New study on the convergence of a formal transformation. II
type: Item
instance of: scholarly article
title: New study on the convergence of a formal transformation. II
zbMATH Open document ID: 0845.34008
author: Masahiro Iwano
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-09-11T00:00:00Z
review text: The paper begins with a brief review of the results presented in part I [Funkc Ekvacioj, Ser. Int. 38, No. 3, 443-479 (1995; Zbl 0845.34007), see the preceding review]. The paper deals with a system of two nonlinear differential equations of the form \[ x^2 {dy\over dx}= (\mu+ \alpha x) y+ f(x, y, z),\quad x^2 {dz\over dx}= (- v+ \beta x) z+ g(x, y, z), \] where (i) \(x\) is an independent variable; (ii) \(\mu\) and \(v\) are positive numbers and their ratio is irrational; (iii) \(\alpha\) and \(\beta\) are complex constants and there is a positive quantity \(K\) satisfying the inequalities \(\mu+ \kappa {\mathfrak R} \alpha> 0\), \(- v+ \kappa {\mathfrak R} \beta> 0\); (iv) \(f(x, y, z)\) and \(g(x, y, z)\) are holomorphic and bounded functions of \((x, y, z)\) for \(|x|< a\), \(|y|< b\), \(|z|< b\), and their Taylor series expansions in \((y, z)\) contain neither the constant terms nor the linear terms, where \(a\) and \(b\) are small. Formal transformations in that earlier paper of the form \[ y= u+ \sum_{j+ k\geq 2} p_{jk}(x) u^j v^k,\quad z= v+ \sum_{j+ k\geq 2} q_{jk}(x) u^j v^k \] are rearranged, whereby the new transformations satisfy nonlinear differential equations of the form \[ x^2 {dP_0\over dx}= (\mu+ \alpha x) P_0+ f(x, P_0, V(x)+ Q_0),\;x^2 {dQ_0\over dx}= (- v+ \beta x) Q_0+ g(x, P_0, V(x)+ Q_0). \] A two parameter family of bounded solutions is constructed implementing fixed point techniques. The domains of holomorphy for the functions contained within the fixed point techniques are given by a family of the product of two circles over every point in the domain of the independent variable. Again this paper gives a very complete proof presenting several clever inequalities and proofs which are elegant.
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 847736
zbMATH Keywords: formal transformations
zbMATH Keywords: two nonlinear differential equations
zbMATH Keywords: two parameter family of bounded solutions
reviewed by: John Schmeelk
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225456
OpenAlex ID: W1976697724
cites work: Q5801562
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cites work: Sur l'étude analytique des solutions d'un système d'équations différentielles dans le voisinage d'un point singulier d'indetermination. 1
DOI: 10.2748/TMJ/1178225456
description: scientific article; zbMATH DE number 847736