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A note on second-order sublinear oscillation theorems. (English) Zbl 0609.34045

The author proves integral criteria for the oscillation of solutions to the differential equation \[ (1)\quad u''+a(t)| u|^{\alpha}sgn u=0,\quad (0<\alpha <1), \] where \(a(t)\in C[t_ 0,\infty)\), \(t_ 0>0\). One of these theorems: Assume that for some \(\beta\in [0,\alpha]\) and \(\gamma\in [1,\infty)\) \[ \limsup_{t\to \infty}t^{- \gamma}\int^{t}_{t_ 0}(t-s)^{\gamma}s^{\beta}a(s)ds=\infty, \] then equation (1) is oscillatory. The theorems of this paper generalize Kura’s results [T. Kura, Proc. Am. Math. Soc. 84, 535-538 (1982; Zbl 0488.34022)].
Reviewer: P.Marušiak

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Citations:

Zbl 0488.34022
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References:

[1] Belohorec, S., Two remarks on the properties of solutions of a nonlinear differential equation, Acta Fac. Rerum Natur, Univ. Comenian. Math., 22, 19-26 (1969) · Zbl 0166.07702
[2] Kamenev, I. V., Some specifically nonlinear oscillation theorems, Mat. Zametki, 10, 129-134 (1971) · Zbl 0266.34042
[3] Kura, T., Oscillation theorems for a second order sublinear ordinary differential equation, (Proc. Amer. Math. Soc., 84 (1982)), 535-538 · Zbl 0488.34022
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