On the location of zeros of second-order differential equations
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- by Vadim Komkov
- Proc. Amer. Math. Soc. 35 (1972), 217-222
- DOI: https://doi.org/10.1090/S0002-9939-1972-0298128-5
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Abstract:
The paper considers the location of zeros of the equation $(\alpha (t)x’)’ + \gamma (t)x = 0,t \in [{t_0},{t_1}]$. The following theorem is proved. Let $[a,a + T],T = na$ (n a positive integer), be a subset of $[{t_0},{t_1}]$. Denote $\omega = \pi /T$. Let the coefficient functions obey the inequality $\smallint _a^{a + T}\{ \gamma (t) - {\omega ^2}\alpha (t){\sin ^2}(\omega t)\} dt > {\omega ^2}\smallint _a^{a + T}\{ \alpha \cos 2\omega t\} dt$. Then every solution of this equation will have a zero on $[a,a + T]$. A more general form of this theorem is also proved.References
- Walter Leighton, Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603–610. MR 140759, DOI 10.1090/S0002-9939-1962-0140759-0
- C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, Vol. 48, Academic Press, New York-London, 1968. MR 0463570
- M. I. El′šin, Qualitative problems on the linear differential equations of the second order, Doklady Akad. Nauk SSSR (N.S.) 68 (1949), 221–224 (Russian). MR 0031161
- Vadim Komkov, A generalization of Leighton’s variational theorem, Applicable Anal. 2 (1972/73), 377–383. MR 414994, DOI 10.1080/00036817208839051
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 217-222
- MSC: Primary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-1972-0298128-5
- MathSciNet review: 0298128