Lower bounds to the zeros of solutions of $y^{”}+ p(x)y=0$
Author:
A. S. Galbraith
Journal:
Proc. Amer. Math. Soc. 26 (1970), 111-116
MSC:
Primary 34.42
DOI:
https://doi.org/10.1090/S0002-9939-1970-0265679-7
MathSciNet review:
0265679
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Abstract: If $p(x)$ is nonnegative, monotonic and concave, no solution of $y'' + p(x)y = 0$ has more than $n + 1$ zeros in the interval $(a,b)$ defined by \[ (b - a)\int _a^b {p(x)dx = {n^2}{\pi ^2}.} \] This is proved by showing that, if $y’(a) = 0$, the $n$th succeeding zero of $y’(x)$ will not precede $b$.
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- Á. Elbert, On the zeros of the solutions of the differential equation $y^{\prime \prime }+q(x)y=0$, where $[q(x)]^{\nu }$ is concave, Studia Sci. Math. Hungar. 2 (1967), 293–298. MR 219808
- A. S. Galbraith, On the zeros of solutions of ordinary differential equations of the second order, Proc. Amer. Math. Soc. 17 (1966), 333–337. MR 190435, DOI https://doi.org/10.1090/S0002-9939-1966-0190435-7
- E. Makai, Über eine Eigenwertabschätzung bei gewissen homogenen linearen Differentialgleichungen zweiter Ordnung, Compositio Math. 6 (1939), 368–374 (German). MR 1557034 H. Prüfer, Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen, Math Ann. 95 (1925/26), 499-518.
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Keywords:
Linear differential equations,
lower bounds to zeros,
estimates of characteristic values,
number of zeros in an interval
Article copyright:
© Copyright 1970
American Mathematical Society