Abstract
In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.
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1 Introduction
Let K be a nonempty subset of a real normed linear space E. A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that
for all and .
In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, 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 self-mapping of K, then T has a fixed point.
Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [2–16]), which extend and improve the result of Goebel and Kirk in several ways.
Recently, Chidume et al. [10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.
Definition 1.1 [10]
Let K be a nonempty subset of a real normed linear space E. Let be a nonexpansive retraction of E onto K. A nonself-mapping is said to be asymptotically nonexpansive if there exists a sequence with as such that
for all and .
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 , where is a sequence in and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.
In 2006, Wang [11] generalized the iteration process (1.3) as follows:
for each , where are two asymptotically nonexpansive nonself-mappings and , are real sequences in , and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).
The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings 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 be a nonexpansive retraction of E onto K. Let be two asymptotically nonexpansive self-mappings and be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:
for each , where , are two sequences in .
If and are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).
We denote the set of common fixed points of , , and by and denote the distance between a point z and a set A in E by .
Now, we recall some well-known concepts and results.
Let E be a real Banach space, be the dual space of E and be the normalized duality mapping defined by
for all , where denotes duality pairing between E and . A single-valued normalized duality mapping is denoted by j.
A subset K of a real Banach space E is called a retract of E [10] if there exists a continuous mapping such that for all . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping is called a retraction if . It follows that if a mapping P is a retraction, then for all y in the range of P.
A Banach space E is said to satisfy Opial’s condition [17] if, for any sequence of E, weakly as implies that
for all with .
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 Kadec-Klee property [19] if for every sequence in E, weakly and , it follows that strongly.
Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping is said to be semi-compact [11] if, for any sequence in K such that as , there exists a subsequence of such that converges strongly to some .
Lemma 2.1 [15]
Let , and be three nonnegative sequences satisfying the following condition:
for each , where is some nonnegative integer, and . Then exists.
Lemma 2.2 [8]
Let E be a real uniformly convex Banach space and for each . Also, suppose that and are two sequences of E such that
hold for some . Then .
Lemma 2.3 [10]
Let E be a real uniformly convex Banach space, K be a nonempty closed convex subset of E and be an asymptotically nonexpansive mapping with a sequence and as . Then is demiclosed at zero, i.e., if weakly and strongly, then , where is the set of fixed points of T.
Lemma 2.4 [16]
Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function with such that, for each mapping with a Lipschitz constant ,
for all and .
Lemma 2.5 [16]
Let X be a uniformly convex Banach space such that its dual space has the Kadec-Klee property. Suppose is a bounded sequence and such that
exists for all , where denotes the set of all weak subsequential limits of . Then .
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 3.1 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let be two asymptotically nonexpansive self-mappings with and be two asymptotically nonexpansive nonself-mappings with such that and for , respectively, and . Let be the sequence defined by (2.1), where and are two real sequences in . Then
-
(1)
exists for any ;
-
(2)
exists.
Proof (1) Set . For any , it follows from (2.1) that
and so
Since and for , we have . It follows from Lemma 2.1 that exists.
-
(2)
Taking the infimum over all in (3.2), we have
for each . It follows from and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □
Lemma 3.2 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let be two asymptotically nonexpansive self-mappings with and be two asymptotically nonexpansive nonself-mappings with such that and for , respectively, and . Let be the sequence defined by (2.1) and the following conditions hold:
-
(a)
and are two real sequences in for some ;
-
(b)
for all and .
Then for .
Proof Set . For any given , exists by Lemma 3.1. Now, we assume that . It follows from (3.2) and that
and
Taking lim sup on both sides in (3.1), we obtain and so
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 by the condition (b). It follows from (3.5) that
Since and is a nonexpansive retraction of E onto K, we have
and so
Furthermore, we have
Thus it follows from (3.5), (3.6) and (3.7) that
Since by the condition (b) and
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 . It follows from (3.6) and the inequality
that
Since
from (3.8), (3.11) and (3.13), it follows that
Again, since , for and , are two asymptotically nonexpansive nonself-mappings, we have
for . It follows from (3.12), (3.14) and (3.15) that
for . Moreover, we have
Using (3.4), (3.8) and (3.12), we have
In addition, we have
for . Thus it follows from (3.6), (3.10), (3.16) and (3.17) that
Finally, we prove that
In fact, by the condition (b), we have
for . Thus it follows from (3.5), (3.6), (3.9) and (3.10) that
This completes the proof. □
Now, we find two mappings, and , satisfying the condition (b) in Lemma 3.2 as follows.
Example 3.1 [20]
Let ℝ be the real line with the usual norm and let . Define two mappings by
and
Now, we show that T is nonexpansive. In fact, if or , then we have
If and or and , then we have
This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with for each . Similarly, we can show that S is an asymptotically nonexpansive mapping with for each .
