Abstract
In this paper, we first introduce a new hybrid iteration method for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces, and then we discuss the strong and weak convergence for the iterative processes. The results presented in this paper extend and improve the corresponding results of Wang and Osilike.
MSC:47H05, 47H09, 49M05.
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1 Introduction and preliminaries
Throughout this paper we assume that E is a real Banach space and is a mapping. We denote by and the set of fixed points and the domain of T, respectively.
Recently, the convergence problems of an implicit (or non-implicit) iterative process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, e.g., [1–24]).
Recall that E is said to satisfy Opial’s condition [11] if for each sequence in E, the condition that the sequence weakly implies that
for all with .
Definition 1.1 Let D be a closed subset of E and be a mapping.
-
(1)
T is said to be demi-closed at the origin if for each sequence in D, the conditions weakly and strongly imply .
-
(2)
T is said to be semi-compact if for any bounded sequence in D such that (), there exists a subsequence such that .
-
(3)
T is said to be asymptotically nonexpansive [3] if there exists a sequence with such that
-
(4)
T is said to be L-Lipschitzian if there exists a constant such that for all .
Proposition 1.1 Let K be a nonempty subset of E, and let be m asymptotically nonexpansive mappings. Then there exists a sequence with such that
Proof Since for each , is an asymptotically nonexpansive mapping, there exists a sequence with () such that
Letting
we have that with () and
for all and for each . □
In 2007, for studying the strong and weak convergence of fixed points of nonexpansive mappings in a Hilbert space H, Wang [19] introduced the following hybrid iteration scheme:
where for all , is an initial point, is an η-strongly monotone and k-Lipschitzian mapping, μ is a positive fixed constant.
In the same year, Osilike et al. [13] extended the results of Wang from Hilbert spaces to arbitrary Banach spaces and proved those theorems by Wang without the strong monotonicity condition.
In this paper, we introduce the following new hybrid iteration method in Banach spaces:
for a finite family of asymptotically nonexpansive mappings , where is an L-Lipschitzian mapping, μ is a positive fixed constant, is a sequence in , and such that .
Especially, if are m nonexpansive mappings, is an L-Lipschitzian mapping, μ is a positive fixed constant, is a sequence in , and such that , then the sequence defined by
is called the hybrid iteration scheme for a finite family of nonexpansive mappings .
The purpose of this paper is to study the weak and strong convergence of an iterative sequence defined by (1.3) and (1.4) to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the main results in [13] and [19].
In order to prove the main results of this paper, we need the following lemmas.
Lemma 1.1 [17]
Let , , be three nonnegative real sequences satisfying the following condition:
If and , then the limit exists.
Lemma 1.2 [15]
Let E be a uniformly convex Banach space, and let b, c be two constants with . Suppose that is a sequence in and , are two sequences in E. Then the conditions
imply that , where is some constant.
Lemma 1.3 [4]
Let E be a uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be an asymptotically nonexpansive mapping with . Then is semi-closed at zero, where I is the identity mapping of E, that is, for each sequence in K, if converges weakly to and converges strongly to 0, then .
2 Main results
We are now in a position to prove our main results in this paper.
Theorem 2.1 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be m asymptotically nonexpansive mappings with (the set of common fixed points of ); is an L-Lipschitzian mapping. Let the hybrid iteration be defined by (1.3), where and are real sequences in , let be the sequence defined by (1.1) satisfying the following conditions:
-
(i)
for some ;
-
(ii)
;
-
(iii)
.
Then
-
(1)
exists for each ,
-
(2)
, ,
-
(3)
converges strongly to a common fixed point of if and only if .
Proof (1) Since , for each , it follows from Proposition 1.1 that
Since (), we know that is bounded, and there exists such that . Let , , by condition (ii) we have . Therefore we have
Taking , , and by using condition (iii) and , it is easy to see that
It follows from Lemma 1.1 that exists.
(2) Since is bounded, there exists such that
We can assume that
where is some number. Since is a convergent sequence, so is a bounded sequence in K. Let
then
By (2.4) we have that
From (2.1) and (2.3) we have
By condition (iii), , and (2.3), (2.4), (2.6), we have that
Thus from (2.4), (2.5), (2.6), (2.8) and Lemma 1.2 we know that
By (2.9), we have that
From (2.10) we obtain that
It follows from (2.7) and (2.9) that
Let , , then from (2.12) we have
It follows from (2.10) and (2.13) that
(3) From (2.2) and (2.3), we have that
where and with and . Hence, we have
It follows from (2.16) and Lemma 1.1 that the limit exists.
