Factorization and disconjugacy of third order differential equations

Author:
Anton Zettl

Journal:
Proc. Amer. Math. Soc. **31** (1972), 203-208

MSC:
Primary 34C10

MathSciNet review:
0296421

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Abstract: Sufficient conditions for the factorization of into a product of first order operators as well as into a product of a first order and a second order operator are given. Factorization into a product of first order factors is known to be equivalent to disconjugacy. These conditions are simple inequalities involving the coefficients.

**[1]**N. V. Azbelev and Z. B. Calyuk,*On the question of the distribution of the zeros of solutions of a third-order linear differential equation*, Mat. Sb. (N.S.)**51 (93)**(1960), 475–486 (Russian). MR**0121529****[2]**John H. Barrett,*Third-order differential equations with nonnegative coefficients*, J. Math. Anal. Appl.**24**(1968), 212–224. MR**0232039****[3]**John H. Barrett,*Oscillation theory of ordinary linear differential equations*, Advances in Math.**3**(1969), 415–509. MR**0257462****[4]**J. M. Dolan,*Oscillatory behavior of solutions of linear ordinary differential equations of third order*, Doctoral Dissertation, University of Tennessee, Knoxville, Tenn., 1967 (unpublished).**[5]**A. M. Fink,*An extension of Polya’s theorem*, J. Math. Anal. Appl.**23**(1968), 625–627. MR**0229904****[6]**Maurice Hanan,*Oscillation criteria for third-order linear differential equations.*, Pacific J. Math.**11**(1961), 919–944. MR**0145160****[7]**Philip Hartman,*Principal solutions of disconjugate 𝑛-𝑡ℎ order linear differential equations*, Amer. J. Math.**91**(1969), 306–362. MR**0247181****[8]**Philip Hartman,*On disconjugacy criteria*, Proc. Amer. Math. Soc.**24**(1970), 374–381. MR**0251304**, 10.1090/S0002-9939-1970-0251304-8**[9]**Lloyd K. Jackson,*Disconjugacy conditions for linear third-order differential equations*, J. Differential Equations**4**(1968), 369–372. MR**0226112****[10]**W. J. Kim,*Oscillatory properties of linear third-order differential equations.*, Proc. Amer. Math. Soc.**26**(1970), 286–293. MR**0264162**, 10.1090/S0002-9939-1970-0264162-2**[11]**A. C. Lazer,*The behavior of solutions of the differential equation 𝑦”’+𝑝(𝑥)𝑦′+𝑞(𝑥)𝑦=0*, Pacific J. Math.**17**(1966), 435–466. MR**0193332****[12]**A. Ju. Levin,*The non-oscillation of solutions of the equation 𝑥⁽ⁿ⁾+𝑝₁(𝑡)𝑥⁽ⁿ⁻¹⁾+\cdots+𝑝_{𝑛}(𝑡)𝑥=0*, Uspehi Mat. Nauk**24**(1969), no. 2 (146), 43–96 (Russian). MR**0254328****[13]**Kenneth S. Miller,*Linear differential equations in the real domain*, W. W. Norton & Co., Inc., New York, 1963. MR**0156014****[14]**Zeev Nehari,*Disconjugacy criteria for linear differential equations*, J. Differential Equations**4**(1968), 604–611. MR**0233006****[15]**G. Pólya,*On the mean-value theorem corresponding to a given linear homogeneous differential equation*, Trans. Amer. Math. Soc.**24**(1922), no. 4, 312–324. MR**1501228**, 10.1090/S0002-9947-1922-1501228-5**[16]**E. Willett,*Asymptotic behavior of disconjugate th order differential equations*, Canad. J. Math.**23**(1971).**[17]**D. Willett,*Generalized de la Vallée Poussin disconjugacy tests for linear differential equations*, Canad. Math. Bull.**14**(1971), 419–428. MR**0348189****[18]**-,*Disconjugacy tests for singular linear differential equations*(preprint).**[19]**Anton Zettl,*Factorization of differential operators*, Proc. Amer. Math. Soc.**27**(1971), 425–426. MR**0273085**, 10.1090/S0002-9939-1971-0273085-5

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DOI:
https://doi.org/10.1090/S0002-9939-1972-0296421-3

Keywords:
Ordinary differential equations,
disconjugacy criteria,
factorization of differential operators,
Frobenius factorization,
Pólya's property ``W",
Wronskians,
oscillation or nonoscillation of solutions,
zeros of solutions,
third order linear homogeneous differential equations,
Tchebycheff property of solutions

Article copyright:
© Copyright 1972
American Mathematical Society