Gollwitzer, H. E. On nonlinear oscillations for a second order delay equation. (English) Zbl 0169.11401 J. Math. Anal. Appl. 26, 385-389 (1969). 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 \((*)\), \(1<\gamma\), \(0<\alpha<1\). Reviewer: Herman E. Gollwitzer (Knoxville) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 35 Documents MSC: 34K11 Oscillation theory of functional-differential equations Keywords:nonlinear oscillations; second order delay equation Citations:Zbl 0065.32001; Zbl 0123.28202 PDFBibTeX XMLCite \textit{H. E. Gollwitzer}, J. Math. Anal. Appl. 26, 385--389 (1969; Zbl 0169.11401) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.