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 \[ (p(t)y')'+q(t)y=0 \tag{*} \] where \(1/p,q\in L_{\text{loc}}([t_0,\infty), \mathbb R)\) and \(p>0\) a.e. on \([t_0,\infty)\). Integral conditions on the functions \(p,q\) are given which guarantee the existence of (disjoint) intervals \([a_i,b_i]\), \(a_i<b_i\leq a_{i+1}\), \(a_i\to\infty\) as \(i\to \infty\), 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.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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