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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 $$(r(t)|x'(t)|^{\alpha-1}x'(t))'+F(t,x(t),x(\tau(t)),x'(t),x'(\tau(t)))=0,\quad t\ge t_0,$$ 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 $\tau(t) \ge t$ delay and $Gt(t) \ge t$ advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.

MSC:
34K11Oscillation theory of functional-differential equations
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References:
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