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The Journal of Analysis

, Volume 26, Issue 1, pp 9–14 | Cite as

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

  • V. Sankar Raj
  • S. Jamal Fathima
Original Research Paper
  • 38 Downloads

Abstract

Let us consider two nonempty subsets A and B of a uniformly convex Banach space X. Let \(T:A\cup B\rightarrow A\cup B\) be a mapping such that \(T(A)\subseteq A,\,T(B)\subseteq B\) and there is a sequence \(\{k_n\}\) in \([1,\infty )\), with \(k_n\rightarrow 1\), satisfying \(\Vert T^nx-T^ny\Vert \le k_n\Vert x-y\Vert \), for all \(x\in A\) and \(y\in B\). 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.

Keywords

Asymptotically relatively nonexpansive mapping UC-property P-property Fixed points 

Mathematics Subject Classification

47H09 47H10 

Notes

Acknowledgements

The authors gratefully acknowledge the Referee for his valuable comments which have improved the presentation of this manuscript. The author S. Jamal Fathima acknowledge the University Grant Commission(UGC) for the financial support she received in the form of Senior Research Fellow (SRF) (F1-17.1/2011-12/MANF-MUS-TAM-1335/(SA-III)) under Maulana Azad National Fellowship.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Forum D'Analystes, Chennai 2017

Authors and Affiliations

  1. 1.Department of MathematicsManonmaniam Sundaranar UniversityTirunelveliIndia

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