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Integral inequalities for second-order linear oscillation. (English) Zbl 0921.34035

The author extends in various directions the classical Lyapunov inequality for solutions to the second-order differential equation

${y}^{\text{'}\text{'}}+q\left(t\right)y=0·\phantom{\rule{2.em}{0ex}}\left(*\right)$

This investigation attracted attention in several recent papers [see e.g. B. Harris and Q. Kong, Trans. Am. Math. Soc. 347, No. 5, 1831-1839 (1995; Zbl 0829.34025) and S. Clark and D. B. Hinton, Math. Inequal. Appl. 1, No. 2, 201-209 (1998; Zbl 0909.24033) and the reference given therein].

The principal role plays the concept of the downswing of a function, which measures (in a certain sense) how much a function can fall down in a given interval. Using this concept, several necessary conditions are obtained for the existence of conjugate/focal points of solutions to (*) in a given interval.

Using these results, the following interesting nonoscillation criterion for (*) is proved. If

$\underset{T\to \infty }{lim sup}{\int }_{0}^{T}tq\left(t\right)dt-\underset{T\to \infty }{lim inf}{\int }_{0}^{T}tq\left(t\right)dt<1,$

then (*) is nonoscillatory.

Reviewer: O.Došlý (Brno)

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 26D10 Inequalities involving derivatives, differential and integral operators