Oscillatory solutions of the system of n integro-differential equations with deviating arguments
type: Item
instance of: scholarly article
title: Oscillatory solutions of the system of n integro-differential equations with deviating arguments
zbMATH Open document ID: 0734.45003
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 1990-01-01T00:00:00Z
review text: Consider a system of n integro-differential equations with advanced arguments of the form \[ x_ k'(t)=\int^{t}_{t_ 0}f_ k(t,s,x_ 1(s),...,x_ n(s),x_ 1(g_ 1(s)),...,x_ n(g_ n(s)))ds, \] \(1\leq k\leq n\), where \(f_ k,g_ k\) are real continuous functions, \(t\geq 0\), \(s\geq 0\), \(g_ k(t)>t\), \(\lim_{t\to \infty}g_ k(t)=+\infty.\) By a solution \(x=(x_ 1,...,y_ n)\) of the system we understand only nontrivial solutions extended to infinity. A solution of the system is called oscillatory (resp. nonoscillatory) if every component \(x_ k(t)\) has an infinite sequence of zeros tending to infinity as \(t\to \infty\) (resp. has a constant sign for sufficiently large \(t\geq T\geq 0).\) Let the functions \(f_ k\) satisfy the following assumptions: \[ (1)\quad f_ k(t,s,x,v)sgn v_{k+1}\geq a_{k+1}(t,s)A_{k+1}(| v_{k+1}|),\quad 1\leq k\leq n-1, \] \[ f_ n(t,s,x,v)sgn v_ 1\leq - a_ 1(t,s)A_ 1(| v_ 1|), \] where \(v=(v_ 1,...,v_ n)\), \(v_ i=x_ i(g_ i(t))\), \(1\leq i\leq n\), \(x_{n+1}=x_ n\), \(a_ k\) are continuous functions and \(A_ k\) are continuous, nondecreasing and nonnegative functions; \[ (2)\quad \lim_{t\to \infty}\int^{t}_{T}\int^{\tau}_{T}a_ k(\tau,s)dsd\tau =\infty,\quad 2\leq k\leq n. \] The author proves three lemmas and two main results. The first one asserts that if (1), (2) and \(\lim_{t\to \infty}\int^{t}_{T}\int^{\tau}_{T}a_ 1(\tau,s)dsd\tau =\infty\) hold, then every bounded solution of the system is oscillatory. The second result asserts that if (1), (2) and a certain condition hold, then every solution of the system is oscillatory for even n, and either oscillatory or tending monotonically to zero as \(t\to \infty\) for odd n.
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45M15
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45G15
zbMATH DE Number: 4214691
zbMATH Keywords: oscillatory solutions
zbMATH Keywords: deviating arguments
zbMATH Keywords: system
zbMATH Keywords: integro-differential equations with advanced arguments
author: Izabela Foltyńska
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4214691