Modified Picard-Mann hybrid iteration process for total asymptotically nonexpansive mappings

Balwant Singh Thakur,Mihai Postolache,Dipti Thakur

doi:10.1186/s13663-015-0391-5

Balwant Singh Thakur, Mihai Postolache + Show 1 more

Open Access

https://doi.org/10.1186/s13663-015-0391-5

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Abstract

AbstractIn this paper, using the modified hybrid Picard-Mann iteration process, we establish Δ-convergence and strong convergence theorems for total asymptotically nonexpansive mappings on a $CAT(0)$ C A T ( 0 ) space. Results established in the paper extend and improve a number of results in the literature. A numerical example is also given to examine the fastness of the proposed iteration process under different control conditions and initial points.

Highlights

Let K be a nonempty, closed and convex subset of a normed linear space E
In the last four decades, many papers have appeared in the literature on the iteration methods to approximate fixed points of a nonexpansive mapping, cf. [ – ] and the references therein
Alber et al [ ] made an effort to unify some generalization of nonexpansive mappings and introduced the notion of total asymptotically nonexpansive mappings

Summary

Introduction

Let K be a nonempty, closed and convex subset of a normed linear space E. A mapping T : K → K is said to be total asymptotically nonexpansive if there exist nonnegative real sequences {kn( )} and {kn( )}, n ≥ , kn( ), kn( ) → as n → ∞, and a strictly increasing and continuous function φ : R+ → R+ with φ( ) = such that They further studied the iterative approximation of fixed point of total asymptotically nonexpansive mappings using a modified Mann iteration process.

Results

Conclusion