Long-time behavior of positive solutions of a differential equation with state-dependent delay

Abstract

The long-time behavior of positive solutions of a differential equation with state-dependent delayẏ(t) = −c(t)y(t − τ (t, y(t))), where c is a positive coefficient, is considered. Sufficient conditions are given for the existence of positive solutions bounded from below and from above by functions of exponential type. As a consequence, criteria for the existence of positive solutions are derived and their lower bounds are given. Relationships are discussed with the existing results on the existence of positive solutions for delayed differential equations.

Existence of a solution of discrete Emden-Fowler equation caused by continuous equation

“…Let us remark as well that the asymptotic behaviour of solutions to linear discrete equations is studied in [10]. For further recent investigations of the properties of solutions to similar classes of discrete and continuous equations, involving new methods and ideas we refer to [3,9,14,15,18,21,30,31].…”