A comparison theorem

Authors:
Walter Leighton and William Oo Kian Ke

Journal:
Proc. Amer. Math. Soc. **28** (1971), 185-188

MSC:
Primary 34.42

MathSciNet review:
0273121

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Abstract: In this paper the authors consider a pair of differential equations , where are positive and continuous, and where solutions and have common consecutive zeros at and . They show that if the curves and have a single intersection (possibly a closed subinterval) and if , the first conjugate point of ( and small) for the second equation precedes that of the first.

**[1]**Maxime Bôcher,*Leçons sur les méthodes de Sturm*, Gauthier-Villars, Paris, 1917.**[2]**A. M. Fink,*The functional 𝑇 ∫₀^{𝑇} 𝑅 and the zeroes of a second order linear differential equation*, J. Math. Pures Appl. (9)**45**(1966), 387–394. MR**0208053****[3]**A. M. Fink,*Comparison theorems for ∫ₐ^{𝑏}𝑝 with 𝑝 an admissible sub or superfunction*, J. Differential Equations**5**(1969), 49–54 (Russian). MR**0232992****[4]**Stanley B. Eliason,*The integral 𝑇∫^{𝑇/2}_{-𝑇/2}𝑝(𝑡)𝑑𝑡 and the boundary value problem 𝑥′′+𝑝(𝑡)𝑥=0,𝑥(-𝑇/2)=𝑥(𝑇/2)=0*, J. Differential Equations**4**(1968), 646–660. MR**0232990****[5]**Walter Leighton,*Some elementary Sturm theory*, J. Differential Equations**4**(1968), 187–193. MR**0224907****[6]**Walter Leighton,*Bounds for conjugate points*, J. Reine Angew. Math.**246**(1971), 126–135. MR**0280799**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0273121-6

Keywords:
Second-order linear differential equation,
conjugate point,
comparison theorem

Article copyright:
© Copyright 1971
American Mathematical Society