- Research Article
- Open Access

# Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi- -Nonexpansive Mapping in Banach Space

- Farrukh Mukhamedov
^{1}Email author and - Mansoor Saburov
^{1}

**2010**:719631

https://doi.org/10.1155/2010/719631

© F. Mukhamedov and M. Saburov. 2010

**Received:**31 August 2009**Accepted:**6 December 2009**Published:**9 December 2009

## Abstract

We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi- -nonexpansive mapping and an asymptotically quasi-nonexpansive mapping , defined on a nonempty closed convex subset of a Banach space.

## Keywords

- Hilbert Space
- Banach Space
- Convex Subset
- Nonexpansive Mapping
- Strong Convergence

## 1. Introduction

Let be a nonempty subset of a real normed linear space and let be a mapping. Denote by the set of fixed points of , that is, . Throughout this paper, we always assume that . Now let us recall some known definitions.

Definition 1.1.

A mapping is said to be

(i)nonexpansive, if for all ;

(ii)asymptotically nonexpansive, if there exists a sequence with such that for all and ;

(iii)quasi-nonexpansive, if for all ;

(iv)asymptotically quasi-nonexpansive, if there exists a sequence with such that for all and .

Note that from the above definitions, it follows that a nonexpansive mapping must be asymptotically nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, but the converse does not hold (see [1]).

If
is a closed nonempty subset of a Banach space and
is nonexpansive, then it is known that
may not have a fixed point (unlike the case if
is a strict contraction), and even when it has, the sequence
defined by
(the so-called *Picard sequence*) may fail to converge to such a fixed point.

In [2, 3] Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the studies of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians (see for more details [1, 4]).

In [5] Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces. Ghosh and Debnath [6] established a necessary and sufficient condition for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banach space. The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied extensively by Goebel and Kirk [7], Liu [8], Wittmann [9], Reich [10], Gornicki [11], Schu [12] Shioji and Takahashi [13], and Tan and Xu [14] in the settings of Hilbert spaces and uniformly convex Banach spaces.

There are many methods for approximating fixed points of a nonexpansive mapping. Xu and Ori [15] introduced implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. Recently, Sun [16] has extended an implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces. In [17] it has been studied the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces, which extends and improves the mentioned papers (see also [18, 19] for applications and other methods of implicit iteration processes).

There are many concepts which generalize a notion of nonexpansive mapping. One of such concepts is -nonexpansivity of a mapping ([20]). Let us recall some notions.

Definition 1.2.

Let , be two mappings of a nonempty subset of a real normed linear space . Then is said to be

(i)
*-* nonexpansive, if
for all
;

(ii)asymptotically
*-* nonexpansive, if there exists a sequence
with
such that
for all
and
;

(iii)asymptotically quasi
*-* nonexpansive mapping, if there exists a sequence
with
such that
for all
and

Remark 1.3.

If then an asymptotically -nonexpansive mapping is asymptotically quasi- -nonexpansive. But, there exists a nonlinear continuous asymptotically quasi -nonexpansive mappings which is asymptotically -nonexpansive.

In [21] a weakly convergence theorem for -asymptotically quasi-nonexpansive mapping defined in Hilbert space was proved. In [22] strong convergence of Mann iterations of -nonexpansive mapping has been proved. Best approximation properties of -nonexpansive mappings were investigated in [20]. In [23] the weak convergence of three-step Noor iterative scheme for an -nonexpansive mapping in a Banach space has been established. Recently, in [24] the weak and strong convergence of implicit iteration process to a common fixed point of a finite family of -asymptotically nonexpansive mappings were studied. Assume that the family consists of one -asymptotically nonexpansive mapping . Now let us consider an iteration method used in [24], for , which is defined by

where and are two sequences in . From this formula one can easily see that the employed method, indeed, is not implicit iterative processes. The used process is some kind of modified Ishikawa iteration.

In this paper we will prove the weak and strong convergences of the implicit iterative process (1.2) to a common fixed point of and . All results presented here generalize and extend the corresponding main results of [15–17] in a case of one mapping.

