Positive solutions of nonlinear delayed differential equations with impulses

Abstract

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations y ̇ ( t ) = − f ( t , y t ) , t ≥ t 0 . It is assumed that f : [ t 0 , ∞ ) × C ↦ R is a continuous mapping satisfying a local Lipschitz condition with respect to the second argument and C ≔ C ( [ − r , 0 ] , R ) , r > 0 is the Banach space of continuous functions. The problem is solved of the existence of positive solutions if the equation is subjected to impulses y ( t s + ) = b s y ( t s ) , s = 1 , 2 , … , where t 0 ≤ t 1 < t 2 < ⋯ and b s > 0 , s = 1 , 2 , … . A criterion for the existence of positive solutions on [ t 0 − r , ∞ ) is proved and their upper estimates are given. Relations to previous results are discussed as well.