- Research Article
- Open Access

# Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces

- AbdulRahim Khan
^{1}Email author, - Hafiz Fukhar-ud-din
^{2}and - AbdulAziz Domlo
^{3}

**2010**:385986

https://doi.org/10.1155/2010/385986

© Abdul Rahim Khan et al. 2010

**Received:**1 October 2009**Accepted:**19 January 2010**Published:**26 January 2010

## Abstract

We study strong convergence of the Ishikawa iterates of qasi-nonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corresponding results existing in the literature for uniformly convex Banach spaces.

## Keywords

- Convex Subset
- Fixed Point Theorem
- Strong Convergence
- Nonempty Subset
- Cauchy Sequence

## 1. Introduction and Preliminaries

Let be a nonempty subset of a metric space and let be a map. Denote the set of fixed points of by The map is said to be (i) quasi-nonexpansive if and for all and , (ii) -Lipschitz if for some we have for all for it becomes nonexpansive, and (iii) generalized nonexpansive (cf. [1] and the references therein) if

The concept of quasi-nonexpansiveness is more general than that of nonexpansiveness. A nonexpansive map with at least one fixed point is quasi-nonexpansive but there are quasi-nonexpansive maps which are not nonexpansive [2].

Mann and Ishikawa type iterates for nonexpansive and quasi-nonexpansive maps have been extensively studied in uniformly convex Banach spaces [1, 3–6]. Senter and Dotson [7] established convergence of Mann type iterates of quais-nonexpansive maps under a condition in uniformly convex Banach spaces. In 1973, Goebel et al. [8] proved that generalized nonexpansive self maps have fixed points in uniformly convex Banach spaces. Based on their work, Bose and Mukerjee [1] proved theorems for the convergence of Mann type iterates of generalized nonexpansive maps and obtained a result of Kannan [9] under relaxed conditions. Maiti and Ghosh [6] generalized the results of Bose and Mukerjee [1] for Ishikawa iterates by using modified conditions of Senter and Dotson [7] (see, also [10]). For the sake of completeness, we state the result of Kannan [9] and its generalization by Bose and Mukerjee [1].

Theorem 1.1 (see [9]).

Let be a nonempty, bounded, closed, and convex subset of a uniformly convex Banach space. Let be a map of into itself such that

(ii) where is any nonempty convex subset of which is mapped into itself by and is the diameter of

Then the sequence defined by converges to the fixed point of where is any arbitrary point of

Theorem 1.2 (see [1]).

for all where and Define a sequence in for , for all , where Then converges to a fixed point of .

In Theorem 1.2, taking , and for all it becomes Theorem 1.1 without requiring condition (ii).

In 1970, Takahashi [11] introduced a notion of convexity in a metric space as follows: a map is a convex structure in if

for all and A metric space together with a convex structure is said to be convex metric space. A nonempty subset of a convex metric space is convex if for all and In fact, every normed space and its convex subsets are convex metric spaces but the converse is not true, in general (see [11]). Later on, Shimizu and Takahashi [12] obtained some fixed point theorems for nonexpansive maps in convex metric spaces. This notion of convexity has been used in [13–15] to study Mann and Ishikawa iterations in convex metric spaces. For other fixed point results in the closely related classes of spaces, namely, hyperbolic and hyperconvex metric spaces, we refer to [16–19].

In the sequel, we assume that is a nonempty convex subset of a convex metric space and is a selfmap on . For an initial value we define the Ishikawa iteration scheme in as follows:

where and are control sequences in

If we choose then (1.3) reduces to the following Mann iteration scheme:

where is a control sequence in

If is a normed space with as its convex subset, then is a convex structure in consequently (1.3) and (1.4), respectively, become

where and are control sequences in

A convex metric space is said to be uniformly convex [11] if for arbitrary positive numbers and , there exists such that

In 1989, Maiti and Ghosh [6] generalized the two conditions due to Senter and Dotson [7]. We state all these conditions in convex metric spaces:

*Let*
*be a map with nonempty fixed point set*
and
. *Then*
*is said to satisfy the following Condotions*.

Condition 1.

If there is a nondecreasing function with and for all such that for .

Condition 2.

If there exists a real number such that for .

Condition 3.

If there is a nondecreasing function with and for all such that for and all corresponding where .

Condition 4.

