Conjugate point properties for an even order linear differential equation

Author:
George W. Johnson

Journal:
Proc. Amer. Math. Soc. **45** (1974), 371-376

MSC:
Primary 34C10

DOI:
https://doi.org/10.1090/S0002-9939-1974-0348186-6

MathSciNet review:
0348186

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Abstract: Consider an even order equation , where is defined recursively by , where is continuous on , and for . The th conjugate point function is defined and is shown to be a strictly increasing continuous function with domain or . Extremal solutions are defined as nontrivial solutions with exactly zeros on , and are shown to exist, to have odd order zeros at and and exactly odd order zeros in , thus establishing that if .

**[1]**J. H. Barrett,*Oscillation theory of ordinary linear differential equations*, Advances in Math. 3 (1969), 415-509. MR**41**#2113. MR**0257462 (41:2113)****[2]**D. B. Hinton,*Disconjugate properties of a system of differential equations*, J. Differential Equations**2**(1966), 420-437. MR**34**#7856. MR**0208046 (34:7856)****[3]**W. Leighton and Z. Nehari,*On the oscillation of solutions of self-adjoint linear differential equations of the fourth order*, Trans. Amer. Math. Soc.**89**(1958), 325-377. MR**21**#1429. MR**0102639 (21:1429)****[4]**L. S. Nicholson,*Disconjugate systems of linear differential equations*, J. Differential Equations**7**(1970), 570-583. MR**0262601 (41:7207)****[5]**A. C. Peterson,*Distribution of zeros of solutions of a fourth order differential equation*, Pacific J. Math.**30**(1969), 751-764. MR**40**#5975. MR**0252758 (40:5975)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0348186-6

Keywords:
Extremal solution,
conjugate point,
zero of multiplicity

Article copyright:
© Copyright 1974
American Mathematical Society