Convergence theorems on asymptotically demicontractive and hemicontractivemappings in the intermediate sense
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Convergence theorems on asymptotically demicontractive and hemicontractive mappings in the intermediate sense JO Olaleru and GA Okeke* *
Correspondence: [email protected] Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria
Abstract In this study, we introduce two classes of nonlinear mappings, the class of asymptotically demicontractive mappings in the intermediate sense and asymptotically hemicontractive mappings in the intermediate sense and prove the convergence of Mann-type and Ishikawa-type iterative schemes to their respective fixed points. Our results are improvements and generalizations of the results of several authors in the literature. MSC: 47H10; 47H09 Keywords: asymptotically demicontractive mappings in the intermediate sense; asymptotically hemicontractive mappings in the intermediate sense; Ishikawa iterative scheme; Mann iterative scheme
1 Introduction and preliminaries In the sequel, we give the following definitions of some of the concepts that will feature prominently in this study. We define C as a convex subset of a normed space E. Definition . Let T : C → C be a mapping. T is said to be () asymptotically nonexpansive [] if there exists a sequence {kn } with kn ≥ and lim kn = such that n T x – T n y ≤ kn x – y
(.)
for all integers n ≥ and all x, y ∈ C; () asymptotically strict pseudocontractive [] if there exist a constant k ∈ [, ) and a sequence {kn } ⊂ [, ∞) with kn → as n → ∞ such that n T x – T n y ≤ kn x – y + k I – T n x – I – T n y ,
∀x, y ∈ C.
(.)
If kn = and T n = T for all n ∈ N in (.), then we obtain the class of strict pseudocontractive mappings. The class of asymptotically strict pseudocontractive mappings was introduced by Qihou in . We remark that the class of asymptotically strict pseudocontractive mappings is a generalization of the class of strict pseudocontractive mappings. Observe that if k = in (.), then we obtain (.); © 2013 Olaleru and Okeke; 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.
Olaleru and Okeke Fixed Point Theory and Applications 2013, 2013:352 http://www.fixedpointtheoryandapplications.com/content/2013/1/352
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() asymptotically strict pseudocontractive in the intermediate sense [] if there exist a constant k ∈ [, ) and a sequence {kn } ⊂ [, ∞) with kn → as n → ∞ such that lim sup sup T n x – T n y – kn x – y – k I – T n x – I – T n y ≤ .
(.)
n→∞ x,y∈C
Put ζn = max , sup T n x – T n y – kn x – y – k I – T n x – I – T n y .
(.)
x,y∈C
It follows that ζn → as n → ∞. Then (.) is reduced to the following: n T x – T n y ≤ kn x – y + k I – T n x – I – T n y + ζn , ∀n ≥ , x, y ∈ C.
(.)
We rem
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