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On nonlinear oscillations for a second order delay equation. (English) Zbl 0169.11401
The author considers the second order differential delay equation $y''(t) + q(t)y(t - \tau(t))^\gamma =0 \tag{$$*$$}$ on a half line $$[a, \infty)$$. The function $$q$$ is non-negative, continuous and $$\gamma$$ is the quotient of odd integers, $$0<\gamma <1$$ or $$1<\gamma$$. The delay $$\tau(t)$$ is non-negative, continuous and bounded for $$t\ge a$$, and only extendable solutions of $$(*)$$ are considered. The author proves, using monotonicity arguments, two oscillation theorems for $$(*)$$ which reduce to known cases if $$\tau\equiv 0$$ (see F. V. Atkinson [Pac. J. Math. 5, 643–647 (1955; Zbl 0065.32001)] and I. Ličko and M. Švec [Czech. Math. J. 13(88), 481–491 (1963; Zbl 0123.28202)]).
Theorem 1: Let $$1<\gamma$$. Equation $$(*)$$ is oscillatory if and only if $$\int^\infty sq = \infty$$.
Theorem 2: Let $$0<\gamma <1$$. Equation $$(*)$$ is oscillatory if and only if $$\int^\infty s^\gamma q = \infty$$.
Using these theorems, the author shows that $y''(t) + q_1(t)y(t - \tau_1(t))^\gamma + q_2(t)y(t - \tau_2 (t))^\alpha = 0 \tag{$$**$$}$ is oscillatory if and only if $$\int^\infty (sq_1 + s^\alpha q_2) = \infty$$, where $$q_1,q_2,\tau_1, \tau_2$$ are as in equation $$(*)$$, $$1<\gamma$$, $$0<\alpha<1$$.

##### MSC:
 34K11 Oscillation theory of functional-differential equations
##### Keywords:
nonlinear oscillations; second order delay equation
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##### References:
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