An Explicit Criterion for the Existence of Positive Solutions of the Linear Delayed Equationx˙(t)=−c(t)x(t−τ(t))

Abstract

The paper investigates an equation with single delayẋ(t)=-c(t)x(t-τ(t)), whereτ:[t0-r,∞)→(0,r],r>0,t0∈R, andc:[t0-r,∞)→(0,∞)are continuous functions, and the differencet-τ(t)is an increasing function. Its purpose is to derive a new explicit integral criterion for the existence of a positive solution in terms ofcandτ. An overview of known relevant criteria is provided, and relevant comparisons are also given.