Convergence of an iterative scheme for multifunctions
type: Item
instance of: scholarly article
title: Convergence of an iterative scheme for multifunctions
zbMATH Open document ID: 1282.47072
author: Sh. Rezapour
author: Nazeer Shahzad
author: S. M. A. Aleomraninejad
published in: Journal of Fixed Point Theory and Applications
publication date: 2013-05-13T00:00:00Z
review text: Let \((X,d)\) be a metric space, \(\mathcal{C}(X)\) the set of all nonempty closed subsets of \(X\) and \(H\) the Pompeiu-Hausdorff metric. Let \(T: X\rightarrow \mathcal{C}(X)\) be a multivalued operator satisfying a Suzuki type contractive condition, i.e., there exist \(\alpha,\beta \in (0,1)\) such that \(\alpha (3-2\alpha=\beta)\leq 1\) and \[ \alpha d(x,Tx)\leq d(x,y) \Rightarrow H(Tx,Ty)\leq \beta d(x,y). \] Let \(\{T_i\}_{i=1}^{\infty}\) be a sequence of approximants of \(T\), that is, \[ H(Tx,T_i x)\leq \epsilon_i,\text{ for all }x\in X, \] where \(\sum\limits_{i=0}^{\infty}\epsilon_i<\infty\). The main result of the paper (Theorem 2.1) is a stability theorem that can be briefly stated as follows: if \[ \frac{\epsilon_i}{1-\alpha}\leq d(x_i,x_{i+1})\leq d(x_i,T_ix_i)+\delta_i,\quad x_{i+1}\in T_i x_i, \] where \(\sum\limits_{i=0}^{\infty}\delta_i<\infty\), then the ``perturbed'' sequence \(\{x_i\}\) still converges to a fixed point of \(T\).
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6162219
zbMATH Keywords: metric space
zbMATH Keywords: multivalued operator
zbMATH Keywords: Suzuki contractive condition
zbMATH Keywords: fixed point
zbMATH Keywords: Picard iteration
zbMATH Keywords: stability
reviewed by: Vasile Berinde
MaRDI profile type: Publication
full work available at URL: s11784-011-0046-z
OpenAlex ID: W1998681272
cites work: Approximating fixed points of contractive set-valued mappings
cites work: Convergence of inexact iterative schemes for nonexpansive set-valued mappings
cites work: Generic existence and approximation of fixed points for nonexpansive set-valued maps
cites work: A generalized Banach contraction principle that characterizes metric completeness
cites work: A new type of fixed point theorem in metric spaces
DOI: 10.1007/S11784-011-0046-Z
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description: scientific article; zbMATH DE number 6162219