Abstract
In this paper, let C be a nonempty closed convex subset of a strictly convex Banach space. Then we prove strong convergence of the modified Ishikawa iteration process when T is an ANI self-mapping such that is contained in a compact subset of C, which generalizes the result due to Takahashi and Kim (Math. Jpn. 48:1-9, 1998).
MSC:47H05, 47H10.
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1 Introduction
Let C be a nonempty closed convex subset of a Banach space E, and let T be a mapping of C into itself. Then T is said to be asymptotically nonexpansive [1] if there exists a sequence , , with , such that
for all and . In particular, if for all , T is said to be nonexpansive. T is said to be uniformly L-Lipschitzian if there exists a constant such that
for all and . T is said to be asymptotically nonexpansive in the intermediate sense (in brief, ANI) [2] provided T is uniformly continuous and
We denote by the set of all fixed points of T, i.e., . We define the modulus of convexity for a convex subset of a Banach space; see also [3]. Let C be a nonempty bounded convex subset of a Banach space E with , where is the diameter of C. Then we define with as follows:
where . When is a sequence in E, then will denote strong convergence of the sequence to x. For a mappings T of C into itself, Rhoades [4] considered the following modified Ishikawa iteration process (cf. Ishikawa [5]) in C defined by :
where and are two real sequences in . If for all , then the iteration process (1.1) reduces to the modified Mann iteration process [6] (cf. Mann [7]).
Takahashi and Kim [8] proved the following result: Let E be a strictly convex Banach space and C be a nonempty closed convex subset of E and be a nonexpansive mapping such that is contained in a compact subset of C. Suppose , and the sequence is defined by , where and are chosen so that and or and for some a, b with . Then converges strongly to a fixed point of T. In 2000, Tsukiyama and Takahashi [9] generalized the result due to Takahashi and Kim [8] to a nonexpansive mapping under much less restrictions on the iterative parameters and .
In this paper, let C be a nonempty closed convex subset of a strictly convex Banach space. We prove that if is an ANI mapping such that is contained in a compact subset of C, then the iteration defined by (1.1) converges strongly to a fixed point of T, which generalizes the result due to Takahashi and Kim [8].
2 Strong convergence theorem
We first begin with the following lemma.
Lemma 2.1 [9]
Let C be a nonempty compact convex subset of a Banach space E with . Let and suppose for some ϵ with . Then, for all λ with ,
Lemma 2.2 [9]
Let C be a nonempty compact convex subset of a strictly convex Banach space E with . If , then .
Lemma 2.3 [10]
Let and be two sequences of nonnegative real numbers such that and
for all . Then exists.
Lemma 2.4 Let C be a nonempty compact convex subset of a Banach space E, and let be an ANI mapping. Put
so that . Suppose that the sequence is defined by (1.1). Then exists for any .
Proof The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. For a fixed , since
we obtain
By Lemma 2.3, we readily see that exists. □
Theorem 2.5 Let C be a nonempty compact convex subset of a strictly convex Banach space E with . Let be an ANI mapping. Put
so that . Suppose , and the sequence defined by (1.1) satisfies and or and for some a, b with . Then .
Proof The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. For any fixed , we first show that if and for some , then we obtain . In fact, let . Then we have since . As in the proof of Lemma 2.4, we obtain
Since
and by (2.1) and Lemma 2.1, we have
Thus
Since
we obtain
By using Lemma 2.2, we obtain
Since
we obtain
Since , we have
From (2.2), (2.3) and (2.4), we obtain
Since
and by (2.2), we obtain
Since
and by the uniform continuity of T, (2.5) and (2.6), we have
Next, we show that if and , then we also obtain (2.7). In fact, let . Then we have . From , there are some positive integer and a positive number a such that for all . Since
and hence
So, we obtain
Since
from Lemma 2.1, we obtain
By using (2.8) and (2.9), we obtain
Hence
We also obtain
similarly to the argument above. Since
and by using (2.10), we obtain
Since
by using (2.10) and (2.11), we obtain
Since
by using (2.11) and (2.12), we obtain
Since
by (2.11) and (2.13), we get
From
and by (2.10) and (2.14), we obtain
Since
and by the uniform continuity of T, (2.11), (2.13) and (2.15), we have
□
Our Theorem 2.6 carries over Theorem 3 of Takahashi and Kim [8] to an ANI mapping.
Theorem 2.6 Let C be a nonempty closed convex subset of a strictly convex Banach space E, and let be an ANI mapping, and let be contained in a compact subset of C. Put
so that . Suppose , and the sequence defined by (1.1) satisfies and or and for some a, b with . Then converges strongly to a fixed point of T.
Proof By Mazur’s theorem [12], is a compact subset of C containing which is invariant under T. So, without loss of generality, we may assume that C is compact and is well defined. The existence of a fixed point of T follows from Schauder’s fixed theorem [11]. If , then the conclusion is obvious. So, we assume . From Theorem 2.5, we obtain
Since C is compact, there exist a subsequence of the sequence and a point such that . Thus we obtain by the continuity of T and (2.16). Hence we obtain by Lemma 2.4. □
Corollary 2.7 Let C be a nonempty closed convex subset of a strictly convex Banach space E, and let be an asymptotically nonexpansive mapping with satisfying , , and let be contained in a compact subset of C. Suppose , and the sequence defined by (1.1) satisfies and or and for some a, b with . Then converges strongly to a fixed point of T.
Proof Note that
where . The conclusion now follows easily from Theorem 2.6. □
We give an example which satisfies all assumptions of T in Theorem 2.6, i.e., is an ANI mapping which is not Lipschitzian and hence not asymptotically nonexpansive.
Example 2.8 Let and . Define by
Note that for all and and . Clearly, T is uniformly continuous, ANI on C, but T is not Lipschitzian. Indeed, suppose not, i.e., there exists such that
for all . If we take and , then
This is a contradiction.
We also give an example of an ANI mapping which is not a Lipschitz function.
Example 2.9 Let and and let . Let be defined by
for each and for all , where ℕ denotes the set of all positive integers. Clearly . Since
we obtain uniformly on C as . Thus
for all . Hence T is an ANI mapping, but it is not a Lipschitz function. In fact, suppose that there exists such that for all . If we take and , then
whereas
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The author would like to express their sincere appreciation to the anonymous referee for useful suggestions which improved the contents of this manuscript.
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Kim, G.E. Strong convergence for asymptotically nonexpansive mappings in the intermediate sense. Fixed Point Theory Appl 2014, 162 (2014). https://doi.org/10.1186/1687-1812-2014-162
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DOI: https://doi.org/10.1186/1687-1812-2014-162