Explicit conditions for the factorization of $n$th order linear differential operators
HTML articles powered by AMS MathViewer
- by Anton Zettl PDF
- Proc. Amer. Math. Soc. 41 (1973), 137-145 Request permission
Abstract:
For any integer $k$ with $1 \leqq k \leqq n$ sufficient conditions on the coefficients ${p_i}$, are given for the factorization of certain classes of operators $Ly = {p_n}{y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y$ into a product $L = PQ$ where $P$ and $Q$ are operators of the same type of orders $n - k$ and $k$, respectively. A special case yields that if ${( - 1)^k}{p_0} \geqq 0$ then ${y^n} + {p_0}y$ is factorable into a product of two regular differential operators of orders $n - k$ and $k$.References
- Philip Hartman, Principal solutions of disconjugate $n-\textrm {th}$ order linear differential equations, Amer. J. Math. 91 (1969), 306–362. MR 247181, DOI 10.2307/2373512
- Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
- Philip Hartman, On disconjugacy criteria, Proc. Amer. Math. Soc. 24 (1970), 374–381. MR 251304, DOI 10.1090/S0002-9939-1970-0251304-8
- Erhard Heinz, Halbbeschränktheit gewöhnlicher Differentialoperatoren höherer Ordnung, Math. Ann. 135 (1958), 1–49 (German). Vorwort von B. L. van der Waerden. MR 101941, DOI 10.1007/BF01350826
- W. J. Kim, Oscillatory properties of linear third-order differential equations, Proc. Amer. Math. Soc. 26 (1970), 286–293. MR 264162, DOI 10.1090/S0002-9939-1970-0264162-2
- A. Ju. Levin, The non-oscillation of solutions of the equation $x^{(n)}+p_{1}(t)x^{(n-1)}+\cdots +p_{n} (t)x=0$, Uspehi Mat. Nauk 24 (1969), no. 2 (146), 43–96 (Russian). MR 0254328
- A. Ju. Levin, Some questions on the oscillation of solutions of linear differential equations, Dokl. Akad. Nauk SSSR 148 (1963), 512–515 (Russian). MR 0146450
- A. Ju. Levin, On the distribution of zeros of solutions of a linear differential equation, Dokl. Akad. Nauk SSSR 156 (1964), 1281–1284 (Russian). MR 0164079
- J. Mikusiński, Sur l’équation $x^{(n)}+A(t)x=0$, Ann. Polon. Math. 1 (1955), 207–221 (French). MR 86201, DOI 10.4064/ap-1-2-207-221
- Kenneth S. Miller, Linear differential equations in the real domain, W. W. Norton & Co. Inc., New York, 1963. MR 0156014
- Zeev Nehari, Disconjugate linear differential operators, Trans. Amer. Math. Soc. 129 (1967), 500–516. MR 219781, DOI 10.1090/S0002-9947-1967-0219781-0
- 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, DOI 10.1090/S0002-9947-1922-1501228-5
- Robert Ristroph, Pólya’s property $W$ and factorization—A short proof, Proc. Amer. Math. Soc. 31 (1972), 631–632. MR 288338, DOI 10.1090/S0002-9939-1972-0288338-5
- D. Willett, Generalized de la Vallée Poussin disconjugacy tests for linear differential equations, Canad. Math. Bull. 14 (1971), 419–428. MR 348189, DOI 10.4153/CMB-1971-073-3
- D. Willett, Disconjugacy tests for singular linear differential equations, SIAM J. Math. Anal. 2 (1971), 536–545. MR 304772, DOI 10.1137/0502055
- Anton Zettl, Factorization and disconjugacy of third order differential equations, Proc. Amer. Math. Soc. 31 (1972), 203–208. MR 296421, DOI 10.1090/S0002-9939-1972-0296421-3
- Anton Zettl, Factorization of differential operators, Proc. Amer. Math. Soc. 27 (1971), 425–426. MR 273085, DOI 10.1090/S0002-9939-1971-0273085-5
Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 41 (1973), 137-145
- MSC: Primary 34A30
- DOI: https://doi.org/10.1090/S0002-9939-1973-0320413-X
- MathSciNet review: 0320413