Abstract
In this paper, we introduce a new twostep iterative scheme of mixed type for two asymptotically nonexpansive selfmappings and two asymptotically nonexpansive nonselfmappings and prove strong and weak convergence theorems for the new twostep iterative scheme in uniformly convex Banach spaces.
Keywords:
mixed type asymptotically nonexpansive mapping; strong and weak convergence; common fixed point; uniformly convex Banach space1 Introduction
Let K be a nonempty subset of a real normed linear space E. A mapping
for all
In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive selfmappings, which is an important generalization of the class of nonexpansive selfmappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive selfmapping of K, then T has a fixed point.
Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive selfmappings in Banach spaces (see [216]), which extend and improve the result of Goebel and Kirk in several ways.
Recently, Chidume et al.[10] introduced the concept of asymptotically nonexpansive nonselfmappings, which is a generalization of an asymptotically nonexpansive selfmapping, as follows.
Definition 1.1[10]
Let K be a nonempty subset of a real normed linear space E. Let
for all
Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.
In 2003, also, Chidume et al.[10] studied the following iteration scheme:
for each
In 2006, Wang [11] generalized the iteration process (1.3) as follows:
for each
The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive selfmappings and two asymptotically nonexpansive nonselfmappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.
2 Preliminaries
Let E be a real Banach space, K be a nonempty closed convex subset of E and
for each
If
We denote the set of common fixed points of
Now, we recall some wellknown concepts and results.
Let E be a real Banach space,
for all
A subset K of a real Banach space E is called a retract of E[10] if there exists a continuous mapping
A Banach space E is said to satisfy Opial’s condition[17] if, for any sequence
for all
A Banach space E is said to have a Fréchet differentiable norm[18] if, for all
exists and is attained uniformly in
A Banach space E is said to have the KadecKlee property[19] if for every sequence
Let K be a nonempty closed subset of a real Banach space E. A nonselfmapping
Lemma 2.1[15]
Let
for each
Lemma 2.2[8]
LetEbe a real uniformly convex Banach space and
hold for some
Lemma 2.3[10]
LetEbe a real uniformly convex Banach space, Kbe a nonempty closed convex subset ofEand
Lemma 2.4[16]
LetXbe a uniformly convex Banach space andCbe a convex subset ofX. Then there exists a strictly increasing continuous convex function
for all
Lemma 2.5[16]
LetXbe a uniformly convex Banach space such that its dual space
exists for all
3 Strong convergence theorems
In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.
Lemma 3.1LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Let
(1)
(2)
Proof (1) Set
and so
Since
(2) Taking the infimum over all
for each
Lemma 3.2LetEbe a real uniformly convex Banach space andKbe a nonempty closed convex subset ofE. Let
(a)
(b)
Then
Proof Set
and
Taking lim sup on both sides in (3.1), we obtain
Using Lemma 2.2, we have
By the condition (b), it follows that
and so, from (3.3), we have
Since
Taking lim inf on both sides in the inequality above, we have
by (3.4) and so
Using (3.1), we have
In addition, we have
and
It follows from Lemma 2.2 that
Now, we prove that
Indeed, since
Since
and so
Furthermore, we have
Thus it follows from (3.5), (3.6) and (3.7) that
Since
Using (3.3) and (3.8), we have
and
It follows from
and (3.3) that
In addition, we have
Using (3.3) and (3.11), we obtain that
Thus, using (3.9), (3.10) and the inequality
we have
that
Since
from (3.8), (3.11) and (3.13), it follows that
Again, since
for
for
Using (3.4), (3.8) and (3.12), we have
In addition, we have
for
Finally, we prove that
In fact, by the condition (b), we have
for
This completes the proof. □
Now, we find two mappings,
Example 3.1[20]
Let ℝ be the real line with the usual norm
and
Now, we show that T is nonexpansive. In fact, if
If
This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with
Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:
Case 1. Let
Case 2. Let
Case 3. Let
Case 4. Let
Therefore, the condition (b) in Lemma 3.2 is satisfied.
Theorem 3.1Under the assumptions of Lemma 3.2, if one of
Proof Without loss of generality, we can assume that
and
by Lemma 3.2, which imply that
as
and
for
Theorem 3.2Under the assumptions of Lemma 3.2, if one of
Proof Since
Theorem 3.3Under the assumptions of Lemma 3.2, if there exists a nondecreasing function
for all
Proof Since
Now, we show that
for each
where
Taking the infimum over all
Thus it follows from
4 Weak convergence theorems
In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.
Lemma 4.1Under the assumptions of Lemma 3.1, for all
exists for all
Proof Set
Define the mapping
for all
for all
for each
for all
Then, using (4.3) and Lemma 2.4, we have
It follows from Lemma 3.1 and
Thus we have
Lemma 4.2Under the assumptions of Lemma 3.1, ifEhas a Fréchet differentiable norm, then, for all
exists, where
Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □
Theorem 4.1Under the assumptions of Lemma 3.2, ifEhas a Fréchet differentiable norm, then the sequence
Proof Since E is a uniformly convex Banach space and the sequence
for
Now, we prove that the sequence
Therefore,
Theorem 4.2Under the assumptions of Lemma 3.2, if the dual space
Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence
Now, we prove that the sequence
exists for all
Theorem 4.3Under the assumptions of Lemma 3.2, ifEsatisfies Opial’s condition, then the sequence
Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence
Now, we prove that the sequence
Thus, by Opial’s condition, we have
which is a contradiction and so
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Acknowledgements
The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 20120008170).
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