Interval criteria for oscillation of second-order functional differential equations

By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation,(r(t)|x′(t)|α−1x′(t))′+F(t,x(t),x(τ(t)),x′(t),x′(τ(t)))=0,t≥t0that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t0, ∞], rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional differential equations, our criteria implies that the τ(t) ≤ t delay and Gt(t) ≥ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.