Explicit integral criteria for the existence of positive solutions of the linear delayed equation x˙(t)=−c(t)x(t−τ(t))

The paper analyzes the linear differential equation with single delay x˙(t)=−c(t)x(t−τ(t)) with continuous τ:[t0,∞)→(0,r]τ:[t0,∞)→(0,r], r>0r>0, t0∈Rt0∈R, and c:[t0−r,∞)→(0,∞)c:[t0−r,∞)→(0,∞). New explicit integral criteria for the existence of a positive solution expressed in terms of c and τ are derived, an overview of known relevant criteria is provided, and relevant comparisons are also given. It is demonstrated that the known criteria are consequences of the new results.