Strong convergence for asymptotically nonexpansive mappings in the intermediate sense
© Kim; licensee Springer. 2014
Received: 13 February 2014
Accepted: 4 July 2014
Published: 23 July 2014
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).
Keywordsstrong convergence fixed point Mann and Ishikawa iteration process ANI
Takahashi and Kim  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  generalized the result due to Takahashi and Kim  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 .
2 Strong convergence theorem
We first begin with the following lemma.
Lemma 2.1 
Lemma 2.2 
Let C be a nonempty compact convex subset of a strictly convex Banach space E with . If , then .
Lemma 2.3 
for all . Then exists.
so that . Suppose that the sequence is defined by (1.1). Then exists for any .
By Lemma 2.3, we readily see that exists. □
so that . Suppose , and the sequence defined by (1.1) satisfies and or and for some a, b with . Then .
Our Theorem 2.6 carries over Theorem 3 of Takahashi and Kim  to an ANI mapping.
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.
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.
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.
This is a contradiction.
We also give an example of an ANI mapping which is not a Lipschitz function.
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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