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Oscillation and nonoscillation criteria for a second-order linear equation. (English) Zbl 0944.34025

The authors investigate oscillatory properties of the second-order linear differential equation \[ u''+p(t)u=0. \tag{*} \] Let \(c(t):={1\over t}\int_1^t\int_1^s p(\xi) d\xi ds\). The classical result of oscillation theory for (*) states that this equation is oscillatory provided \(-\infty<\liminf_{t\to \infty} c(t)< \limsup_{t\to \infty}\) or if \(\lim_{t\to \infty} c(t)=\infty\) [see e.g. P. Hartman, Ordinary differential equations, J. Wiley, New York (1964; Zbl 0125.32102)]. The authors treat the “complementary” case when there exists a finite limit \(c_0:=\lim_{t\to \infty} c(t)\). A typical result is the following oscillation criterion:
Let \(\limsup_{t\to \infty}{t\over \lg t}\left(c_0-c(t)\right)>{1\over 4}\) then (*) is oscillatory.
Oscillation and nonoscillation criteria involving upper and lower limits of the quantities \[ Q(t):=t\left(c_0-\int_1^t p(s) ds\right), \quad H(t):=\tfrac 1t \int_1^t s^2 p(s) ds, \] are given as well. Note also that one of the corollaries of the oscillation criteria given in the paper coincides with a result of M. K. Kwong [Math. Inequal. Appl. 2, No. 1, 55-71 (1999; Zbl 0921.34035)], which states that (*) is oscillatory provided \[ -\infty<\limsup_{t\to\infty}\int_1^t sp(s) ds<\liminf_{t\to\infty}\int_1^t sp(s) ds +1<\infty \] and that this result was proved independently using different methods.
Reviewer: O.Došlý (Brno)

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34K25 Asymptotic theory of functional-differential equations
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