Linearized oscillation theory for a nonlinear equation with a distributed delay

We obtain linearized oscillation theorems for the equation with distributed delays ẋ(t)+ m ∑ k=1 rk(t) ∫ t −∞ fk(x(s)) ds Rk(t, s) = 0. (1) 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)+ ( m ∑ k=1 rk(t) f ′ k(ξk(t)) ) x(g(t)) = 0. c © 2007 Elsevier Ltd. All rights reserved.