## 2. Preliminaries

Throughout this paper, we always assume that
is a real Banach space. We denote by
and
the set of fixed points and the domain of a mapping
respectively. Recall that a Banach space
is said to satisfy *Opial condition* [25], if for each sequence
in
converging weakly to
implies that

for all with It is well known that (see [26]) inequality (2.1) is equivalent to

Definition 2.1.

Let be a closed subset of a real Banach space and let be a mapping.

(i)A mapping is said to be semiclosed (demiclosed) at zero, if for each bounded sequence in the conditions converges weakly to and converges strongly to imply

(ii)A mapping is said to be semicompact, if for any bounded sequence in such that then there exists a subsequence such that strongly.

(iii) is called a uniformly -Lipschitzian mapping, if there exists a constant such that for all and

The following lemmas play an important role in proving our main results.

Lemma 2.2 (see [12]).

holds some Then

Lemma 2.3 (see [14]).

Let and be two sequences of nonnegative real numbers with If one of the following conditions is satisfied:

(i)

(ii)

then the limit exists.

## 3. Main Results

In this section we will prove our main results. To formulate one, we need some auxiliary results.

Lemma 3.1.

Let be a real Banach space and let be a nonempty closed convex subset of Let be an asymptotically quasi -nonexpansive mapping with a sequence and be an asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

If is the implicit iterative sequence defined by (1.2), then for each the limit exists.

Proof.

exists, where is a constant. This completes the proof.

Now we prove the following result.

Theorem 3.2.

Let be a real Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi- -nonexpansive mapping with a sequence and let be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

Proof.

The necessity of condition (3.12) is obvious. Let us proof the sufficiency part of theorem.

Since are uniformly -Lipschitzian mappings, so and are continuous mappings. Therefore the sets and are closed. Hence is a nonempty closed set.

For any given we have (see (3.8))

Let us prove that the sequence converges to a common fixed point of and In fact, due to for all and from (3.13), we obtain

for all , where

Since then for any given there exists a positive integer number such that

which means that the strong convergence of the sequence is a common fixed point of and This proves the required assertion.

We need one more auxiliary result.

Proposition 3.3.

Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi- -nonexpansive mapping with a sequence and let be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in which satisfy the following conditions:

(i)

(ii)

(iii)

Proof.

According to Lemma 3.1 for any we have . It follows from (1.2) that

Now we are ready to formulate one of main results concerning weak convergence of the sequence .

Theorem 3.4.

Let be a real uniformly convex Banach space satisfying Opial condition and let be a nonempty closed convex subset of Let be an identity mapping, let be a uniformly -Lipschitzian asymptotically quasi- -nonexpansive mapping with a sequence and, be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in satisfying the following conditions:

(i)

(ii)

(iii)

If the mappings and are semiclosed at zero, then the implicitly iterative sequence defined by (1.2) converges weakly to a common fixed point of and

Proof.

Since the mappings and are semiclosed at zero, therefore, we find and which means

Finally, let us prove that converges weakly to In fact, suppose the contrary, that is, there exists some subsequence such that converges weakly to and . Then by the same method as given above, we can also prove that

Taking and and using the same argument given in the proof of (3.11), we can prove that the limits and exist, and we have

This is a contradiction. Hence This implies that converges weakly to This completes the proof of Theorem 3.4.

Now we formulate next result concerning strong convergence of the sequence .

Theorem 3.5.

Let be a real uniformly convex Banach space and let be a nonempty closed convex subset of Let be a uniformly -Lipschitzian asymptotically quasi- -nonexpansive mapping with a sequence and be a uniformly -Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence such that Suppose and and are two sequences in satisfying the following conditions:

(i)

(ii)

(iii)

If at least one mapping of the mappings and is semicompact, then the implicitly iterative sequence defined by (1.2) converges strongly to a common fixed point of and

Proof.

which means that converges to This completes the proof.

Note that all results presented here generalize and extend the corresponding main results of [15–17] in a case of one mapping.

## Declarations

### Acknowledgment

The authors acknowledge the MOSTI Grant 01-01-08-SF0079.

