Kwong, Man Kam Integral inequalities for second-order linear oscillation. (English) Zbl 0921.34035 Math. Inequal. Appl. 2, No. 1, 55-71 (1999). The author extends in various directions the classical Lyapunov inequality for solutions to the second-order differential equation \[ y''+ q(t)y=0. \tag{*} \] 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 \[ \limsup_{T\to \infty}\int_0^T tq(t) dt- \liminf_{T\to \infty}\int_0^T tq(t) dt<1, \] then (*) is nonoscillatory. Reviewer: O.Došlý (Brno) Cited in 1 ReviewCited in 3 Documents MSC: 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 26D10 Inequalities involving derivatives and differential and integral operators Keywords:Lyapunov inequality; focal point; conjugate point; downswing of a function Citations:Zbl 0829.34025; Zbl 0909.34033; Zbl 0909.24033 PDFBibTeX XMLCite \textit{M. K. Kwong}, Math. Inequal. Appl. 2, No. 1, 55--71 (1999; Zbl 0921.34035) Full Text: DOI