Ordinary differential equations in the complex domain via topological transversality
type: Item
instance of: scholarly article
title: Ordinary differential equations in the complex domain via topological transversality
zbMATH Open document ID: 0663.34005
DOI: 10.1016/0022-0396(87)90141-0
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: In this paper the author studies the differential equation \(y^{(n)}=f(t,y,y',...,y^{(n-1)}),\) \(t\in U_ T\), with initial conditions \(y(0)=\alpha_ 0,...,v_{(0)}^{(n-1)}=\alpha_{n-1}\), where \(\alpha_ j\in {\mathbb{C}}\), \(j=0,1,...,n-1\), and \(f: \overline{U}_ T\times {\mathbb{C}}^ n\to {\mathbb{C}}\) is analytic in \(t,y,...,y^{(n-1)}\) for \(t\in U_ T\) and continuous in \(t,y,...,y^{(n-1)}\) for \(t\in \overline{U}_ T\), where \(U_ T=\{z: | z| <T\}\), \(T\geq 0\). For a given integer \(k\geq 0\) and a constant \(T>0\), let \(A^ k(\overline{U}_ T)\) denote the Banach space of functions g: \(\overline{U}_ T\to {\mathbb{C}}\) which are analytic on \(U_ T\) with \(g,...,g^{(k)}\) continuous on \(\overline{U}_ T\) with the norm \(\| g\|_ k\) defined as \(\max \{\| g\|_ 0,\| g'\|_ 0,...,\| g^{(k)}\|_ 0\}\), where \(\| g\|_ 0=\sup_{t\in \overline{U}_ T}| g(t)|\). Imposing upon \(| f(t,y,...,y^{(n-1)}|\) different types of growth conditions the author shows how to determine \(T>0\) such that the above initial value problem has at least one solution y in \(A^ n(\overline{U}_ T)\). In the case \(\alpha_ j=0\), \(j=0,...,n-1\), the author shows that if \[ | f(t,u_ 0,u_ 1,...,u_{n-1})| \leq A_ 0(t)| u_ 0| +...+A_{n-1}(t)| u_{n-1}| +C(t), \] where \(A_ 0(t),...,A_{n-1}(t)\), C(t)\(\geq 0\) are functions bounded on bounded sets, then the initial value problem has at least one solution in \(A^ n(\overline{U}_ T)\) provided \(0<T<T_ 0\), where \(T_ 0\) is the smallest positive root of \(A_ 0T^ n+A_ 1T^{n-1}+...+A_{n-1}T- 1=0\) and \(A_ 0,...,A_{n-1}\) are the upper bounds on \(\overline{U}_ T\) for \(A_ 0(t),...,A_{n-1}(t)\), respectively. The method used is extended also to the case where the values of \(y(t),y'(t),...,y^{(n- 1)}(t)\) are prescribed at different points as exemplified by the following result for \(n=2:\) The problem \[ y''=f(t,y,y'),\quad t\in U_ T,\quad y'(0)=0,\quad y(c)=0,\quad c\in \overline{U}_ T \] has at least one solution in \(A^ 2(\overline{U}_ T)…
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 4084074
zbMATH Keywords: topological transversality
zbMATH Keywords: growth conditions
author: Donal O'Regan
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90141-0
OpenAlex ID: W2066274331
cites work: Q3257872
cites work: Q4421936
cites work: Q4118379
cites work: Nonlinear boundary value problems for some classes of ordinary differential equations
cites work: Applications of topological transversality to differential equations. I: Some nonlinear diffusion problems
cites work: Q3676458
cites work: Q5843703
cites work: Q5526190
cites work: Q4051163
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
description: scientific article; zbMATH DE number 4084074