## Authors’ Affiliations

## References

- Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Browder FE:
**Nonexpansive nonlinear operators in a Banach space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**54:**1041–1044. 10.1073/pnas.54.4.1041MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces.***Archive for Rational Mechanics and Analysis*1967,**24:**82–90.MathSciNetView ArticleMATHGoogle Scholar - Chidume Ch:
*Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics*.*Volume 1965*. Springer, London, UK; 2009:xviii+326.MATHGoogle Scholar - Diaz JB, Metcalf FT:
**On the structure of the set of subsequential limit points of successive approximations.***Bulletin of the American Mathematical Society*1967,**73:**516–519. 10.1090/S0002-9904-1967-11725-7MathSciNetView ArticleMATHGoogle Scholar - Ghosh MK, Debnath L:
**Convergence of Ishikawa iterates of quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1997,**207**(1):96–103. 10.1006/jmaa.1997.5268MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Kirk WA:
**A fixed point theorem for asymptotically nonexpansive mappings.***Proceedings of the American Mathematical Society*1972,**35:**171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleMATHGoogle Scholar - Liu Q:
**Iterative sequences for asymptotically quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2001,**259**(1):1–7. 10.1006/jmaa.2000.6980MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Gornicki J:
**Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces.***Commentationes Mathematicae Universitatis Carolinae*1989,**30**(2):249–252.MathSciNetMATHGoogle Scholar - Schu J:
**Weak and strong convergence to fixed points of asymptotically nonexpansive mappings.***Bulletin of the Australian Mathematical Society*1991,**43**(1):153–159. 10.1017/S0004972700028884MathSciNetView ArticleMATHGoogle Scholar - Shioji N, Takahashi W:
**Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces.***Nonlinear Analysis: Theory, Methods & Applications*1998,**34**(1):87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleMATHGoogle Scholar - Tan K-K, Xu H-K:
**Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process.***Journal of Mathematical Analysis and Applications*1993,**178**(2):301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleMATHGoogle Scholar - Xu H-K, Ori RG:
**An implicit iteration process for nonexpansive mappings.***Numerical Functional Analysis and Optimization*2001,**22**(5–6):767–773. 10.1081/NFA-100105317MathSciNetView ArticleMATHGoogle Scholar - Sun Z-H:
**Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2003,**286**(1):351–358. 10.1016/S0022-247X(03)00537-7MathSciNetView ArticleMATHGoogle Scholar - Gu F, Lu J:
**A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces.***Fixed Point Theory and Applications*2006,**2006:**-11.Google Scholar - Li HY, Li HZ:
**Strong convergence of an iterative method for equilibrium problems and variational inequality problems.***Fixed Point Theory and Applications*2009,**2009:**-21.Google Scholar - Zhang F, Su Y:
**Strong convergence of modified implicit iteration processes for common fixed points of nonexpansive mappings.***Fixed Point Theory and Applications*2007,**2007:**-9.Google Scholar - Shahzad N:
**Generalized -nonexpansive maps and best approximations in Banach spaces.***Demonstratio Mathematica*2004,**37**(3):597–600.MathSciNetMATHGoogle Scholar - Temir S, Gul O:
**Convergence theorem for -asymptotically quasi-nonexpansive mapping in Hilbert space.***Journal of Mathematical Analysis and Applications*2007,**329**(2):759–765. 10.1016/j.jmaa.2006.07.004MathSciNetView ArticleMATHGoogle Scholar - Rhodes BH, Temir S:
**Convergebce thorems for -nonexpansive mapping.***International Journal of Mathematics and Mathematical Sciences*2006,**2006:**-4.Google Scholar - Kumam P, Kumethong W, Jewwaiworn N:
**Weak convergence theorems of three-step Noor iterative scheme for -quasi-nonexpansive mappings in Banach spaces.***Applied Mathematical Sciences*2008,**2**(57–60):2915–2920.MathSciNetMATHGoogle Scholar - Temir S:
**On the convergence theorems of implicit iteration process for a finite family of -asymptotically nonexpansive mappings.***Journal of Computational and Applied Mathematics*2009,**225**(2):398–405. 10.1016/j.cam.2008.07.049MathSciNetView ArticleMATHGoogle Scholar - Opial Z:
**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleMATHGoogle Scholar - Lami Dozo E:
**Multivalued nonexpansive mappings and Opial's condition.***Proceedings of the American Mathematical Society*1973,**38:**286–292.MathSciNetView ArticleMATHGoogle Scholar

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