Oscillation of equations with an infinite distributed delay

Abstract

For the equation with a finite or infinite distributed delay ẋ(t)+∫−∞tx(s)dsR(t,s)=0 the existence of nonoscillatory solutions is studied. A general comparison theorem is obtained which allows to compare oscillation properties of equations with concentrated delays to integrodifferential equations. Sharp nonoscillation conditions are deduced for some autonomous integrodifferential equations. Using comparison theorems, an example is constructed where oscillation properties of an integrodifferential equation are compared to equations with several concentrated delay which can be treated as its finite difference approximations.