OSCILLATION AND NONOSCILLATION CRITERIA FOR DIFFERENTIAL EQUATIONS OF SECOND ORDER

We give necessary and sufficient conditions such that the homogeneous differential equations of the type: (r(t)x' (t))' + q(t) x' (t) + p(t) x(t) = 0 are nonoscillatory where r(t) > 0 for l is an element of I = [alpha, infinity), alpha > 0. Under the suitable conditions we show that the above equation is nonoscillatory if and only if for gamma > 0, (r(t) x'(t))' + q(t) x' (t) + p(t) x (t-gamma) = 0 is nonoscillatory. We obtain several comparison theorems.