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Asymptotics of Nonoscillatory Solutions of Some Second‐Order Linear Differential Equations
Author(s) -
Howard H.,
Marić V.
Publication year - 1994
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/26.4.373
Subject(s) - mathematics , order (exchange) , mathematical analysis , differential equation , comparison theorem , pure mathematics , mathematical physics , economics , finance
In this paper we prove a theorem on the existence and asymptotic behaviour of nonoscillatory solutions of the equationx ″ + p ( t ) x = 0 , where p ( t ) = (−λ 2 + h ( t )) t −2α with λ > 0 and 0 < α < 1. The coefficient p need not be onesigned. Examples show that the same asymptotic formula can hold either when p ( t ) is eventually negative, or when it oscillates. Moreover, the result can be applied to cases where p is negative, but is such that the classical Liouville‐Green approximation formula cannot be used.