Asymptotic and locally asymptotic funnels for systems from the Osgood class for the inner product
type: Item
instance of: scholarly article
title: Asymptotic and locally asymptotic funnels for systems from the Osgood class for the inner product
zbMATH Open document ID: 0671.34029
DOI: 10.1007/BF01056827
author: S. K. Norkin
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let \(x\in R_+\), \(w=colon(y,z)\in R^ k\times R^{n-k}\), \(n>k\), \[ F(x,w)=colon(f(x,y,z),g(x,y,z)): R_+\times R^ k\times R^{n-k}\to R^ k\times R^{n-k\quad}. \] The author considers the nonlinear n dimensional system (1) \(w'=F(x,w)\) defined in the domain \[ S\{(x,w):\quad x\in (0,a),\quad \| y\| \in [0,b),\quad \| z\| \in [0,d)\}, \] (a, b, d, are fixed real numbers, \(\| \cdot \|\) is the Euclidean norm). The solution \(w(x;x^ 0,w^ 0)\) of (1) satisfying \(w(x^ 0;x^ 0,w^ 0)=w^ 0\) is called an 0-solution if there holds \(\lim \| w(x;x^ 0,w^ 0)\| =0,\) \(x\to +0\). The set M of such solutions is studied, its dimension, asymptotics and a representation is determined under the assumption that the scalar product for components of F(x,w) belongs to the Osgood class as defined by the author. This extends previous results of the author, A. F. Andreev and of N. A. Bodunov, where the mentioned scalar product belongs to the Lipschitz class.
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34E99
zbMATH DE Number: 4097830
zbMATH Keywords: asymptotic funnels
zbMATH Keywords: 0-solution
zbMATH Keywords: Osgood class
reviewed by: Vojislav Marić
MaRDI profile type: Publication
cites work: Q4067423
cites work: Q3853969
cites work: Q3338593
description: scientific article; zbMATH DE number 4097830