## 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.

### MSC:

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

### Citations:

Zbl 0661.34030; Zbl 0836.34033; Zbl 0408.34031
Full Text:

### 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) · JFM 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), Birkhäuser: Birkhäuser Basel · Zbl 0125.32102 [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
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