Strong convergence theorems for three-steps iterations for asymptotically nonexpansive mappings in Banach spaces
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RESEARCH
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Strong convergence theorems for three-steps iterations for asymptotically nonexpansive mappings in Banach spaces Cui Yun An1 , Zuo Zhan Fei2 and Henryk Hudzik3* *
Correspondence: [email protected] 3 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614, Poznan, Poland Full list of author information is available at the end of the article
Abstract We consider the problem of the convergence of the three-steps iterative sequences for asymptotically nonexpansive mappings in a real Banach space. Under suitable conditions, it has been proved that the iterative sequence converges strongly to a fixed point, which is also a solution of certain variational inequality. The results presented in this paper extend and improve some recent results. Keywords: asymptotically nonexpansive mapping; three-steps iteration; fixed points; uniformly Gâteaux differentiable norm
1 Introduction ∗ Let X be a real Banach space with dual X ∗ , J : X → X denotes the normalized duality mapping from X into X ∗ given by J(x) = f ∈ X ∗ : x, f = xf ∧ f = x ,
∀x ∈ X.
Let C be a subset of X. A mapping T : C → C is called contraction if there exists a constant α ∈ (, ) such that Tx – Ty ≤ αx – y for any x, y ∈ C. The mapping T is called nonexpansive if Tx – Ty ≤ x – y for any x, y ∈ C, and it is called asymptotically nonexpansive if there exists a sequence {kn } in the interval [, ∞) with limn→∞ kn = and such that n T x – T n y ≤ kn x – y
()
for all x, y ∈ C and all n ∈ N , where N is the set of natural numbers. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [] as an important generalization of nonexpansive mappings. They proved that if C is a nonempty, bounded, closed and convex subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self-mapping of C, then T has a fixed point in C. In , Noor [] introduced a three-steps iterative scheme and studied the approximate solutions of a variational inclusion in Hilbert spaces. In , Xu and Noor [] introduced and studied a new class of three-steps iterative schemes for solving the nonlinear equation Tx = x for asymptotically nonexpansive mappings T in uniformly convex Banach spaces. In , Nilsrakoo and Saejung [] defined a three-steps mean value iterative scheme and © 2013 An et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
An et al. Fixed Point Theory and Applications 2013, 2013:150 http://www.fixedpointtheoryandapplications.com/content/2013/1/150
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extended the results of Xu and Noor []. In , Yao and Noor [] made a refinement and improvement of the previously known results. Now we define a new three-steps iteration scheme for asymptotically nonexpansive mappings a
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