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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:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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##### References:
 [1] Butler, G. J.; Erbe, L. H.; Mingarelli, A. B.: Riccati techniques and variational principles in oscillation theory for linear systems. Trans. amer. Math. soc. 303, 263-282 (1987) · Zbl 0648.34031 [2] Byers, R.; Harris, B. J.; Kwong, M. K.: Weighted means and oscillation conditions for second order matrix differential equations. J. differential equations 61, 164-177 (1986) · Zbl 0609.34042 [3] El-Sayed, M. A.: An oscillation criterion for a forced second-order linear differential equation. Proc. amer. Math. soc. 118, 813-817 (1993) · Zbl 0777.34023 [4] Erbe, L. H.; Kong, Q.; Ruan, S.: Kamenev type theorems for second order matrix differential systems. Proc. amer. Math. soc. 117, 957-962 (1993) · Zbl 0777.34024 [5] Fite, W. B.: Concerning the zeros of the solutions of certain differential equations. Trans. amer. Math. soc. 19, 341-352 (1918) · Zbl 46.0702.02 [6] Hartman, P.: On nonoscillatory linear differential equations of second order. Amer. J. Math. 74, 389-400 (1952) · Zbl 0048.06602 [7] Hartman, P.: Ordinary differential equations. (1982) · Zbl 0476.34002 [8] Kamenev, I. V.: Integral criterion of linear differential equations of second order. Mat. zametki 23, 249-251 (1978) · Zbl 0386.34032 [9] Kwong, M. K.: On Lyapunov’s inequality for disfocality. J. math. Anal. appl. 83, 486-494 (1981) · Zbl 0504.34020 [10] Kwong, M. K.; Zettl, A.: Integral inequalities and second linear oscillation. J. differential equations 45, 16-33 (1982) · Zbl 0498.34022 [11] Li, H. J.: Oscillation criteria for second order linear differential equations. J. math. Anal. appl. 194, 217-234 (1995) · Zbl 0836.34033 [12] Philos, Ch.G.: Oscillation theorems for linear differential equations of second order. Arch. math. (Basel) 53, 483-492 (1989) · Zbl 0661.34030 [13] Wintner, A.: A criterion of oscillatory stability. Quart. appl. Math. 7, 115-117 (1949) · Zbl 0032.34801