Oscillation criteria for second order linear differential equations with damping (Q2785597)

The authors investigate oscillatory properties of solutions of the second order differential equation with damping NEWLINE\[NEWLINE[r(t)x'(t)]'+p(t)x'(t)+q(t)x(t)=0,\quad t\geq t_0,\tag{*}NEWLINE\]NEWLINE where \(r(t)>0\) and \(p,q\) are allowed to change sign in \([t_0,\infty)\). NEWLINENEWLINENEWLINEThe obtained oscillation criteria are based on the generalized Riccati transformation NEWLINE\[NEWLINEw(t)=a(t)r(t)\left({x'(t)\over x(t)}+ {a'(t)\over 2a(t)}\right),NEWLINE\]NEWLINE where \(a\in C^2[t_0,\infty)\) is a nonzero function, which relates (*) to a certain Riccati-type equation, and in case \(p\equiv 0\) these criteria reduce essentially to the results of the paper \textit{H. J. Li} [J. Math. Anal. Appl. 194, No. 1, 217-234, (1995; Zbl 0836.34033)] NEWLINENEWLINENEWLINERecently, \textit{Y. Rogovchenko} [J. Math. Anal. Appl. 203, 560-563 (1996; Zbl 0862.34024)] observed that certain assumptions in oscillation criteria of Li may be omitted. The present paper already reflects this note of Rogovchenko.