to have strong convergence. Also, we append this paper by obtaining weak convergence theorems for Mann's doubly sequence iteration with errors in a uniformly convex Banach space by a Fréchet differentiable norm. Our results in this paper extend, generalize, and improve a lot of known results (see, e.g., [4, 7, 8, 17]). Our generalizations and improvements are in the use of doubly sequence settings as well as by adding the error part in the iteration processes.

3. A Fixed Point of Nonexpansive Mappings

In this section, we propose a modification of doubly Mann's iteration method with errors to have strong convergence. Modified doubly Mann's iteration process is a convex combination of a fixed point in , and doubly Mann's iteration process with errors can be defined as
(3.1)

The advantage of this modification is that not only strong convergence is guaranteed, but also computations of iteration processes are not substantially increased.

Now, we will generalize and extend Theorem A by using scheme (3.1).

Theorem 3.1.

Let be a closed convex subset of a uniformly smooth Banach space and let be a nonexpansive mapping such that Given a point and given sequences and in the following conditions are satisfied.

(i) ,

(ii)

Define a sequence in by (3.1). Then, converges strongly to a fixed point of

Proof.

First, we observe that is bounded. Indeed, if we take a fixed point of noting that
(3.2)
we obtain
(3.3)
Now, an induction yields
(3.4)
Hence, is bounded, so is . As a result, we obtain by condition (i)
(3.5)
We next show that
(3.6)
It suffices to show that
(3.7)
Indeed, if (3.7) holds, in view of (3.5), we obtain
(3.8)
Hence, (3.6) holds. In order to prove (3.7), we calculate
(3.9)
It follows that
(3.10)

Hence, by assumptions (i)-(ii), we obtain

Next, we claim that
(3.11)
where with being the fixed point of the contraction In order to prove (3.11), we need some more information on , which is obtained from that of (cf. [18]). Indeed, solves the fixed point equation
(3.12)
Thus we have
(3.13)
We apply Lemma 2.3 to get
(3.14)
(3.15)
It follows that
(3.16)
Letting in (3.16) and noting (3.15) yield
(3.17)
where is a constant such that for all and . Since the set is bounded, the duality map is norm-to-norm uniformly continuous on bounded sets of (Lemma 2.2), and strongly converges to By letting in (3.17), thus (3.11) is therefore proved. Finally, we show that strongly and this concludes the proof. Indeed, using Lemma 2.3 again, we obtain
(3.18)

Now we apply Lemma 2.5, and using (3.11) we obtain that

We support our results by giving the following examples.

Example 3.2.

Let be given by Then, the modified doubly Mann's iteration process with errors converges to the fixed point , and both Picard and Mann iteration processes converge to the same point too.

Proof.
  1. (I)

    Doubly Picards iteration converges.

     
For every point in is a fixed point of Let be a point in then
(3.19)
Hence,
(3.20)
Let for all Take and Thus
(3.21)
  1. (II)

    Doubly Mann's iteration converges.

     
Let be a point in then
(3.22)
Since doubly Mann's iteration is defined by
(3.23)
Take to obtain
(3.24)
  1. (III)
    Modified doubly Mann's iteration process with errors converges because the sequence as we can see and by using (3.1), we obtain
    (3.25)
     
In (3.1), we suppose that ,
(3.26)

Let and using Theorem 3.1 ( we obtain

Example 3.3.

Let be given by Then the doubly Mann's iteration converges to the fixed point of but modified doubly Mann's iteration process with errors does not converge.

Proof.
  1. (I)
    Doubly Mann's iteration converges because the sequence as we can see,
    (3.27)
     

The last inequality is true because for all and

(II)The origin is the unique fixed point of

(III)Note that, modified doubly Mann's iteration process with errors does not converge to the fixed point of because the sequence as we can see and by using (3.1), we obtain
(3.28)
Putting ,
(3.29)

Letting we deduce that

4. Convergence to a Zero of Accretive Operator

In this section, we prove a convergence theorem for -accretive operator in Banach spaces. Let be a real Banach space. Recall that, the (possibly multivalued) operator with domain and range in is accretive if, for each and there exists a such that
(4.1)
An accretive operator is m-accretive if for each . Throughout this section, we always assume that is -accretive and has a zero. The set of zeros of is denoted by Hence,
(4.2)

For each , we denote by the resolvent of that is, Note that if is -accretive, then is nonexpansive and for all We need the resolvent identity (see [19, 20] for more information).

