On oscillation of equations with distributed delay

For the scalar delay differential equation with a distributed delay ẋ(t) + ∫ _-∞^t x(s) d_s R(t, s) = f(t) (t > t₀) a connection between the properties non-oscillation positiveness of the fundamental function existence of a non-negative solution for a certain nonlinear integral inequality is established. This enables to obtain comparison theorems and explicit non-oscillation and oscillation conditions being generalizations of some known results for delay equations and integro-differential equations and leads to oscillation results for equations with infinite number of delays.