On a general class of multi-valued weakly Picard mappings
type: Item
instance of: scholarly article
title: On a general class of multi-valued weakly Picard mappings
zbMATH Open document ID: 1117.47039
author: Mădălina Berinde
author: Vasile Berinde
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-14T00:00:00Z
review text: Let \((X,d)\) be a metric space and \(T:X\rightarrow\mathcal{P}(X)\) be a multi-valued operator. \(T\) is said to be a generalized \((\alpha ,L)\)-weak contraction if there exist a constant \(L\geq 0\) and a function \(\alpha:[0,\infty)\rightarrow[0,1)\), with \(\lim \sup_{r\rightarrow t^+} \alpha (r)<1\) for every \(t\in [ 0,\infty )\), such that \[ H(Tx, Ty)\leq \alpha (d(x,y)) d(x,y)+LD(y,Tx) \] for all \(x,y\in X\). When \(\alpha (t)=\theta\in[0,1)\) for all \(t\in[0,\infty)\), we say that \(T\) is a \((\theta ,L)\)-weak contraction. The authors prove that a generalized \((\alpha ,L)\)-weak contraction \(T\) has a fixed point whenever \(X\) is complete and \(T\) has nonempty bounded and closed values. Moreover, if \(T\) is a \((\theta,L)\)-weak contraction, then for any \(x_0\in X\) there exists an orbit \(\{x_n\}_{n\geq 0}\) converging to a fixed point \(u\) of \(T\) for which the following estimate holds: \[ d(x_n,u)\leq\min\left\{\frac{h^n}{1-h}d(x_1,x_0),\frac{h}{1-h}d(x_{n-1},x_n)\right\} \] for a certain constant \(h<1\).
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 5125934
zbMATH Keywords: metric space
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: multivalued mapping
zbMATH Keywords: fixed point
zbMATH Keywords: weak contraction
zbMATH Keywords: weakly Picard operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.03.016
OpenAlex ID: W2061856266
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DOI: 10.1016/J.JMAA.2006.03.016
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description: scientific article; zbMATH DE number 5125934