Skip to main content
  • Research Article
  • Open access
  • Published:

Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings

Abstract

A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are studied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.

1. Introduction and Preliminaries

Let be a metric space and . A mapping is said to be nonexpansive if

(1.1)

and it is said to be weakly contractive if

(1.2)

where is continuous and nondecreasing such that is positive on , and .

It is evident that is contractive if it is weakly contractive with , where , and it is nonexpansive if it is weakly contractive.

As an important extension of the class of contractive mappings, the class of weakly contractive mappings was introduced by Alber and Guerre-Delabriere [1]. In Hilbert and Banach spaces, Alber et al. [1–4] and Rhoades [5] established convergence theorems on iteration of fixed point for weakly contractive single mapping.

Inspired by [2, 5, 6], the purpose of this paper is to study a family of commuting nonexpansive mappings, one of which is weakly contractive, in arbitrary complete metric spaces and Banach spaces.

We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error estimates.

Throughout this paper, we assume that is the set of fixed points of a mapping , that is, ; is defined by the antiderivative (indefinite integral) of on , that is, , and is the inverse function of .

We define iterations which will be needed in the sequel.

Suppose that is a metric space and , is a family of commuting self-mappings of and . The iteration of type Krasnoselski-Mann (see [7, 8]) is cyclically defined by

(1.3)

For convenience, we write

(1.4)

where the function takes values in .

Let be a closed convex subset of the normed space . Then the iteration of type Kirk (see [5, 9]) is defined by

(1.5)

Again, the iteration of type lshikawa with error (see [10–12]) is defined by

(1.6)

where , , , and

(1.7)

We will make use of following result in the proof of Theorem 2.4.

Lemma 1.1 (see [12]).

Suppose that , are two sequences of nonnegative numbers such that , for all . If , then exists.

2. Main Result

Theorem 2.1.

Let be a complete metric space and let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive, then there is a unique common fixed point and the iteration of type Krasnoselski-Mann generated by (1.4) converges in metric to , with the following error estimate:

(2.1)

where is the Gauss integer of .

Proof.

The uniqueness of fixed point of is clear from (1.2). Hence, the common fixed point of is unique. Let be an arbitrary point in and let be an iteration of type Krasnoselski-Mann generated by (1.4). Since is commutative, then we have . Suppose that and . Then,

(2.2)

Write for fixed . Then is a subsequence of . Since is nonexpansive and is weakly contractive, then we obtain

(2.3)

which shows , that is, is a nonincreasing sequence of nonnegative real numbers. Therefore, it tends to a limit . If , then, by nondecreasity of , Thus, from (2.3) it follows that

(2.4)

a contradiction for large enough. Therefore,

(2.5)

By (2.5), for any given , there exists such that

(2.6)

We claim that

(2.7)

In fact, from (2.6) we see that (2.7) holds when . Suppose that . If , then from (2.6) we get

(2.8)

If , then we also get

(2.9)

Therefore, by induction we derive that (2.7) holds. Since is arbitrary, is a Cauchy sequence. As is complete, we have

(2.10)

Observe that are all continuous, so is . From (2.10), it follows that

(2.11)
(2.12)

By (1.1), (1.2), and (2.11), we deduce

(2.13)

which shows

(2.14)

From (2.12), it implies that is a common fixed point of that is, . Hence, . By (2.10) and (2.14), we conclude . Set From (2.3), we have

(2.15)

Since is continuous and nondecreasing, using (2.15), it yields

(2.16)

Observe that

(2.17)

From (2.16) and (2.17), we obtain the error estimate (2.1). This completes the proof.

Remark 2.2.

If in Theorem 2.1, where is the identity mapping of , then we conclude that the sequence converges to the unique common fixed point of weakly contractive mapping , with the error estimate , where . Thus, our Theorem 2.1 is a generalization of the corresponding theorem of Rhoades [5].

Theorem 2.3.

Let be a Banach space and let be a nonempty closed convex set. Let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive. Then, for any , the iteration of type Kirk generated by (1.5) converges strongly to a unique common fixed point , with the following error estimate:

(2.18)

Proof.

