Oscillation for equations with positive and negative coefficients and with distributed delay I: General results

Abstract

We study a scalar delay differential equation with a bounded distributed delay, ẋ(t) + ∫th(t) x(s) dsR(t, s) - ∫tg(t) x(s) d_s T (t, s) = 0, where R(t, s), T(t, s) are nonnegative nondecreasing in s for any t, R(t, h(t)) = T(t, g(t)) = 0, R(t, s) ≥ T(t, s). We establish a connection between non-oscillation of this differential equation and the corresponding differential inequalities, and between positiveness of the fundamental function and the existence of a nonnegative solution for a nonlinear integral inequality that constructed explicitly. We also present comparison theorems, and explicit non-oscillation and oscillation results. In a separate publication (part II), we will consider applications of this theory to differential equations with several concentrated delays, integrodifferential, and mixed equations.