If there exists a real number such that for and all corresponding where

Note that if satisfies Condition 1 (resp., 3), then it satisfies Condition 2 (resp., 4). We also note that Conditions 1 and 2 become Conditions A and B, respectively, of Senter and Dotson [7] while Conditions 3 and 4 become Conditions I and II, respectively, of Maiti and Ghosh [6] in a normed space. Further, Conditions 3 and 4 reduce to Conditions 1 and 2, respectively, when

In this note, we present results under relaxed control conditions which generalize the corresponding results of Kannan [9], Bose and Mukerjee [1], and Maiti and Ghosh [6] from uniformly convex Banach spaces to uniformly convex metric spaces. We present sufficient conditions for the convergence of Ishikawa iterates of Lipschitz maps to their fixed points in convex metric spaces and improve [3, Lemma 2]. A necessary and sufficient condition is obtained for the convergence of a sequence to fixed point of a generalized nonexpansive map in metric spaces.

We need the following fundamental result for the developmant of our results.

Theorem 1.3 (see [20]).

## 2. Convergence Analysis

We prove a lemma which plays key role to establish strong convergence of the iterative schemes (1.3) and (1.4).

Lemma 2.1.

Let be a uniformly convex metric space. Let be a nonempty closed convex subset of a quasi-nonexpansive map and as in (1.3). If and then

Proof.

This implies that the sequence is nonincreasing and bounded below. Thus exists. We may assume that

Since exists, so is bounded and hence exists. We show that . Assume that

Then

This is contradiction and hence

In the light of above result, we can construct subsequences and of and , respectively, such that and hence

Now we state and prove Ishikawa type convergence result in uniformly convex metric spaces.

Theorem 2.2.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 3. If is as in (1.3), where and , then converges to a fixed point of .

Proof.

Using the properties of we have . As exists, therefore

Now, we show that is a Cauchy sequence. For there exists a constant such that for all we have In particular, That is, There must exist such that Now, for , we have that

This proves that is a Cauchy sequence in . Since is a closed subset of a complete metric space therefore it must converge to a point in .

Finally, we prove that is a fixed point of

Since

Choose for all in the above theorem; it reduces to the following Mann type convergence result.

Theorem 2.3.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous quasi-nonexpansive map of into itself satisfying Condition 1. If is as in (1.4), where , then converges to a fixed point of .

Next we establish strong convergence of Ishikawa iterates of a generalized nonexpansive map.

Theorem 2.4.

Let and be as in Theorem 2.3. Let be a continuous generalied nonexpansive map of into itself with at least one fixed point. If is as in (1.3), where and then converges to a fixed point of .

Proof.

where Thus satisfies Condition 4 (and hence Condition 3). The result now follows from Theorem 2.2.

Remark 2.5.

In the above theorem, we have assumed that the generalied nonexpansive map has a fixed point. It remains an open questions: what conditions on , and in (*) are sufficient to guarantee the existence of a fixed point of even in the setting of a metric space.

Choose for all in Theorem 2.4 to get the following Mann type convergence result.

Theorem 2.6.

Let and be as in Theorem 2.4. If is as in (1.4), where then converges to a fixed point of .

Proof.

Thus satisfies Condition 2 (and hence Condition 1) and so the result follows from Theorem 2.3.

The analogue of Kannan result in uniformly convex metric space can be deduced from Theorem 2.6 (by taking , and for all ) as follows.

Theorem 2.7.

Let be a uniformly convex complete metric space with continuous convex structure and let be its nonempty closed convex subset. Let be a continuous map of into itself with at least one fixed point such that for all . Then the sequence where and converges to a fixed point of

Next we give sufficient conditions for the existence of fixed point of a -Lipschitz map in terms of the Ishikawa iterates.

Theorem 2.8.

Let be a convex metric space and let be its nonempty convex subset. Let be a -Lipschitz selfmap of Let be the sequence as in (1.3), where and satisfy (i) for all (ii) and (iii) If and then is a fixed point of

Proof.

Taking on both the sides in the above inequality and using the condition , we have

Finally, using a generalized nonexpansive map on a metric space , we provide a necessary and sufficient condition for the convergence of an arbitrary sequence in to a fixed point of in terms of the approximating sequence

Theorem 2.9.

for all Then a sequence in converges to a fixed point of if and only if

Proof.

That is,

Since and therefore from the above inequality, it follows that is a Cauchy sequence in In view of closedness of this sequence converges to an element of Also gives that Now using the continuity of we have Hence is a fixed point of

Conversely, suppose that converges to a fixed point of Using the continuity of we have that Thus

Remark 2.10.

Theorem 2.8 improves Lemma 2 in [3] from real line to convex metric space setting. Theorem 2.9 is an extension of Theorem 4 in [21] to metric spaces. If we choose in Theorem 2.9, it is still an improvement of [21, Theorem 4].

Remark 2.11.

We have proved our results (2.1)–(2.8) in convex metric space setting. All these results, in particular, hold in Banach spaces if we set

## Declarations

### Acknowledgment

The author A. R. Khan is grateful to King Fahd University of Petroleum & Minerals for support during this research.

