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Oscillatory property for second order linear delay differential equations. (English) Zbl 0626.34077

This interesting paper deals with the equation (1) \(x''(t)+a(t)x(g(t))=0,\) where \(a\in C[0,\infty)\to [0,\infty)\), a(t)\(\not\equiv 0\) on \([t_ 0,\infty)\) \((t_ 0\geq 0)\); \(g\in C[0,\infty)\to [0,\infty)\); \(0\leq g(t)\leq t\), \(t\geq 0\), \(\lim_{t\to \infty}g(t)=\infty\). The function sequence \(\{\alpha_ n(t)\}\) for \(n=1,2,..\). and \(t\geq t_ 0\), where \(\alpha_ 0(t)=\epsilon \int^{\infty}_{t}\frac{g(s)}{s}a(s)ds,\alpha_ n(t)=\int^{\infty}_{t}\alpha^ 2_{n-1}(s)ds+\alpha_ 0(t),\) \(n=1,2,..\). and \(0<\epsilon <1\) is introduced here. Sufficient conditions for (1) to be oscillatory are formulated by the functions \(\alpha_ n(t)\). The main result of the paper extends the well-known oscillation criteria of Hille, Kneser and Opial in the ordinary differential equation case and Erbe in the delay differential equation case.
Reviewer: J.Ohriska

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

[1] Erbe, L., Oscillation criteria for second order nonlinear delay equations, Canad. Math. Bull., 16, 49-56 (1973) · Zbl 0272.34095
[2] Hille, E., Nonoscillation theorems, Trans. Amer. Math. Soc., 64, 234-252 (1948) · Zbl 0031.35402
[3] Kneser, A., Untersuchungen über die reelen Nullstellen der Integrale linearer Differentialgleichungen, Math. Ann., 42, 409-435 (1893) · JFM 25.0522.01
[4] Opial, Z., Sur les integrales oscillantes de l’equation differentielle \(u\)″ + \(f(t) u = 0\), Ann. Polon. Math., 4, 308-313 (1958) · Zbl 0083.07701
[5] Wong, J. S.W, Second order oscillation with retarded arguments, (Ordinary Differential Equation (1972), Academic Press: Academic Press New York, London), 581-596 · Zbl 0167.38003
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