Integral criteria for the existence of positive solutions of first-order linear differential advanced-argument equations

A linear differential equation with advanced-argument y′(t)−c(t)y(t+τ)=0 is considered where c:[t0,∞)→[0,∞), t0∈R is a bounded and locally Lipschitz continuous function and τ>0. The well-known explicit integral criterion ∫tt+τc(s)ds≤1/e,t∈[t0,∞) guarantees the existence of a positive solution on [t0,∞). The paper derives new integral criteria involving the coefficient c. Their independence of the previous result is discussed as well.