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Convergence theorems of a new iteration for two nonexpansive mappings
Journal of Inequalities and Applications volume 2014, Article number: 82 (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.
(I) Let C be a nonempty convex subset of E and let be a mapping. For any given the sequence defined by
is called the Ishikawa iteration sequence, where and are two real sequences in satisfying some conditions. In particular, if for all , then defined by
is called the Mann iteration sequence.
In , Liu introduced the concepts of Ishikawa and Mann iterative processes with errors as follows.
(II) For a nonempty subset C of a Banach space E and a mapping , the sequence defined by
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.
(III) Let C be a nonempty convex subset of E and let be a mapping. For any given the sequence defined by
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.
In the present paper, we consider the following scheme:
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.
A Banach space E is said to satisfy Opial’s condition if whenever is a sequence in E which converges weakly to x, then
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 
Let , be two sequences of nonnegative real numbers satisfying
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.
Lemma 2.1 Let E be a real uniformly convex Banach space and C its nonempty convex subset. Let be nonexpansive mappings. Let be the sequence as defined in (1) with the following conditions:
, , , as ;
for some ;
for some .
If , then we have
exists for all and , and are all bounded;
Proof For any , we have
However, as the proof of the above inequality, it follows that
Thus from (2) and (3), it is easy to check
which implies that
and the limit exists for all . Furthermore is bounded and and are both bounded also. Now suppose for some . By the inequalities (3) and (5), we have
From the iterative process (1), we have
Since , , are both bounded and (6), it follows from Lemma 1.3 that
Next, by the inequality (2) and (5), we have
Taking liminf on both sides in the above inequality and by (6) and for some , we have
It is easy to see that
Since , , and are both bounded, then by Lemma 1.3 we have
Moreover, by the iterative process (1) again, we get
which, by as and (8), implies that
Hence from (8) and (13), we have
as . Note that
then it follows from the above inequality that
That is to say,
which, from (8), (12), (14), and , means that
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.
Proof Let . As the proof of Lemma 2.1, exists. Since E is uniformly convex, every bounded subset of E is weakly compact, so that there exists a subsequence of the bounded sequence such that converges weakly to a point . Therefore, it follows from (ii) in Lemma 2.1 that
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 .
Because we have proved that for any , the limit exists, we can let
By the Opial condition of E, we have
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.
Proof Since one of T and S is semicompact and by (ii) in Lemma 2.1, then there exists a subsequence of the sequence such that converges strongly to u. Since C is closed, . Continuity of S and T gives and as . Then by Lemma 2.1,
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.
Proof By Lemma 2.1, exists for all . Let it be c for some . If , there is nothing to prove. Suppose . By Lemma 2.1, . Moreover, from (4), we have which gives
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. □
Remark 2.5 From the proof of the above results, it is easy to see that we can extend our theorems to the iterative process (1) with errors as follows: let C be a bounded closed convex subset of E and the sequence be defined by
where , , , , , , , , , are sequences in with , and are both nonexpansive mappings, .
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The authors are very grateful to the referees for their critical and valuable comments, which allowed us to improve the presentation of this article.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript, read and approved the final manuscript.
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Cite this article
Hou, X., Du, H. Convergence theorems of a new iteration for two nonexpansive mappings. J Inequal Appl 2014, 82 (2014). https://doi.org/10.1186/1029-242X-2014-82
- nonexpansive mapping
- common fixed points
- weak and strong convergence
- condition ()