A comparison theorem
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- by Walter Leighton and William Oo Kian Ke PDF
- Proc. Amer. Math. Soc. 28 (1971), 185-188 Request permission
Abstract:
In this paper the authors consider a pair of differential equations ${y''_1} + {p_1}(x){y_1} = 0,{y''_2} + {p_2}(x){y_2} = 0$, where ${p_i}(x)$ are positive and continuous, and where solutions ${y_1}(x)$ and ${y_2}(x)$ have common consecutive zeros at $x = a$ and $x = b$. They show that if the curves $y = {p_1}(x)$ and $y = {p_2}(x)$ have a single intersection (possibly a closed subinterval) and if ${p_1}(a) > {p_2}(a),{p_2}(b) > {p_1}(b)$, the first conjugate point of $a + {\varepsilon }$ (${\varepsilon } > 0$ and small) for the second equation precedes that of the first.References
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Maxime BΓ΄cher, LeΓ§ons sur les mΓ©thodes de Sturm, Gauthier-Villars, Paris, 1917.
- A. M. Fink, The functional $T$ $\int _{0}^{T}$ $R$ and the zeroes of a second order linear differential equation, J. Math. Pures Appl. (9) 45 (1966), 387β394. MR 208053
- A. M. Fink, Comparison theorems for $\int _{a}^{b}\,p$ with $p$ an admissible sub or superfunction, J. Differential Equations 5 (1969), 49β54 (Russian). MR 232992, DOI 10.1016/0022-0396(69)90103-X
- Stanley B. Eliason, The integral $T\,\int ^{T/2}_{-T/2}\,p(t)\,dt$ and the boundary value problem $x^{\prime \prime }+p(t)x=0,\,x(-T/2)=x(T/2)=0$, J. Differential Equations 4 (1968), 646β660. MR 232990, DOI 10.1016/0022-0396(68)90014-4
- Walter Leighton, Some elementary Sturm theory, J. Differential Equations 4 (1968), 187β193. MR 224907, DOI 10.1016/0022-0396(68)90035-1
- Walter Leighton, Bounds for conjugate points, J. Reine Angew. Math. 246 (1971), 126β135. MR 280799, DOI 10.1515/crll.1971.246.126
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 28 (1971), 185-188
- MSC: Primary 34.42
- DOI: https://doi.org/10.1090/S0002-9939-1971-0273121-6
- MathSciNet review: 0273121