- Research Article
- Open Access

# Approximating Fixed Points of Non-Lipschitzian Mappings by Metric Projections

- Hossein Dehghan
^{1}Email author, - Amir Gharajelo
^{2}and - Davood Afkhamitaba
^{3}

**2011**:976192

https://doi.org/10.1155/2011/976192

© Hossein Dehghan et al. 2011

**Received:**22 November 2010**Accepted:**8 February 2011**Published:**1 March 2011

## Abstract

We define and study a new iterative algorithm inspired by Matsushita and Takahashi (2008). We establish a strong convergence theorem of the proposed algorithm for asymptotically nonexpansive in the intermediate sense mappings in uniformly convex and smooth Banach spaces by using metric projections. This theorem generalizes and refines Matsushita and Takahashi's strong convergence theorem which was established for nonexpansive mappings.

## Keywords

- Banach Space
- Iterative Algorithm
- Nonexpansive Mapping
- Lipschitzian Mapping
- Real Banach Space

## 1. Introduction and Preliminaries

*asymptotically nonexpansive*if there exists a sequence in with such that

*nonexpansive*mapping. is said to be

*asymptotically nonexpansive in the intermediate sense*[1] provided is uniformly continuous and

for all .

It follows from the above definitions that every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense and Lipschitzian mapping but the converse does not hold such as in the following example.

Example 1.1.

Thus, is asymptotically nonexpansive in the intermediate sense.

This contradiction shows that is not Lipschitzian mapping and so it is not asymptotically nonexpansive mapping. Another example of an asymptotically nonexpansive in the intermediate sense mapping which is not asymptotically nonexpansive can be found in [2].

*strictly convex*if for all with and . A Banach space is also said to be

*uniformly convex*if for any two sequences and in such that and . A Banach space is said to have Kadec-Klee property if for every sequence in , and imply that . Every uniformly convex Banach space has the Kadec-Klee property [4]. Let be the unit sphere of . Then the Banach space is said to be

*smooth*if

*metric projection*from onto . Let and . Then, it is known that if and only if

Fixed points of nonlinear mappings play an important role in solving systems of equations and inequalities that often arise in applied sciences. Approximating fixed points of asymptotically nonexpansive and nonexpansive mappings with implicit and explicit iterative schemes has been studied by many authors (see, e.g., [8–14]).

On the other hand, using the metric projection, Nakajo and Takahashi [15] introduced an iterative algorithm in the framework of Hilbert spaces and gave strong convergence theorem for nonexpansive mappings. Xu [16] extended Nakajo and Takahashi's theorem to Banach spaces by using the generalized projection. Recently, Matsushita and Takahashi [17] introduced an iterative algorithm for nonexpansive mappings in Banach spaces as follows.

where denotes the convex closure of the set and is a sequence in with . They proved that generated by (1.14) converges strongly to a fixed point of .

where is a sequence in with and is the metric projection from onto .

In the sequel, the following lemmas are needed to prove our main convergence theorem.

Lemma 1.2 (see [18, Lemma 1.5]).

for any numbers with .

Lemma 1.3 (see [18, Lemma 1.6]).

then is demiclosed at zero; that is, for each sequence in , if for some and , then .

## 2. Main Results

In this section, we study the iterative algorithm (1.15) to find fixed points of asymptotically nonexpansive in the intermediate sense mappings in a uniformly convex and smooth Banach space. We first prove that the sequence generated by (1.15) is well-defined. Then, we prove that converges strongly to , where is the metric projection from onto .

Lemma 2.1.

Let be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space , and let be a mapping which is asymptotically nonexpansive in the intermediate sense. If , then the sequence generated by (1.15) is well-defined.

Proof.

for all and so for all , that is . Thus, . By mathematical induction, we obtain that for all . Therefore, is well-defined.

In order to prove our main result, the following lemma is needed.

Lemma 2.2.

for all integers .

Proof.

This completes the proof.

Now, we state and prove the strong convergence theorem of the iterative algorithm (1.15).

