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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 $$ (p(t)y')'+q(t)y=0 \tag{*} $$ where $1/p,q\in L_{\text{loc}}([t_0,\infty), \Bbb R)$ 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 {\it Ch. G. Philos} [Arch. Math. 53, 483--492 (1989; Zbl 0661.34030)] and by {\it 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 {\it I. V. Kamenev} [Mat. Zametki 23, 249--251 (1978; Zbl 0408.34031)]. Examples illustrating the oscillation criteria are given, too.

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