If converges strongly to a common fixed point p of , then it follows from (2.3) and Lemma 1.2 that the limit . Since , we know that , and so .
Conversely, suppose , then .
Next we prove that the sequence is a Cauchy sequence in K. In fact, since , for all , from (2.15) we have
Hence, for any positive integers n, m, from (2.17) it follows that
where .
Since and , for any given , there exists a positive integer such that
Therefore there exists such that
Consequently, for any and for all , we have
This implies that is a Cauchy sequence in K. By the completeness of K, we can assume that . Then from (2) and Lemma 1.3 we have , and so is a common fixed point of . This completes the proof of Theorem 2.1. □
Theorem 2.2 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be m asymptotically nonexpansive mappings with , and at least there exists , , which is semi-compact. is an L-Lipschitzian mapping. Let and be real sequences in , be the sequence defined by (1.1) satisfying the following conditions:
-
(i)
for some ;
-
(ii)
;
-
(iii)
.
Then the hybrid iterative process defined by (1.3) converges strongly to a common fixed point of in K.
Proof From the proof of Theorem 2.1, is bounded, and , . Especially, we have
By the assumption of Theorem 2.2, we may assume that is semi-compact, without loss of generality. Then it follows from (2.18) that there exists a subsequence of such that converges strongly to , and we have
This implies that . In addition, since exists, therefore , that is, converges strongly to a fixed point of in K. This completes the proof of Theorem 2.2. □
Theorem 2.3 Under the conditions of Theorem 2.1, if E satisfies Opial’s condition, then the hybrid iterative process defined by (1.3) converges weakly to a common fixed point of in K.
Proof From the proof of Theorem 2.1, we know that is a bounded sequence in K. Since E is uniformly convex, it must be reflexive, so every bounded subset of E is weakly compact. Therefore, there exists a subsequence such that converges weakly to . From (2.14) we have
By Lemma 1.3, we know that . By the arbitrariness of , we have that .
Suppose that there exists some subsequence such that weakly and . From Lemma 1.3, . By (2.2) we know that and exist. By the virtue of Opial’s condition of E, we have
which is a contraction. Hence . This implies that converges weakly to a common fixed point of in K. This completes the proof of Theorem 2.3. □
Remark 2.1 Theorems 2.1, 2.2 and 2.3 extend the results of [13] and [19] from a nonexpansive mapping to a finite family of asymptotically nonexpansive mappings.
Theorem 2.4 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be m nonexpansive mappings with ; is an L-Lipschitzian mapping. Let a hybrid iterative sequence be defined by (1.4), where and are real sequences in satisfying the following conditions:
-
(i)
for some ;
-
(ii)
.
Then
-
(1)
exists for each ,
-
(2)
, ,
-
(3)
converges strongly to a common fixed point of if and only if .
Theorem 2.5 Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be m nonexpansive mappings with , and at least there exists , , which is semi-compact. is an L-Lipschitzian mapping. Let and be real sequences in satisfying the following conditions:
-
(i)
for some ;
-
(ii)
.
Then the hybrid iterative process defined by (1.4) converges strongly to a common fixed point of in K.
Theorem 2.6 Under the conditions of Theorem 2.4, if E satisfies Opial’s condition, then the hybrid iterative process defined by (1.4) converges weakly to a common fixed point of in K.
The proofs of Theorems 2.4, 2.5 and 2.6 can be obtained from those of Theorems 2.1, 2.2 and 2.3 with the condition that are m nonexpansive mappings.
Remark 2.2 Theorems 2.4, 2.5 and 2.6 extend the results of [13] and [19] from a nonexpansive mapping to a finite family of nonexpansive mappings.
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Acknowledgements
The study was supported by the National Natural Science Foundation of China (11271105, 11071169) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).
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Gu, F. A new hybrid iteration method for a finite family of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2013, 322 (2013). https://doi.org/10.1186/1687-1812-2013-322
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DOI: https://doi.org/10.1186/1687-1812-2013-322