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 . Then we have
Case 2. Let . Then we have
Case 3. Let and . Then we have
Case 4. Let and . Then we have
Therefore, the condition (b) in Lemma 3.2 is satisfied.
Theorem 3.1 Under the assumptions of Lemma 3.2, if one of , , and is completely continuous, then the sequence defined by (2.1) converges strongly to a common fixed point of , , and .
Proof Without loss of generality, we can assume that is completely continuous. Since is bounded by Lemma 3.1, there exists a subsequence of such that converges strongly to some . Moreover, we know that
and
by Lemma 3.2, which imply that
as and so . Thus, by the continuity of , , and , we have
and
for . Thus it follows that . Furthermore, since exists by Lemma 3.1, we have . This completes the proof. □
Theorem 3.2 Under the assumptions of Lemma 3.2, if one of , , and is semi-compact, then the sequence defined by (2.1) converges strongly to a common fixed point of , , and .
Proof Since for by Lemma 3.2 and one of , , and is semi-compact, there exists a subsequence of such that converges strongly to some . Moreover, by the continuity of , , and , we have and for . Thus it follows that . Since exists by Lemma 3.1, we have . This completes the proof. □
Theorem 3.3 Under the assumptions of Lemma 3.2, if there exists a nondecreasing function with and for all such that
for all , where , then the sequence defined by (2.1) converges strongly to a common fixed point of , , and .
Proof Since for by Lemma 3.2, we have . Since is a nondecreasing function satisfying , for all and exists by Lemma 3.1, we have .
Now, we show that is a Cauchy sequence in K. In fact, from (3.2), we have
for each , where and . For any m, n, , we have
where . Thus, for any , we have
Taking the infimum over all , we obtain
Thus it follows from that is a Cauchy sequence. Since K is a closed subset of E, the sequence converges strongly to some . It is easy to prove that , , and are all closed and so F is a closed subset of K. Since , , the sequence converges strongly to a common fixed point of , , and . This completes the proof. □
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 4.1 Under the assumptions of Lemma 3.1, for all , the limit
exists for all , where is the sequence defined by (2.1).
Proof Set . Then and, from Lemma 3.1, exists. Thus it remains to prove Lemma 4.1 for any .
Define the mapping by
for all . It is easy to prove that
for all , where . Letting , it follows from and that . Setting
for each , from (4.1) and (4.2), it follows that
for all and , for any . Let
Then, using (4.3) and Lemma 2.4, we have
It follows from Lemma 3.1 and that uniformly for all m. Observe that
Thus we have , that is, exists for all . This completes the proof. □
Lemma 4.2 Under the assumptions of Lemma 3.1, if E has a Fréchet differentiable norm, then, for all , the limit
exists, where is the sequence defined by (2.1). Furthermore, if denotes the set of all weak subsequential limits of , then for all and .
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 4.1 Under the assumptions of Lemma 3.2, if E has a Fréchet differentiable norm, then the sequence defined by (2.1) converges weakly to a common fixed point of , , and .
Proof Since E is a uniformly convex Banach space and the sequence is bounded by Lemma 3.1, there exists a subsequence of which converges weakly to some . By Lemma 3.2, we have
for . It follows from Lemma 2.3 that .
Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some . Then, by the same method given above, we can also prove that . So, . It follows from Lemma 4.2 that
Therefore, , which shows that the sequence converges weakly to q. This completes the proof. □
Theorem 4.2 Under the assumptions of Lemma 3.2, if the dual space of E has the Kadec-Klee property, then the sequence defined by (2.1) converges weakly to a common fixed point of , , and .
Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence of which converges weakly to some .
Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some . Then, as for q, we have . It follows from Lemma 4.1 that the limit
exists for all . Again, since , by Lemma 2.5. This shows that the sequence converges weakly to q. This completes the proof. □
Theorem 4.3 Under the assumptions of Lemma 3.2, if E satisfies Opial’s condition, then the sequence defined by (2.1) converges weakly to a common fixed point of , , and .
Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence of which converges weakly to some .
Now, we prove that the sequence converges weakly to q. Suppose that there exists a subsequence of such that converges weakly to some and . Then, as for q, we have . Using Lemma 3.1, we have the following two limits exist:
Thus, by Opial’s condition, we have
which is a contradiction and so . This shows that the sequence converges weakly to q. This completes the proof. □
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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: 2012-0008170).
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Guo, W., Cho, Y.J. & Guo, W. Convergence theorems for mixed type asymptotically nonexpansive mappings. Fixed Point Theory Appl 2012, 224 (2012). https://doi.org/10.1186/1687-1812-2012-224
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DOI: https://doi.org/10.1186/1687-1812-2012-224