Canonical forms and principal systems for general disconjugate equations

Author:
William F. Trench

Journal:
Trans. Amer. Math. Soc. **189** (1974), 319-327

MSC:
Primary 34C10

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330632-X

MathSciNet review:
0330632

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Abstract: It is shown that the disconjugate equation (1) , *a* , where , and (2) , can be written in essentially unique canonical forms so that for . From this it follows easily that (1) has solutions which are positive in (*a, b*) near and satisfy for . Necessary and sufficient conditions are given for (1) to have solutions such that for . Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition .

**[1]**W. A. Coppel,*Disconjugacy*, Lecture Notes in Math., vol. 220, Springer-Verlag, Berlin, 1971. MR**0460785 (57:778)****[2]**P. Hartman,*Disconjugate nth order differential equations and principal solutions*, Bull. Amer. Math. Soc.**74**(1968), 125-129. MR**36**#5440. MR**0222388 (36:5440)****[3]**-,*Principal solutions of disconjugate n-th order linear differential equations*, Amer. J. Math.**91**(1969), 306-362. MR**40**#450. MR**0247181 (40:450)****[4]**-,*Corrigendum and addendum*:*Principal solutions of disconjugate n-th order linear differential equations*, Amer. J. Math.**93**(1971), 439-451. MR**45**#648. MR**0291557 (45:648)****[5]**A. Ju. Levin,*Non-oscillation of solutions of the equation*, Uspehi Mat. Nauk**24**(1969), no. 2 (146), 43-96 = Russian Math. Surveys**24**(1969), no. 2, 43-99. MR**40**#7537. MR**0254328 (40:7537)****[6]**G. Polya,*On the mean-value theorem corresponding to a given linear homogeneous differential equation*, Trans. Amer. Math. Soc.**24**(1924), 312-324. MR**1501228****[7]**D. Willett,*Asymptotic behaviour of disconjugate nth order differential equations*, Canad. J. Math.**23**(1971), 293-314. MR**45**#2275. MR**0293196 (45:2275)****[8]**-,*Disconjugacy tests for singular linear differential equations*, SIAM J. Math. Anal.**2**(1971), 536-545. MR**0304772 (46:3904)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330632-X

Keywords:
Disconjugacy,
principal systems

Article copyright:
© Copyright 1974
American Mathematical Society