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A fixed point theorem for asymptotically nonexpansive type mappings in uniformly convex Banach spaces

A fixed point theorem for asymptotically nonexpansive type mappings in uniformly convex Banach... Let us consider two nonempty subsets A and B of a uniformly convex Banach space X. Let $$T:A\cup B\rightarrow A\cup B$$ T : A ∪ B → A ∪ B be a mapping such that $$T(A)\subseteq A,\,T(B)\subseteq B$$ T ( A ) ⊆ A , T ( B ) ⊆ B and there is a sequence $$\{k_n\}$$ { k n } in $$[1,\infty )$$ [ 1 , ∞ ) , with $$k_n\rightarrow 1$$ k n → 1 , satisfying $$\Vert T^nx-T^ny\Vert \le k_n\Vert x-y\Vert $$ ‖ T n x - T n y ‖ ≤ k n ‖ x - y ‖ , for all $$x\in A$$ x ∈ A and $$y\in B$$ y ∈ B . We investigate sufficient conditions for the existence of fixed points x in A and y in B in such a way that the distance between x and y is optimum in some sense. Our main result provides a natural and simple proof for a particular case of Rajesh and Veeramani (Numer Funct Anal Optim 37:80–91, 2016) fixed point theorem for asymptotically relatively nonexpansive mappings. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Journal of Analysis Springer Journals

A fixed point theorem for asymptotically nonexpansive type mappings in uniformly convex Banach spaces

The Journal of Analysis , Volume 26 (1) – Dec 8, 2017

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Forum D'Analystes, Chennai
Subject
Mathematics; Analysis; Functional Analysis; Abstract Harmonic Analysis; Special Functions; Fourier Analysis; Measure and Integration
ISSN
0971-3611
eISSN
2367-2501
DOI
10.1007/s41478-017-0062-5
Publisher site
See Article on Publisher Site

Abstract

Let us consider two nonempty subsets A and B of a uniformly convex Banach space X. Let $$T:A\cup B\rightarrow A\cup B$$ T : A ∪ B → A ∪ B be a mapping such that $$T(A)\subseteq A,\,T(B)\subseteq B$$ T ( A ) ⊆ A , T ( B ) ⊆ B and there is a sequence $$\{k_n\}$$ { k n } in $$[1,\infty )$$ [ 1 , ∞ ) , with $$k_n\rightarrow 1$$ k n → 1 , satisfying $$\Vert T^nx-T^ny\Vert \le k_n\Vert x-y\Vert $$ ‖ T n x - T n y ‖ ≤ k n ‖ x - y ‖ , for all $$x\in A$$ x ∈ A and $$y\in B$$ y ∈ B . We investigate sufficient conditions for the existence of fixed points x in A and y in B in such a way that the distance between x and y is optimum in some sense. Our main result provides a natural and simple proof for a particular case of Rajesh and Veeramani (Numer Funct Anal Optim 37:80–91, 2016) fixed point theorem for asymptotically relatively nonexpansive mappings.

Journal

The Journal of AnalysisSpringer Journals

Published: Dec 8, 2017

References