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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
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[2] Coffman, C.V; Ullrich, D.F, On the continuation of solutions of a certain nonlinear differential equation, Monatshefte für math., 71, 385-392, (1967) · Zbl 0153.40204
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[7] Ličko, Imrich; Švec, Marko, Le caractère oscillatoire des solutions de l’équation y(n) + f(x)yα = 0, n > 1, Czech. math. J., 13, 481-491, (1963) · Zbl 0123.28202
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