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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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References:

[1] Atkinson, F. V., On second-order nonlinear oscillations, Pacific J. Math., 5, 643-647 (1955) · Zbl 0065.32001
[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
[3] El’sgol’ts, L. E., Introduction to the Theory of Differential Equations with Deviating Arguments (1966), Holden-Day: Holden-Day San Francisco · Zbl 0133.33502
[4] J. W. HeidelProc. Amer. Math. Soc.; J. W. HeidelProc. Amer. Math. Soc. · Zbl 0169.42203
[5] J. W. Heidel; J. W. Heidel
[6] Kiguradze, I. T., Soviet Math. Dokl., 3, 649-652 (1962), Translated as · Zbl 0144.11201
[7] Ličko, Imrich; Švec, Marko, Le caractère oscillatoire des solutions de l’équation \(y^{(n)} + f(x)y^{α\) · Zbl 0123.28202
[8] Utz, W. R., Properties of Solutions of \(u\)″ + \(g(t)u^{2n − 1} = 0\), II, Monatshefte für Math., 69, 353-361 (1965) · Zbl 0144.10701
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