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Journal of Applied Mathematics

Volume 2012 (2012), Article ID 294718, 11 pages

http://dx.doi.org/10.1155/2012/294718

## Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

^{1}Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, Pakistan^{2}Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 20 April 2012; Accepted 2 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Arif Rafiq and Shin Min Kang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.

#### 1. Introduction

Let be a nonempty subset of an arbitrary Banach space and let be its dual space. The symbols and stand for the domain and the set of fixed points of (for a single-valued mapping is called a *fixed point* of iff ). We denote by the *normalized duality mapping* from to defined by
where denotes the duality pairing. In a smooth Banach space, is singlevalued (we denoted by ).

*Remark 1.1. *(1) is called uniformly smooth if is uniformly convex.

(2) In a uniformly smooth Banach space, is uniformly continuous on bounded subsets of .

Let be a mapping.

*Definition 1.2. *The mapping is called *Lipshitz* if there exists a constant such that

for all . If , then is called *nonexpansive* and if , then is called *contractive*.

*Definition 1.3 (see [1, 2]). *(1) The mapping is said to be *pseudocontractive* if
for all and .

(2) The mapping is said to be *strongly pseudocontractive* if there exists a constant such that
for all and .

(3) The mapping is said to be *local strongly pseudocontractive* if for each there exists a constant such that
for all and .

(4) The mapping is said to be *strictly hemicontractive* if and if there exists a constant such that
for all , and .

Clearly, each strongly pseudocontractive mapping is local strongly pseudocontractive.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of in case is a Lipschitz strongly pseudocontractive mapping from a bounded closed convex subset of (or ) into itself. Schu [3] generalized the result in [1] to both uniformly continuous strongly pseudocontractive mappings and real smooth Banach spaces. Park [4] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [5] proved that the Mann and Ishikawa iteration methods may exhibit different behaviors for different classes of nonlinear mappings. Afterwards, several generalizations have been made in various directions (see, e.g., [6–13]).

In 2001, Xu and Ori [14] introduced the following implicit iteration process for a finite family of nonexpansive mappings (here ) with a real sequence in and an initial point : which can be written in the following compact form: where (here the function takes values in ). Xu and Ori [14] proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters are sufficient to guarantee the strong convergence of the sequence .

In [11], Osilike proved the following results.

Theorem 1.4. *Let be a real Banach space and let be a nonempty closed convex subset of . Let be strictly pseudocontractive mappings from to with . Let be a real sequence satisfying the following conditions: *(i)*,
*(ii)*,
*(iii)*. **From arbitrary , define the sequence by the implicit iteration process (1.8). Then converges strongly to a common fixed point of the mappings if and only if .*

*Remark 1.5. *One can easily see that for , . Hence the results of Osilike [11] are needed to be improved.

Let be a nonempty closed bounded convex subset of an arbitrary smooth Banach space and let be a continuous strictly hemicontractive mapping. We proved that the implicit Mann type iteration method converges strongly to a unique fixed point of .

The results presented in this paper extend and improve the corresponding results particularly in [1, 3, 4, 7, 8, 10, 11, 13, 15].

#### 2. Preliminaries

We need the following results.

Lemma 2.1 (see [4]). *Let be a smooth Banach space. Suppose that one of the following holds: *(a)* is uniformly continuous on any bounded subsets of ,*(b)* for all in ,*(c)*for any bounded subset of , there is a function such that** for all , where satisfies .**Then for any and any bounded subset , there exists such that**
for all and .*

*Remark 2.2. *(1) If is uniformly smooth, then in Lemma 2.1 holds.

(2) If is a Hilbert space, then in Lemma 2.1 holds.

Lemma 2.3 (see [8]). *Let be a mapping with . Then is strictly hemicontractive if and only if there exists a constant such that for all and , there exists satisfying*

Lemma 2.4 (see [10]). *Let be an arbitrary normed linear space and let be a mapping. *(a)*If is a local strongly pseudocontractive mapping and , then is a singleton and is strictly hemicontractive.*(b)*If is strictly hemicontractive, then is a singleton.*

Lemma 2.5 (see [10]). *Let ,, and be nonnegative real sequences and let be a constant satisfying
**
where , for all and . Then .*

#### 3. Main Results

We now prove our main results.

Lemma 3.1. *Let be a smooth Banach space. Suppose that one of the following holds:*(a)* is uniformly continuous on any bounded subsets of ,*(b)* for all , in ,*(c)*for any bounded subset of , there is a function such that** for all , where satisfies .**Then for any and any bounded subset , there exists such that**
for all and ;.**Proof. *For ;, by using (2.2), consider
This completes the proof.