Applying Theorem 2.1, we can suppose that is a unique common fixed point of . Since

(2.19)

we derive that is a fixed point of . Since are all nonexpansive, is weakly contractive, and , then we have

(2.20)

The inequality (2.20) shows that is weakly contractive. Thus, is a unique fixed point of . Set Then,

(2.21)

and converges to with the following error estimate (see Remark 2.2):

(2.22)

Observe that

(2.23)

From (2.21)–(2.23), we obtain (2.18). This completes the proof.

Theorem 2.4.

Let be a Banach space and let be a nonempty closed convex set. Let be a family of commuting self-mappings, where are all nonexpansive and is weakly contractive. For any , let be the iteration of type Ishikawa generated by (1.6), where

(2.24)

and are all bounded. Then, converges strongly to a unique common fixed point with the following estimate:

(2.25)

where .

Proof.

Applying Theorem 2.1, we can suppose that is a unique common fixed point of . Since are all bounded, we have . Since are all nonexpansive and is weakly contractive, we obtain in proper order that

(2.26)

Write Then and (2.26) yields

(2.27)
(2.28)

From (2.27) and Lemma 1.1, it implies that exists, and so does by the continuity of . From (2.28), it implies that Since we conclude that Therefore, that is, converges strongly to . To establish the error estimate, we set and Then, (2.26) yields

(2.29)

Set From (2.29) we have

(2.30)

Since is nondecreasing, from (2.30) we deduce

(2.31)

Thus,

(2.32)

Hence, the estimate (2.25) holds. This completes the proof.

References

  1. Alber YI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications. Volume 98. Birkhäuser, Basel, Switzerland; 1997:7-22.

    Chapter  Google Scholar 

  2. Alber YI, Chidume CE, Zegeye A: Approximating fixed points of total asymptotically nonexpansive mappings. Fixed Point Theory and Applications 2006, 2006:-20.

    Google Scholar 

  3. Alber YI, Guerre-Delabriere S: On the projection methods for fixed point problems. Analysis 2001, 21(1):17-39.

    Article  MATH  MathSciNet  Google Scholar 

  4. Alber YI, Guerre-Delabriere S, Zelenko L: The principle of weakly contractive mappings in metric spaces. Communications on Applied Nonlinear Analysis 1998, 5(1):45-68.

    MATH  MathSciNet  Google Scholar 

  5. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Analysis: Theory, Methods and Applications 2001, 47(4):2683-2693. 10.1016/S0362-546X(01)00388-1

    Article  MATH  MathSciNet  Google Scholar 

  6. Hussain N, Khan AR: Common fixed-point results in best approximation theory. Applied Mathematics Letters 2003, 16(4):575-580. 10.1016/S0893-9659(03)00039-9

    Article  MATH  MathSciNet  Google Scholar 

  7. Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996, 202(1):150-159. 10.1006/jmaa.1996.0308

    Article  MATH  MathSciNet  Google Scholar 

  8. Borwein J, Reich S, Shafrir I: Krasnoselski-Mann iterations in normed spaces. Canadian Mathematical Bulletin 1992, 35(1):21-28. 10.4153/CMB-1992-003-0

    Article  MATH  MathSciNet  Google Scholar 

  9. Kirk WA: On successive approximations for nonexpansive mappings in Banach spaces. Glasgow Mathematical Journal 1971, 12: 6-9. 10.1017/S0017089500001063

    Article  MATH  MathSciNet  Google Scholar 

  10. Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44(1):147-150. 10.1090/S0002-9939-1974-0336469-5

    Article  MATH  MathSciNet  Google Scholar 

  11. Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 1976, 59(1):65-71. 10.1090/S0002-9939-1976-0412909-X

    Article  MATH  MathSciNet  Google Scholar 

  12. Tan KK, Xu HK: 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.1993.1309

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10671094).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jian-Zhong Xiao.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Xiao, JZ., Zhu, XH. Common Fixed Point Theorems on Weakly Contractive and Nonexpansive Mappings. Fixed Point Theory Appl 2008, 469357 (2007). https://doi.org/10.1155/2008/469357

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2008/469357

Keywords