Linearized oscillation theory for a nonlinear equation with a distributed delay

We obtain linearized oscillation theorems for the equation with distributed delays equation(1)ẋ(t)+∑k=1mrk(t)∫−∞tfk(x(s))dsRk(t,s)=0. The results are applied to logistic, Lasota–Wazewska and Nicholson’s blowflies equations with a distributed delay. In addition, the “Mean Value Theorem” is proved which claims that a solution of (1) also satisfies the linear equation with a variable concentrated delay ẋ(t)+(∑k=1mrk(t)fk′(ξk(t)))x(g(t))=0.