Convergence and Stability of the Ishikawa Iterative Process for a class of ϕ-quasinonexpansive Mappings

Authors

  • F. D. Ajibade Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria
  • G. Akinbo Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria
  • M. O. Ajisope Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria
  • M. O. Fatai Department of Mathematics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria

Keywords:

Ishikawa iterative Process, Φ-quasinonexpansive mappings, Uniformly convex Banach space, Stability of the iteration

Abstract

The paper analyzes the convergence of Ishikawa iteration to the fixed point of a class of '-quasinonexpansive mappings in uniformly convex Banach spaces, as well as the stability of the Ishikawa iteration used in approximating the fixed point. The work not only confirmed Ishikawa iteration’s convergence and stability to the fixed point of '-quasinonexpansive mappings, but it also pointed the way for future research in the estimate of fixed points of ϕ-quasinonexpansive mappings.

Dimensions

W. G. Dotson, “On the mann iterative process”, Trans. Amer. Math. Soc. 4 (1970) 506.

T. Zamfirescu, “Fixed point theorems in metric spaces”, Arch. Math. 11 (1972) 292.

L. B. Ciric, “Quasi-contraction non-self mappings on Banach spaces”, Bull. Acad. Serbe Sci. Arts. 23 (1998) 25.

V. Berinde, “On the convergence of the Ishikawa iteration in the class of Quasi-contractive operators”, Acta Math. Univ. Comenianae. 73 (2004) 119.

M. O. Olatinwo, “Some stability results for nonexpansive and Quasi-nonexpansive operators uniformly convex Banach spaces using Ishikawa iteration process”, Carpathian J. Math. 24 (2008) 82.

D. Ariza-Ruiz, “Convergence and stability of some iterative processes for a class of Quasi-nonexpansive type mappings”, J. Nonlinear Sci. Appl. 5 (2012) 93.

W. Takahashi, “Aconvexity in metric space and nonexpansive mapping”, Kodai Math. Sem. Rep. 22 (1970) 142.

Z-H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptotically Quasi-nonexpansive mappings”, J. Math.

Anl. Appl. 286 (2003) 351.

L. B. Ciric, “A generalization of Banach‘s contraction principle”, Proc. Am. Math. Soc. 45 (1974) 267.

C. W. Groetsch, “A note on segmenting mann iterates”, Journal of Mathematical Analysis and Applications 40 (1972) 369.

C. E. Chidume, “Geometric properties of Banach spaces and nonlinear iterations”, The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy (2000) 145.

M. O. Osiliki, “Some stability results for fixed point iteration procedures” J. of Nigeria Math. Japonica. 33 (1995) 693.

A. M. Harder & T. L. Hicks ”A Stable Iteration Procedure for Nonexpansive Mappings” Math. Japon. 33 (1988) 687.

M. O. Olatinwo, “Convergence and Stability Results for Some Iterative Schemes” Acta University Apulensis 26 (2011) 225.

C. E. Chidume, “Approximation of fixed point for Quasi-nonexpansive mappings in Lp spaces”, Indian J. Pure Appl. Math. 22 (1991) 273.

R. Jahed, H. Vaezi & H. Piri, “Strong convergence of the iterations of -quasinonexpansive mappings and its applications in Banach spaces” Sahand Communication in Mathematical Analysis 17 (2020) 71.

S. Atailia, N, Redjel & A. Dehici, “Some fixed point results for generalized contractions of Suzuki type in Banach spaces”, J. Fixed Point Theory Appl. 21 (2019) 1.

G. E. Hardy & T. D. Rogers, “A generalization of a fixed point theorem of reich”, Can. Math. Bull. 16 (1973) 201.

R. Pant, P. Patel, R. Shukla & Manuel De la Sen ”Fixed point theorems for nonexpansive type mappings in Banach spaces”, Symmetry 13 (2021) 585. https://doi.org/10.3390/sym13040585

Published

2022-08-29

How to Cite

Convergence and Stability of the Ishikawa Iterative Process for a class of ϕ-quasinonexpansive Mappings. (2022). African Scientific Reports, 1(2), 73–80. https://doi.org/10.46481/asr.2022.1.2.21

Issue

Volume 1, Issue 2, August 2022

Section

Original Research
Copyright & Licensing

Copyright (c) 2022 F. D. Ajibade, G. Akinbo, M. O. Ajisope, M. O. Fatai

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Convergence and Stability of the Ishikawa Iterative Process for a class of ϕ-quasinonexpansive Mappings. (2022). African Scientific Reports, 1(2), 73–80. https://doi.org/10.46481/asr.2022.1.2.21