Comparison of Harder stability and Rus stability of Mann iteration procedure and their equivalence

Document Type : Research Paper

Authors

1 Department of Mathematics, Andhra University, Visakhapatnam-530 003, India

2 Department of Mathematics, Andhra University, Visakhapatnam-530 003, India.

Abstract

In this paper, we study the stability of Mann iteration procedure in two directions, namely one due to Harder and the second one due to Rus with respect to a map $T:K\to K$ where $K$ is a nonempty closed convex subset of a normed linear space $X$ and there exist $\delta\in(0,1)$ and $L\geq 0$ such that $||Tx-Ty||\leq\delta||x-y||+L||x-Tx||$ for $x,y\in K$. Also, we show that the Mann iteration procedure is stable in the sense of Rus may not imply that it is stable in the sense of Harder for weak contraction maps. Further, we compare and study the equivalence of these two stabilities and provide examples to illustrate our results.

Keywords

[1] G.V.R. Babu, M.L. Sandhya and M.V.R. Kameswari, A note on a fixed point theorem of Berinde on weak
contractions, Carpathian J. Math. 24(1) (2008) 8–12.
[2] V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum
9(1) (2004) 43–53.
[3] T. Eirola, O. Nevanlinna and S.Yu. Pilyugin, Limit Shadowing property, Numer Funct. Anal. Opt. 18 (1997)
75–92.
[4] A.M. Harder and T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japonica 33 (1988)
693–706.
[5] L. Qihou, A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings, J. Math.
Anal. Appl. 146 (1990) 301–305.
[6] W.R. Mann, Mean value methods in iteration, Proc. of Amer. Math. Soc. 44 (1953) 506–510.
[7] M.O. Osilike, Stability results for fixed point iteration procedure, J. Nigerian Math. Soc. 14 (1995) 17–29.
[8] M. O. Osilike, A stable iteration proceedure for quasi-contractive maps, Indian J. pure appl. Math. 27(1) (1996)
25–34.
[9] M. PĖ‡acurar, Iterative Methods for Fixed Point Approximations, Risoprint, Cluj-Napoca, 2009.
[10] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure Appl.
Math. 21(1) (1990) 1–9.
[11] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures II, Indian J. Pure
Appl. Math. 24(11) (1993) 691–703.
[12] I.A. Rus, An abstract point of view on iterative approximation of fixed points: Impact on the theory fixed Point
equations, Fixed Point Theory 13(1) (2012) 179–192.
[13] I. Timis, New stability results of Picard iteration for contractive type mappings, Fasc. Math. 56(1) (2016) 167–184.
Volume 13, Issue 1
March 2022
Pages 409-420
  • Receive Date: 25 March 2019
  • Accept Date: 12 April 2021
  • First Publish Date: 15 September 2021