Theorem 3.2. *Let be a smooth Banach space satisfying any one of the Axioms – of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a continuous strictly hemicontractive mapping. Let , and be real sequences in satisfying conditions *(iv)*,
,
*(v)*,
*(vi)* and .**For a sequence in , suppose that is the sequence generated from an arbitrary by**
satisfying .**Then the sequence converges strongly to a unique fixed point of .*

*Proof. *By [2, Corollary 1], has a unique fixed point in . It follows from Lemma 2.4 that is a singleton. That is, for some .

Set . It is easy to verify that
Also
Consider
where the first inequality holds by the convexity of .

Now we put , where satisfies (2.3). Using (3.4) and Lemma 3.1, we infer that
where
Also, we have
implies
as , and consequently
as . Since is uniformly continuous on any bounded subsets of , we have
For any given and the bounded subset , there exists a satisfying (2.2). Note that (3.13) and (vi) ensure that there exists an such that
Now substituting (3.6) in (3.8) to obtain
by using (3.7), implies
for all .

Put
and we have from (3.16)
For , set . Because , we imply and . Now observe that , for all and . It follows from Lemma 2.5 that
Letting , we obtain that , which implies that as . This completes the proof.

Corollary 3.3. *Let be a smooth Banach space satisfying any one of the Axioms – of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a Lipschitz strictly hemicontractive mapping. Let , and be real sequences in satisfying the conditions (iv)–(vi).**From arbitrary , define the sequence by the implicit iteration process (3.4). Then the sequence converges strongly to a unique fixed point of .*

Corollary 3.4. *Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a continuous strictly hemicontractive mapping. Suppose that be a real sequence in satisfying the conditions (v) and .**From arbitrary , define the sequence by the implicit iteration process (1.8). Then the sequence converges strongly to a unique fixed point of .*

Corollary 3.5. *Let be a smooth Banach space satisfying any one of the Axioms – of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be a Lipschitz strictly hemicontractive mapping. Suppose that be a real sequence in satisfying the conditions (v) and .**From arbitrary , define the sequence by the implicit iteration process (1.8). Then the sequence converges strongly to a unique fixed point of .*

*Remark 3.6. *Similar results can be found for the iteration processes involved error terms; we omit the details.

*Remark 3.7. *Theorem 3.2 and Corollary 3.3 extend and improve Theorem 1.4 in the following directions*. *

We do not need the assumption as in Theorem 1.4.

#### 4. Applications for Multistep Implicit Iterations

Let be a nonempty closed convex subset of a smooth Banach space and let , be a family of mappings.

*Algorithm 4.1. *For a given , compute the sequence by the implicit iteration process of arbitrary fixed order
which is called the *multistep implicit iteration process,* where , , , and , are real sequences in and , .

For , we obtain the following three-step implicit iteration process.

*Algorithm 4.2. *For a given , compute the sequence by the iteration process
where , , , and are real sequences in satisfying some certain conditions.

For , we obtain the following two-step implicit iteration process.

*Algorithm 4.3. *For a given , compute the sequence by the iteration process
where , , and are real sequences in satisfying some certain conditions.

If , and in (4.3), we obtain the following implicit Mann iteration process.

*Algorithm 4.4. *For any given , compute the sequence by the iteration process
where is a real sequence in satisfying some certain conditions.

Theorem 4.5. *Let be a smooth Banach space satisfying any one of the Axioms – of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be mappings. Let be continuous strictly hemicontractive mappings. Let , , and , be real sequences in satisfying the conditions (iv)–(vi) and . For arbitrary , define the sequence by (4.1). Then converges strongly to the common fixed point of .*

*Proof. *By applying Theorem 3.2 under assumption that and are continuous strictly hemicontractive mappings, we obtain Theorem 4.5 which proves strong convergence of the iteration process defined by (4.1). Consider by taking and ,
From (4.5) and the condition , we obtain
This completes the proof.

Corollary 4.6. *Let be a smooth Banach space satisfying any one of the Axioms of Lemma 3.1. Let be a nonempty closed bounded convex subset of and let be mappings. Let be Lipschitz strictly hemicontractive mappings. Let , , and , be real sequences in satisfying the conditions (iv)-(vi) and . For arbitrary , define the sequence by (4.1). Then converges strongly to the common fixed point of .*

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