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Interval criteria for oscillation of second-order functional differential equations. (English) Zbl 1091.34036

Summary: By using averaging functions, new interval oscillation criteria are established for the second-order functional-differential equation

(r(t)|x ' (t)| α-1 x ' (t)) ' +F(t,x(t),x(τ(t)),x ' (t),x ' (τ(t)))=0,tt 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 ,], 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.

34K11Oscillation theory of functional-differential equations