Explicit integral criteria for the existence of positive solutions of the linear delayed equation \(\dot{x}(t) = -c(t) x(t -\tau(t))\) (Q2346032)

Dealing with linear differential equations of the form \[ \dot{x}(t)=-c(t)\,x(t-\tau(t)) \] subject to a continuous delay function \(\tau:[t_0,\infty)\to(0,r]\), \(r>0\), the author establishes an explicit integral criteria for the existence of positive solutions. With a thorough comparative explanation, it is shown that the new criteria encompass some known results and a class of functions satisfying the assumptions of the main result (Theorem 3) is also provided. Moreover, the applicability of this theory is illustrated by two examples.