On the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\)
type: Item
instance of: scholarly article
title: On the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\)
zbMATH Open document ID: 0707.26002
author: A. Zitarosa
author: Giovanni Di Lena
author: Basilio Messano
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108114
review text: Summary: In this paper we consider the iterative process \(x_{n+1}=f(x_ n,x_{n-1})\), \(n\in N\), where f is a continuous function from \([0,1]^ 2\) in \([0,1]\) and we prove that the condition of the non existence of a pair \((x,y)\) of distinct points of \([0,1]\) such that \(f(x,y)=y\) and \(f(y,x)=x,\) obviously necessary for the global convergence, is sufficient if f is decreasing with respect to both variables and whatever the point \((x_ 1,x_ 0)\) of \([0,1]^ 2\) be, the following implications: \[ \max \{x_ 0,x_ 1\}<x_ 2\Rightarrow \min \{x_ 0,x_ 1\}<\min \{x_ 3,x_ 4\}; \] \[ x_ 2<\min \{x_ 0,x_ 1\}\Rightarrow \max \{x_ 3,x_ 4\}<\max \{x_ 0,x_ 1\}, \] are true.
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4161020
zbMATH Keywords: iterative process
zbMATH Keywords: global convergence
MaRDI profile type: Publication
cites work: Continuous and discrete dynamics near manifolds of equilibria
cites work: Conditions for the convergence of iterative processes on the real axis
cites work: On continuous functions, commuting functions, and fixed points
cites work: On the Behavior of Successive Approximations
cites work: On the global convergence of Halley's iteration formula
cites work: Q3923517
cites work: Global convergence of the method of successive approximations on \(S^ 1\)
cites work: Q3871822
cites work: Sulla convergenza del metodo delle approssimazioni successive in \(R^ 1\).
cites work: Q4748506
cites work: Q5185697
cites work: Chaos in \(C^ 0-\)endomorphism of interval
cites work: Q5614464
description: scientific article; zbMATH DE number 4161020