Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces
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Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces Hukmi Kiziltunc* and Esra Yolacan *
Correspondence: [email protected] Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey
Abstract In this paper, we define and study the convergence theorems of a new two-steps iterative scheme for two total asymptotically nonexpansive nonself-mappings in Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of (Shahzad in Nonlinear Anal. 61:1031-1039, 2005; Thianwan in Thai J. Math. 6:27-38, 2008; Ozdemir et al. in Discrete Dyn. Nat. Soc. 2010:307245, 2010) and all the others. MSC: 47H09; 47H10; 46B20 Keywords: total asymptotically nonexpansive mappings; common fixed point; uniformly convex Banach space
1 Introduction Let E be a real normed space and K be a nonempty subset of E. A mapping T : K → K is called nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ K . A mapping T : K → K is called asymptotically nonexpansive if there exists a sequence {kn } ⊂ [, ∞) with kn → such that n T x – T n y ≤ kn x – y
(.)
for all x, y ∈ K and n ≥ . Goebel and Kirk [] proved that if K is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point. A mapping T is said to be asymptotically nonexpansive in the intermediate sense (see, e.g., []) if it is continuous and the following inequality holds: lim sup sup T n x – T n y – x – y ≤ .
(.)
n→∞ x,y∈K
If F(T) := {x ∈ K : Tx = x} = ∅ and (.) holds for all x ∈ K , y ∈ F(T), then T is called asymptotically quasi-nonexpansive in the intermediate sense. Observe that if we define an := sup T n x – T n y – x – y
and
σn = max{, an },
(.)
x,y∈K
© 2013 Kiziltunc and Yolacan; 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.
Kiziltunc and Yolacan Fixed Point Theory and Applications 2013, 2013:90 http://www.fixedpointtheoryandapplications.com/content/2013/1/90
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then σn → as n → ∞ and (.) is reduced to n T x – T n y ≤ x – y + σn ,
for all x, y ∈ K, n ≥ .
(.)
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. []. It is known in [] that if K is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is a self-mapping of K which is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains, properly, the class of asymptotically nonexpansive mappings. Albert et al. [] introduced a more general class of as
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