New discontinuity results at fixed point on metric spaces
type: Item
instance of: scholarly article
title: New discontinuity results at fixed point on metric spaces
zbMATH Open document ID: 1467.54012
author: Nihal Taş
author: Nihal Yilmaz
published in: Journal of Fixed Point Theory and Applications
publication date: 2021-06-09T00:00:00Z
review text: Let \((X,d)\) be a complete metric space and \(T\colon X\to X\). For \(u,v\in X\), denote \[ M(u,v)=\max\left\{d(u,v),d(u,Tu),d(v,Tv),\left[\frac{d(u,Tv)+d(v,Tu)}{1+d(u,Tu)+d(v,Tv)}\right]d(u,v)\right\}. \] The following result is proved: If (1) there exists a function \(\phi: \mathbb R^+\to \mathbb R\) such that \(\phi(t)<t\) for each \(t>0\) and \(d(Tu,Tv)\leq\phi(M(u,v))\), for all \(u,v\in X\); (2) for a given \(\varepsilon>0\), there exists a \(\delta>0\) such that \(\varepsilon<M(u,v)<\varepsilon+\delta\) implies \(d(Tu,Tv)\leq\varepsilon\), then \(T\) has a fixed point \(z\in X\) and \(T^nu\to z\) as \(n\to\infty\), for each \(u\in X\). Moreover, \(T\) is discontinuous at \(z\) if and only if \(\lim_{u\to z}M(u,z)\ne0\). Variants of this result for common fixed point problems of two self-mappings and for fixed circle problems are also obtained. Some appropriate examples are provided for each of these results.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 7356818
zbMATH Keywords: fixed point
zbMATH Keywords: common fixed point
zbMATH Keywords: fixed circle
zbMATH Keywords: discontinuity
MaRDI profile type: Publication
full work available at URL: s11784-021-00863-3
OpenAlex ID: W3154212811
cites work: Fixed-discs in rectangular metric spaces
cites work: A remark on discontinuity at fixed point
cites work: Contractive definitions and discontinuity at fixed point
cites work: Generalized Meir-Keeler type contractions and discontinuity at fixed point
cites work: Geometric properties of discontinuous fixed point set of \(( \epsilon -\delta )\) contractions and applications to neural networks
cites work: A REMARK ON CONVERGENCE THEORY FOR ITERATIVE PROCESSES OF PROINOV CONTRACTION
cites work: Common fixed point theorems for some families of maps
cites work: Equivalent conditions and the Meir-Keeler type theorems
cites work: Some identities of \(q\)-Euler polynomials arising from \(q\)-umbral calculus
cites work: Two new types of fixed point theorems for F-contraction
cites work: A new contractive condition approach to \(\varphi\)-fixed point results in metric spaces and its applications
cites work: Wardowski type contractions and the fixed-circle problem on \(S\)-metric spaces
cites work: On the fixed-circle problem and Khan type contractions
cites work: Some fixed-circle theorems on metric spaces
cites work: FIXED-CIRCLE PROBLEM ON S-METRIC SPACES WITH A GEOMETRIC VIEWPOINT
cites work: Q4629032
cites work: Some fixed-circle theorems and discontinuity at fixed circle
cites work: Generalizations of metric spaces: from the fixed-point theory to the fixed-circle theory
cites work: Fixed-disc results via simulation functions
cites work: Discontinuity and fixed points
cites work: Fixed points and continuity of contractive maps
cites work: Discontinuity at fixed points with applications
cites work: On discontinuity problem at fixed point
cites work: New results on discontinuity at fixed point
cites work: Q3794884
cites work: A new contribution to discontinuity at fixed point
cites work: New types of \(F_c\)-contractions and the fixed-circle problem
cites work: Fixed points of a new type of contractive mappings in complete metric spaces
cites work: Weak theta-phi-contraction and discontinuity
DOI: 10.1007/S11784-021-00863-3
description: scientific article; zbMATH DE number 7356818