Theorem 2.3.

Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space , let be a mapping which is asymptotically nonexpansive in the intermediate sense and let be the sequence generated by (1.15). Then converges strongly to the element of .

Proof.

Since is uniformly convex, by Kadec-Klee property, we obtain that . It follows that . This completes the proof.

Since every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically nonexpansive in the intermediate sense, we have the following result which generalizes and refines the strong convergence theorem of Matsushita and Takahashi [17, Theorem 3.1].

Corollary 2.4.

Let be a nonempty bounded closed convex subset of a uniformly convex and smooth Banach space , let be a nonexpansive self-mapping of , and let be the sequence generated by (1.15). Then converges strongly to the element of .

## Declarations

### Acknowledgment

The authors thank the referees and the editor for their careful reading of the manuscript and their many valuable comments and suggestions for the improvement of this paper.

## Authors’ Affiliations

## References

- Bruck R, Kuczumow T, Reich S:
**Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property.***Colloquium Mathematicum*1993,**65**(2):169–179.MATHMathSciNetGoogle Scholar - Kim GE, Kim TH:
**Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces.***Computers & Mathematics with Applications*2001,**42**(12):1565–1570. 10.1016/S0898-1221(01)00262-0MATHMathSciNetView ArticleGoogle Scholar - Kirk WA:
**Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type.***Israel Journal of Mathematics*1974,**17:**339–346. 10.1007/BF02757136MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, O'Regan D, Sahu DR:
*Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Topological Fixed Point Theory and Its Applications*.*Volume 6*. Springer, New York, NY, USA; 2009:x+368.MATHGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications*.*Volume 62*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. Fixed Point Theory and Its ApplicationsMATHGoogle Scholar - Alber YI:
**Metric and generalized projection operators in Banach spaces: properties and applications.**In*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.*.*Volume 178*. Dekker, New York, NY, USA; 1996:15–50.Google 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.MATHMathSciNetView ArticleGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MATHMathSciNetView ArticleGoogle Scholar - Schu J:
**Iterative construction of fixed points of asymptotically nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1991,**158**(2):407–413. 10.1016/0022-247X(91)90245-UMATHMathSciNetView ArticleGoogle Scholar - Chidume CE, Shahzad N, Zegeye H:
**Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense.***Numerical Functional Analysis and Optimization*2004,**25**(3–4):239–257.MATHMathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N:
**Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(1):325–329. 10.1016/j.na.2009.06.056MATHMathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N:
**Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4496–4503. 10.1016/j.na.2007.11.005MATHMathSciNetView ArticleGoogle Scholar - Zegeye H, Shahzad N:
**Corrigendum to: "Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups".***Nonlinear Analysis: Theory, Methods and Applications*2010,**73**(6):1905–1907. 10.1016/j.na.2010.04.032MATHMathSciNetView ArticleGoogle Scholar - Nakajo K, Takahashi W:
**Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups.***Journal of Mathematical Analysis and Applications*2003,**279**(2):372–379. 10.1016/S0022-247X(02)00458-4MATHMathSciNetView ArticleGoogle Scholar - Xu H-K:
**Strong convergence of approximating fixed point sequences for nonexpansive mappings.***Bulletin of the Australian Mathematical Society*2006,**74**(1):143–151. 10.1017/S0004972700047535MATHMathSciNetView ArticleGoogle Scholar - Matsushita S-Y, Takahashi W:
**Approximating fixed points of nonexpansive mappings in a Banach space by metric projections.***Applied Mathematics and Computation*2008,**196**(1):422–425. 10.1016/j.amc.2007.06.006MATHMathSciNetView ArticleGoogle Scholar - Yang L-P, Xie X, Peng S, Hu G:
**Demiclosed principle and convergence for modified three step iterative process with errors of non-Lipschitzian mappings.***Journal of Computational and Applied Mathematics*2010,**234**(4):972–984. 10.1016/j.cam.2009.01.022MATHMathSciNetView ArticleGoogle Scholar

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