Long-time behaviour of solutions of delayed-type linear differential equations

Abstract

This paper investigates the asymptotic behaviour of the solutions of the retarded-type linear differential functional equations with bounded delays $$ \dot x(t)=-L(t,x_t) $$ when $t\to\infty$. The main results concern the existence of two significant positive and asymptotically different solutions $x=\varphi^*(t)$, $x=\varphi^{**}(t)$ such that $\lim_{t\to\infty}\varphi^{**}(t)/\varphi^{*}(t)=0$. These solutions make it possible to describe the family of all solutions by means of an asymptotic formula. The investigation basis is formed by an auxiliary linear differential functional equation of retarded type $\dot y(t)=L^{\ast}(t,y_t)$ such that $L^{\ast}(t,y_t)\equiv 0$ for an arbitrary constant initial function $y_t$. A commented survey of the previous results is given with illustrative examples.

Keywords

• linear differential functional equation of retarded-type
• asymptotic behaviour of solutions
• asymptotic convergence
• asymptotic divergence