On \(n\)th order differential equations over Hardy fields. (English) Zbl 0822.34033

Ramayyan, A.

Kybernetika 30, No. 4, 461-470 (1994).

A Hardy field is a set of germs of real valued functions on positive half lines in \(\mathbb{R}\) that is closed under differentiation and forms a field with respect to the addition and multiplication. The author’s results concern \(n\)th order differential equations over Hardy fields; for example, there is given a necessary and sufficient condition for a nonhomogeneous \(n\)th order linear ordinary differential equation over a perfect Hardy field to be nonoscillatory.

Reviewer: M.-C.Anisiu (Cluj-Napoca)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
26A12	Rate of growth of functions, orders of infinity, slowly varying functions

Keywords:

linear ordinary differential equation; Hardy field; nonhomogeneous \(n\)th order; nonoscillatory

References:

[1]	M. Boshernitzan: An extension of Hardy’s class of orders of infinity. J. Analyse Math. 39 (1981), 235-255. · Zbl 0539.26002 · doi:10.1007/BF02803337
[2]	M. Boshernitzan: New orders of finity. J. Analyse Math. 41 (1982), 130-167. · Zbl 0539.26003 · doi:10.1007/BF02803397
[3]	M. Boshernitzan: Second order differential equations over Hardy fields. J. London Math. Soc. 35 (1987), 2, 109-120. · Zbl 0616.26002 · doi:10.1112/jlms/s2-35.1.109
[4]	M. Boshernitzan: Orders of infinity generated by difference equations. Amer. J. Math. 106 (1984), 1067-1089. · Zbl 0602.26002 · doi:10.2307/2374273
[5]	M. Boshernitzan: Hardy fields and existence of trans exponential functions. Acquationes Math. 30 (1986), 258-280. · Zbl 0593.26003 · doi:10.1007/BF02189932
[6]	E. A. Coddington: An Introduction to Ordinary Differential Equations. Prentice Hall of India Private Ltd., New Delhi 1974.
[7]	G. H. Hardy: Orders of Infinity. Cambridge University Press, London 1954.
[8]	A. R. Forsyth: A Treatise on Differential Equations. Sixth edition. Mac Millan and Co. Ltd., New York 1954.
[9]	V. Maric: Asymptotic behaviour of solutions of a nonlinear differential equation of the first order. J. Math. Anal. Appl. 38 (1972), 187-192.
[10]	M. Rosenlicht: Hardy fields. J. Math. Anal. Appl. 93 (1983), 297-311. · Zbl 0518.12014 · doi:10.1016/0022-247X(83)90175-0
[11]	M. Rosenlicht: The rank of a Hardy field. Trans. Amer. Math. Soc. 280 (1983), 659-671. · Zbl 0536.12015 · doi:10.2307/1999639
[12]	M. Rosenlicht: Rank change on adjoint real powers to Hardy fields. Trans. Amer. Math. Soc. 284 (1984), 829-836. · Zbl 0544.34052 · doi:10.1090/S0002-9947-1984-0743747-5
[13]	M. Rosenlicht: Growth properties of functions in Hardy fields. Trans. Amer. Math. Soc. 299 (1987), 261-272. · Zbl 0619.34057 · doi:10.2307/2000493
[14]	M. Rosenlicht: Asymptotic solutions of \(y''=F(x)y\). · Zbl 0824.34068 · doi:10.1006/jmaa.1995.1042