Positive solutions of nonlinear delayed differential equations with impulses

The paper is concerned with the long-term behavior of solutions to scalar nonlinear functional delayed differential equations ẏ(t)=−f(t,yt),t≥t0.It is assumed that f:[t0,∞)×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(ts+)=bsy(ts), s=1,2,…, where t0≤t10, s=1,2,…. A criterion for the existence of positive solutions on [t0−r,∞) is proved and their upper estimates are given. Relations to previous results are discussed as well.