Interval Criteria for Oscillation of Second-Order Functional Differential Equations

Abstract

By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation, (r (t) vertical bar x' (t)vertical bar(alpha-1) x' (t))' + F (t, x (t), x (tau (t)), x' (t), x' (tau (t))) = 0, t >= t(0), that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [t(0), infinity), 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 tau(t) <= t delay and tau(t) >= t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results. (c) 2005 Elsevier Ltd. All rights reserved.