## Authors’ Affiliations

## References

- Bose RK, Mukherjee RN:
**Approximating fixed points of some mappings.***Proceedings of the American Mathematical Society*1981,**82**(4):603–606. 10.1090/S0002-9939-1981-0614886-7MathSciNetView ArticleMATHGoogle Scholar - Petryshyn WV, Williamson TE Jr.:
**Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings.***Journal of Mathematical Analysis and Applications*1973,**43:**459–497. 10.1016/0022-247X(73)90087-5MathSciNetView ArticleMATHGoogle Scholar - Deng L, Ding XP:
**Ishikawa's iterations of real Lipschitz functions.***Bulletin of the Australian Mathematical Society*1992,**46**(1):107–113. 10.1017/S0004972700011710MathSciNetView ArticleMATHGoogle Scholar - Fukhar-ud-din H, Khan AR:
**Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces.***Computers & Mathematics with Applications*2007,**53**(9):1349–1360. 10.1016/j.camwa.2007.01.008MathSciNetView ArticleMATHGoogle Scholar - Khan AR, Domlo A-A, Fukhar-ud-din H:
**Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(1):1–11. 10.1016/j.jmaa.2007.06.051MathSciNetView ArticleMATHGoogle Scholar - Maiti M, Ghosh MK:
**Approximating fixed points by Ishikawa iterates.***Bulletin of the Australian Mathematical Society*1989,**40**(1):113–117. 10.1017/S0004972700003555MathSciNetView ArticleMATHGoogle Scholar - Senter HF, Dotson WG Jr.:
**Approximating fixed points of nonexpansive mappings.***Proceedings of the American Mathematical Society*1974,**44:**375–380. 10.1090/S0002-9939-1974-0346608-8MathSciNetView ArticleMATHGoogle Scholar - Goebel K, Kirk WA, Shimi TN:
**A fixed point theorem in uniformly convex spaces.***Bollettino dell'Unione Matematica Italiana*1973,**7:**67–75.MathSciNetMATHGoogle Scholar - Kannan R:
**Some results on fixed points. III.***Fundamenta Mathematicae*1971,**70**(2):169–177.MathSciNetMATHGoogle 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 - Takahashi W:
**A convexity in metric space and nonexpansive mappings. I.***Kōdai Mathematical Seminar Reports*1970,**22:**142–149. 10.2996/kmj/1138846111View ArticleMathSciNetMATHGoogle Scholar - Shimizu T, Takahashi W:
**Fixed point theorems in certain convex metric spaces.***Mathematica Japonica*1992,**37**(5):855–859.MathSciNetMATHGoogle Scholar - Cirić LB:
**On some discontinuous fixed point mappings in convex metric spaces.***Czechoslovak Mathematical Journal*1993,**43(118)**(2):319–326.MathSciNetMATHGoogle Scholar - Ding XP:
**Iteration processes for nonlinear mappings in convex metric spaces.***Journal of Mathematical Analysis and Applications*1988,**132**(1):114–122. 10.1016/0022-247X(88)90047-9MathSciNetView ArticleMATHGoogle Scholar - Talman LA:
**Fixed points for condensing multifunctions in metric spaces with convex structure.***Kōdai Mathematical Seminar Reports*1977,**29**(1–2):62–70.MathSciNetView ArticleMATHGoogle Scholar - Espinola R, Hussain N:
**Common fixed points for multimaps in metric spaces.***Fixed Point Theory and Applications*2010, Article ID 204981**2010:**-14.Google Scholar - Khamsi MA, Kirk WA:
*An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics*. Wiley-Interscience, New York, NY, USA; 2001:x+302.View ArticleMATHGoogle Scholar - Khamsi MA, Kirk WA, Yañez CM:
**Fixed point and selection theorems in hyperconvex spaces.***Proceedings of the American Mathematical Society*2000,**128**(11):3275–3283. 10.1090/S0002-9939-00-05777-4MathSciNetView ArticleMATHGoogle Scholar - Kirk WA:
**Krasnoselskiĭ's iteration process in hyperbolic space.***Numerical Functional Analysis and Optimization*1981/82,**4**(4):371–381.MathSciNetView ArticleMATHGoogle Scholar - Shimizu T: A convergence theorem to common fixed points of families of nonexpansive mappings in convex metric spaces. Proceedings of the International Conference on Nonlinear and Convex Analysis, 2005 575–585.Google Scholar
- Rassias ThM:
**Some theorems of fixed points in nonlinear analysis.***Bulletin of the Institute of Mathematics. Academia Sinica*1985,**13**(1):5–12.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.