On oscillation properties of delay differential equations with positive and negative coefficients

For a scalar delay differential equation x(t) + a(t)x(h(t)) - b(t)x(g(t)) = 0, a(t) ≥ 0, b(t) ≥ 0, h(t) ≤ t, g(t) ≤ t, a connection between the following properties is established: nonoscillation of the differential equation and the corresponding differential inequalities, positiveness of the fundamental function and existence of a nonnegative solution for a certain explicitly constructed nonlinear integral inequality. A comparison theorem and explicit nonoscillation and oscillation results are presented.