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Interval criteria for oscillation of second-order linear ordinary differential equations. (English) Zbl 0924.34026

New oscillation criteria are established for the second-order differential equation

(p(t)y ' ) ' +q(t)y=0(*)

where 1/p,qL loc ([t 0 ,),) and p>0 a.e. on [t 0 ,). Integral conditions on the functions p,q are given which guarantee the existence of (disjoint) intervals [a i ,b i ], a i <b i a i+1 , a i as i, such that any nontrivial solution to (*) has at least one zero in (a i ,b i ), which implies oscillation of (*). These integral conditions use “H-function” technique introduced by Ch. G. Philos [Arch. Math. 53, 483-492 (1989; Zbl 0661.34030)] and by H. J. Li [J. Math. Anal. Appl. 194, 217-234 (1995; Zbl 0836.34033)]. Some of them are extensions of Kamenev’s and Philos’ type criteria; see I. V. Kamenev [Mat. Zametki 23, 249-251 (1978; Zbl 0408.34031)]. Examples illustrating the oscillation criteria are given, too.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory