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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 $(r(t)|x'(t)|^{\alpha-1}x'(t))'+F(t,x(t),x(\tau(t)),x'(t),x'(\tau(t)))=0,\quad t\geq t_0,$ that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of $$[t_0,\infty]$$, 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 $$\tau(t) \geq t$$ delay and $$Gt(t) \geq t$$ advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations
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##### References:
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