Existence, uniqueness, and stability theorems for bounded solutions of nonlinear differential systems
type: Item
instance of: scholarly article
title: Existence, uniqueness, and stability theorems for bounded solutions of nonlinear differential systems
zbMATH Open document ID: 0581.34007
DOI: 10.1007/BF01776854
author: Zhivko S. Athanassov
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1985-01-01T00:00:00Z
review text: The author investigates the nature and the existence of bounded solutions of the coupled system \[ x'=f_ 1(t,x)+g_ 1(t,x,y)+h_ 1(t),\quad y'=f_ 2(t,x)+g_ 2(t,x,y)+h_ 2(t) \] regarded as perturbation of the uncoupled system \(x'=f_ 1(t,x)\), \(y'=f_ 2(t,y)\) where \(x,f_ 1,g_ 1,h_ 1\) and \(y,f_ 2,g_ 2,h_ 2\) are \(n_ 1\)- and \(n_ 2\)- vectors, respectively, and \(t\geq 0.\) One of the problems considered by the author is to determine the behavior of the solutions as t tends to infinity. The behavior turns out to depend upon the initial conditions. Some of the results of the author himself and of Perron are generalized in this paper. The basic tools used for investigation are Lyapunov functions and the comparison principle. The paper is well organized and is remarkable both in content and style.
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 3929461
zbMATH Keywords: bounded solutions
zbMATH Keywords: perturbation
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: comparison principle
reviewed by: Prem N. Bajaj
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01776854
OpenAlex ID: W2090149056
description: scientific article; zbMATH DE number 3929461