On uniformly generalized Lipschitzian mappings
type: Item
instance of: scholarly article
title: On uniformly generalized Lipschitzian mappings
zbMATH Open document ID: 1208.54016
DOI: 10.1155/2010/692401
author: Ahmed H. Soliman
author: Mohammad Imdad
published in: Fixed Point Theory and Applications
publication date: 2010-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/222935
review text: In [\textit{J. S. Jung} and \textit{B. S. Thakur}, Int. J. Math. Math. Sci. 28, No.~1, 41--50 (2001; Zbl 1017.47042)] the notion of \(G_1\)-generalized mapping was introduced. Definition 1: A mapping \(T: X\to X\) is said to be generalized \(G_1\)-Lipschitzian if \[ d(T^n x,T^n y)\leq a_nd(x, y)+ b_n(d(x, T^n x)+ d(y, T^n y))+ c_n(d(x, T^n y)+ d(y, T^n x)), \] for each \(x,y\in X\) and \(n\geq 1\), where \(a_n\), \(b_n\), \(c_n\) are nonnegative constants such that there exists an integer \(n_0\) such that \(b_n+ c_n< 1\) for all \(n> n_0\). Recently \textit{A. H. Soliman} [Sci. Math. Jpn. 69, No.~3, 323--328 (2009; Zbl 1178.54027)] defined another class of generalized Lipschitzian mappings on metric spaces. Definition 2: A mapping \(T: X\to X\) is said to be a generalized \(G_2\)-Lipschitzian mapping if for each integer \(n\geq 1\) there exists a constant \(k_n> 0\) (depending on \(n\)) such that \[ d(T^n x, T^n y)\leq k_n\max\left\{d(x,y),\tfrac12\,d(x, T^n x),\tfrac12 d(y, T^n y)\right\} \] for every \(x,y\in X\). If \(k_n= k\) for all \(n\geq 1\), then \(T\) is called uniformly \(G_2\)-Lipschitzian. In the present paper another class of generalized Lipschitzian mappings on metric spaces is introduced. Definition 3: A mapping \(T: X\to X\) is said to be a generalized \(G_3\)-Lipschitzian mapping if for each integer \(n\geq 1\) there exists a constant \(k_n> 0\) (depending on \(n\)) such that \[ d(T^nx, T^ny)\leq k_n\max\left\{d(x,y),\tfrac12\, d(x, T^n x),\tfrac12\, d(y, T^ny),\tfrac12\, d(x, T^ny),\tfrac12\, d(y, T^nx)\right\} \] for every \(x,y\in X\). If \(k_n= k\) for all \(n\geq 1\), then \(T\) is called uniformly \(G_2\)-Lipschitzian. Definition 5: A semigroup \({\mathcal I}= \{T(t): t\in G\}\) of self-mappings defined on \(X\) is called a uniformly \(G_1\)-Lipschitzian semigroup if \[ \begin{multlined} d(T(t)x, T(t)y)\leq a(t) d(x,y)+ b(t)(d(x, T(t) x)+ d(y, T(t)y))+\\ c(t)(d(x, T(t)y)+ d(y, T(t)x)),\end{multlined} \] for each \(x,y\in X\), where \(a(t),b(…
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5792456
zbMATH Keywords: \(G_1\), \(G_2\), \(G_3\) Lipschitzian mappings
zbMATH Keywords: uniformly \(G_1\), \(G_2\), \(G_3\) Lipschitzian semigroup
zbMATH Keywords: fixed point
Wikidata QID: Q59254178
MaRDI profile type: Publication
OpenAlex ID: W1965706507
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description: scientific article; zbMATH DE number 5792456