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

Abstract

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