Convergence theorems of a new iteration for two nonexpansive mappings
© Hou and Du; licensee Springer. 2014
Received: 29 November 2013
Accepted: 7 February 2014
Published: 19 February 2014
The purpose of this paper is to introduce the following new general implicit iteration scheme for approximating the common fixed points of a pair of nonexpansive mappings in a uniformly convex Banach space: for any , the iterative process defined by , , where , , , , , , are seven sequences of real numbers satisfying , , and are two nonexpansive mappings. We approximate the common fixed points of these two mappings by weak and strong convergence of the scheme.
Let C be a nonempty subset of a real Banach space E. A mapping T of C into itself is called nonexpansive if holds for all . We first recall the following two iterative processes due to Ishikawa  and Mann , respectively.
is called the Mann iteration sequence.
In , Liu introduced the concepts of Ishikawa and Mann iterative processes with errors as follows.
where and are two summable sequences in E. In particular, if , , the sequence is called the Mann iteration sequence with errors.
Unfortunately, the definitions of Liu, which depend on the convergence of the error terms, are against the randomness of errors. Xu  studied the following new iteration process.
is called the Ishikawa iteration sequence with errors. Here and are two bounded sequences in C and , , , , , are six sequences in satisfying the following conditions: , , . In particular, if , , the sequence is called the Mann iteration sequence with errors. Chidume and Moore  studied the above schemes in 1999.
Recently Khan and Fukhar  considered the above iterative process with bounded errors.
Approximating fixed points is an important subject in the theory of nonexpansive mappings and its applications in numerous applied areas. One is the convergence of iteration schemes constructed through nonexpansive mappings. In this paper, we study the iterative scheme given in (1) for weak and strong convergence for a pair of nonexpansive mappings in a uniformly convex Banach space. Before our discussions, we first recall the following definitions.
It is well known that every Hilbert space satisfies the Opial condition (see for example ).
A mapping T is said to be semicompact (see, e.g., ) if for any sequence in C such that , there exists a subsequence of such that converges strongly to some .
A mapping T with domain and range in E is said to be demiclosed at a point if whenever is as sequence in such that converges weakly to and converges strongly to p, then .
We shall make use of the following results.
Lemma 1.1 
If , then exists.
If and has a subsequence converging to zero, then .
Lemma 1.2 
Let E be a uniformly convex Banach space satisfying Opial’s condition and let C be a nonempty closed convex subset of E. Let T be a nonexpansive mapping of C into itself. Then is demiclosed with respect to zero.
Lemma 1.3 
Suppose that E is a uniformly convex Banach space and for all positive integers n. Also suppose that and are two sequences of E such that , and hold for some . Then .
2 Main results
We first give the following key lemma.
, , , as ;
for some ;
for some .
exists for all and , and are all bounded;
The proof of the lemma is completed. □
Now we give the weak convergence first.
Theorem 2.2 Let E be a uniformly convex Banach space satisfying Opial’s condition and C, S, T, and be as taken in Lemma 2.1. If , then converges weakly to a common fixed point of S and T.
By Lemma 1.2, we know is demiclosed, then it is easy to see that . With a similar proof, it follows that also. Next we prove uniqueness. Suppose that this is not true, then there must exist a subsequence such that converges weakly to another and . Then by the same method given above, we can also prove .
This is a contradiction, hence . This implies that converges weakly to a common fixed point of S and T. □
Next we give several strong convergence results.
Theorem 2.3 Let E be a uniformly convex Banach space and C, be as taken in Lemma 2.1. If one of the nonexpansive mappings T and S is semicompact and , then converges strongly to a common fixed point of S and T.
This yields . By Lemma 2.1 again, exists for all , therefore must itself converge to . This completes the proof. □
Recall that a mapping , where C is a subset of E, is said to satisfy condition (A) if there exists a nondecreasing function with , for all such that for all where .
In , Senter and Dotson approximated fixed points of a nonexpansive mapping T by Mann iteration. Recently Maiti and Ghosh  and Tan and Xu  considered the approximation of fixed points of a nonexpansive mapping T by Ishikawa iteration under the same condition (A) which is weaker than the requirement that T is semicompact. Khan and Fukhar  modified this condition for two mappings as follows.
Let C be a subset of a Banach space E. Two mappings are said to satisfy condition () if there exists a nondecreasing function with , for all such that for all , where .
Note that condition () reduces to condition (A) when . We use condition () to study the strong convergence of defined in (1).
Theorem 2.4 Let E be a uniformly convex Banach space and C, be as taken in Lemma 2.1. Let T, S satisfy the condition () and , then converges strongly to a common fixed point of S and T.
That is to say, shows that exists by virtue of Lemma 1.1. Now by condition (), . By the properties of f, therefore . Next we can take a subsequence of and such that . Then following the method of proof of Tan and Xu , we find that is a Cauchy sequence in F and so it converges. Let . Since F is closed, therefore and then . As exists, . This completes the proof. □
where , , , , , , , , , are sequences in with , and are both nonexpansive mappings, .
The authors are very grateful to the referees for their critical and valuable comments, which allowed us to improve the presentation of this article.
- Ishikawa S: Fixed point and iteration of a nonexpansive mapping in a Banach spaces. Proc. Am. Math. Soc. 1976, 73: 65–71.MathSciNetView ArticleGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
- Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleGoogle Scholar
- Xu YG: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987MathSciNetView ArticleGoogle Scholar
- Chidume CE, Moore C: Fixed point iteration for pseudocontractive maps. Proc. Am. Math. Soc. 1999,127(4):1163–1170. 10.1090/S0002-9939-99-05050-9MathSciNetView ArticleGoogle Scholar
- Das G, Debata JP: Fixed points of quasi-nonexpansive mappings. Indian J. Pure Appl. Math. 1986, 17: 1263–1269.MathSciNetGoogle Scholar
- Takahashi W, Tamura T: Convergence theorems for a pair of nonexpansive mappings. J. Convex Anal. 1998,5(1):45–58.MathSciNetGoogle Scholar
- Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme with errors for two nonexpansive mappings. Proc. Am. Math. Soc. 2005,61(8):1295–1301.MathSciNetGoogle Scholar
- Kim GE: Weak and strong convergence for quasi-nonexpansive mappings in Banach spaces. Bull. Korean Math. Soc. 2012,49(4):799–813. 10.4134/BKMS.2012.49.4.799MathSciNetView ArticleGoogle Scholar
- Zhang F, Zhang H, Zhang YL: New iterative algorithm for two infinite families of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces. J. Appl. Math. 2013., 2013: Article ID 649537Google Scholar
- Zhang JL, Su YF, Cheng QQ: Strong convergence theorems for a common fixed point of two countable families of relatively quasi nonexpansive mappings and applications. Abstr. Appl. Anal. 2012., 2012: Article ID 956950Google Scholar
- Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
- Sun ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings. J. Math. Anal. Appl. 2003, 286: 351–358. 10.1016/S0022-247X(03)00537-7MathSciNetView ArticleGoogle Scholar
- Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleGoogle Scholar
- Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. 18. Proceedings of the Symposium on Pure Mathematics 1976.Google Scholar
- Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView ArticleGoogle Scholar
- Senter HF, Dotson WG: Approximating fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 1974,44(2):375–380. 10.1090/S0002-9939-1974-0346608-8MathSciNetView ArticleGoogle Scholar
- Maiti M, Gosh MK: Approximating fixed points by Ishikawa iterates. Bull. Aust. Math. Soc. 1989, 40: 113–117. 10.1017/S0004972700003555View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.