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.  2002 Elsevier Science (USA). All rights reserved.