Oscillations of linear integro-differential equations (Q1767425)

The authors consider the linear integro-differential equation of the type \[ \dot x(t) +\int_0^t x(t-s)dr_s(t,s)\;\;t\geq 0 \] where the function \(r: [0, \infty)^2\to [0, \infty)\) is increasing with respect to \(s\leq t\). Sufficient conditions which guarantee that this equation cannot have a positive solution are established.