Lemma 4.1 ([7] (the resolvent identity)).

For , and
(4.3)

Theorem 4.2.

Assume that is a uniformly smooth Banach space, and is an -accretive operator in such that Let be defined by
(4.4)

Suppose and satisfy the conditions,

(i) ,

(ii) ,

(iii) for some and for all Also assume that
(4.5)

Then, converges strongly to a zero of

Proof.

First of all we show that is bounded. Take It follows that
(4.6)
By induction, we get that
(4.7)
This implies that is bounded. Then, it follows that
(4.8)
A simple calculation shows that
(4.9)
The resolvent identity (4.3) implies that
(4.10)
which in turn implies that
(4.11)
Combining (4.9) and (4.11), we obtain
(4.12)

where is a constant such that for all and By assumptions (i)–(iii) in the theorem, we have that and Hence, Lemma 2.5 is applicable to (4.12), and we conclude that

Take a fixed number such that Again from the resolvent identity (4.3), we find
(4.13)
It follows that
(4.14)
Hence,
(4.15)
Since in a uniformly smooth Banach space the sunny nonexpansive retract from onto the fixed point set of is unique, it must be obtained from Reich's theorem (Lemma 2.4). Namely, where and solve the fixed point equation
(4.16)
Applying Lemma 2.3, we get
(4.17)
where by (4.15). It follows that
(4.18)
Therefore, letting in (4.18), we get
(4.19)
where is a constant such that for all and . Since strongly and the duality map is norm-to-norm uniformly continuous on bounded sets of it follows that (by letting in (4.19))
(4.20)
(4.21)

Now we apply Lemma 2.5 and using (4.20), we obtain that

5. Weakly Convergence Theorems

We next introduce the following iterative scheme. Given an initial , we define by
(5.1)

Theorem 5.1.

Let be a uniformly convex Banach space with a Frechét differentiable norm. Assume that has a weakly continuous duality map with gauge . Assume also that

(i) ,

(ii)

Then, the scheme (5.1) converges weakly to a point in

Proof.

First, we observe that for any , the sequence is nonincreasing.

Indeed, we have by nonexpansivity of ,
(5.2)

In particular, is bounded, so is . Let be the set of weak limit point of the sequence

Note that we can rewrite the scheme (5.1) in the form
(5.3)
where is the nonexpansive mapping given by
(5.4)
Then, we have for Hence, by Lemma 2.7, we get
(5.5)
Therefore, will converge weakly to a point in if we can show that To show this, we take a point in Then we have a subsequence of such that . Noting that
(5.6)
we obtain
(5.7)

By Lemma 2.6, we conclude that that is, .

Theorem 5.2.

Let be a uniformly convex Banach space which either has a Frechét differentiable norm or satisfies Opial's property. Assume for some

(i) for

(ii) for

Then, the scheme (5.1) converges weakly to a point in

Proof.

We have shown that exists for all Applying Lemma 2.7(i), we have a strictly increasing continuous function such that
(5.8)
This implies that
(5.9)
Since we obtain by (5.9) that
(5.10)

For any fixed by Lemma 4.1, we have

(5.11)
We deduce that
(5.12)
Therefore we obtain by (5.9) that
(5.13)

Apply Lemma 2.9 to find out that It remains to show that is a singleton set. Towards this end, we take and distinguish the two cases.

In case has a Frechét differentiable norm, we apply Lemma 2.8 to get
(5.14)
hence, In case satisfies Opial's condition, we can find two subsequences such that If , Opial's property creates the contradiction,
(5.15)

In either case, we have shown that consists of exact one point, which is clearly the weak limit of

Remark 5.3.

The schemes (3.1), (4.4), and (5.1) generalize and extend several iteration processes from literature (see [7, 8, 17, 2125] and others).

Authors’ Affiliations

(1)
Mathematics Department, Faculty of Science, Sohag University
(2)
Mathematics Department, Faculty of Science, Taif University
(3)
Mathematics Department, The High Institute of Computer Science

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Copyright

© A. El-Sayed Ahmed and A. Kamal. 2009

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