On the generalized successive approximations method
type: Item
instance of: scholarly article
title: On the generalized successive approximations method
zbMATH Open document ID: 0677.65044
DOI: 10.1007/BF02575947
author: A. Zitarosa
author: Giovanni Di Lena
author: Basilio Messano
published in: Calcolo
publication date: 1988-01-01T00:00:00Z
review text: The authors discuss the question of global convergence of the iterative process \(x_{n+1}=f(x_ n,x_{n-1}),\) \(n\in {\mathbb{N}}\), \(x\in [0,1]\) where f is a continuous function from \([0,1]^ 2\) into [0,1]. Any point \((x,y)\in [0,1]^ 2\) is (i) a fixed point of f if \(x=y=f(x,y)\) and (ii) a periodic point of f of period two if \(f(y,x)=x\neq y=f(x,y).\) The authors show that if f is decreasing with respect to the first variable, and there does not exist a point (x,y) of \([0,1]^ 2\) such that there are no periodic points, then one cannot find a point \((x,y)\in [0,1]^ 2\) such that \(f(y,x)\leq x<y\leq f(x,y).\) Use of this result shows that f has a unique fixed point. The authors then present necessary conditions for global convergence when \(f\in A_ 1\cap A_ 2\) and necessary and sufficient conditions for global convergence when \(f\in A_ 1\cap S\). \(A_ 1(A_ 2)\) is the set of all continuous functions from \([0,1]^ 2\) into [0,1] which are decreasing with respect to the first (second) variable and S is the set of all continuous functions from \([0,1]^ 2\) into [0,1] which have no periodic point of order two.
Mathematics Subject Classification ID: 65H05
zbMATH DE Number: 4109959
zbMATH Keywords: fixed point iteration
zbMATH Keywords: successive approximations
zbMATH Keywords: global convergence
zbMATH Keywords: periodic point
MaRDI profile type: Publication
cites work: On the Henon transformation
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description: scientific article; zbMATH DE number 4109959