Note on ``Oscillation criteria for second order linear differential equations''
\textit{Horng Jaan Li} [J. Math. Anal. Appl. 194, 217-234 (1995; Zbl 0836.34033)] proved some oscillation criteria for the equation \([r(t)u'(t)]' + c(t)u(t) =0\) by using a generalized Riccati transformation due to Yu. He improved oscillation criteria of Yan and Philos and obtained, as a special case, the criterion of Kamenev. One of his assumptions has the form \((C_1)\) \(\int^t_{t_0} a(s)r(s) h^2(t,s) ds < \infty\) for all \(t\geq t_0\) where \(a(t) >0\), \(r(t) >0\) are continuous on \([t_0, \infty)\) and \(h(t,s)\) is continuous on \(D_0= \{(t,s): t>s \geq t_0\}\). By the author of this note assumption \((C_1)\) means that there exists a constant \(K>0\) such that \(\int^t_{t_0} a(s) r(s)h^2(t,s) ds<K\) for all \(t\geq t_0\). Then he shows that Horng Jaan Li's results do not improve the results of Yan and Philos and that we do not obtain the Kamenev criterion as a special case. Moreover, he states theorems and corollaries of Horng Jaan Li in which assumption \((C_1)\) should be omitted. \{Reviewer's remark: The function \(h(t,s)\) in \((C_1)\) may be discontinuous for \(t=s\). Hence assumption \((C_1)\) by Horng Jaan Li only means that \(\int^t_{t_0} a(s) r(s)h^2(t,s)ds\) is a Lebesgue integrable function on any interval \([t_0,T]\), \(T>0\}\).
scientific article; zbMATH DE number 944370