The Convergence of Ishikawa Iteration for Generalized Φ-contractive Mappings

Abstract

Charles[1] proved the convergence of Picard-type iterative for generalized Φ− accretive non-self maps in a real uniformly smooth Banach space. Based on the theorems of the zeros of strongly Φ− quasi- accretive and fixed points of strongly Φ− hemi-contractions, we extend the results to Ishikawa iterative and Ishikawa iteration process with er- rors for generalized Φ− hemi-contractive maps .

Keywords

• Ishikawa iteration process with errors
• unique solution
• strongly Φ−quasi-accretive

References

1. [1] Charles,Chidume.;Geometric Properties of Banach Spaces and Nonlinear Itera- tions.(2009)
2. [2] Zhiqun Xue,Guiwen Lvand BE Rhoades;the convergence theorems of Ishikawa itera- tive process with errors for hemi-contractive mappings in uniformly smooth Banach spaces,Xue et al. Fixed Point Theory and Applications 2012, 2012:206.
3. [3] Phayap Katchang, Poom Kumam;Strong convergence of the modified Ishikawa itera- tive method for infinitely many nonexpansive mappings in Banach spaces,Computers and Mathematics with Applications 59 (2010) 1473–1483.
4. [4] Abebe R. Tufa and H. Zegeye;Mann and Ishikawa-Type Iterative Schemes for Ap- proximating Fixed Points of Multi-valued Non-Self Mappings,Springer International Publishing 2016.
5. [5] Godwin Amechi Okeke;Convergence analysis of the Picard–Ishikawa hybrid iterative process with applications,African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019.
6. [6] Xu, YG: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 224, 91-101 (1998).
7. [7] Liu, L.; Ishikawa and Mann iterative process with errors for nonlinear strongly accre- tive mappings in Banach spaces, J. Math. Anal. Appl. 194(1995), no. 1, 114–125.
8. [8] Xu, Y.; Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), 91–101.

9. [9] Cudia, D. F.; The theory of Banach spaces: Smoothness, Trans. Amer. Math. Soc. 110 (1964), 284–314.
10. [10] Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In: Proc. Of Symposia in Pure Math., Vol. XVIII, Part 2 (1976).