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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotic integrations of nonoscillatory second order differential equations

Author: Shao Zhu Chen
Journal: Trans. Amer. Math. Soc. 327 (1991), 853-865
MSC: Primary 34E10; Secondary 34C99
MathSciNet review: 1028756
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Abstract: The linear differential equation (1) $ (r(t)x^{\prime})^{\prime} + (f(t) + q(t))x= 0$ is viewed as a perturbation of the equation (2) $ (r(t)y^{\prime})^{\prime} + (f(t)y = 0$, where $ r > 0$, $ f$ and $ q$ are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and $ {y_1}$, $ {y_2}$ are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution $ {x_1}$ of (1). Under some mild assumptions, we characterize that (1) has a principal solution $ {x_1}$ satisfying $ {x_1}= {y_1}(1 + o(1))$. Sufficient (sometimes necessary) conditions under which the nonprincipal solution $ {x_2}$ of (1) behaves, in three different degrees, like $ {y_2}$ as $ t \to \infty $ are also established.

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Keywords: Nonoscillatory equations, linear perturbations, asymptotic integrations, Riccati techniques
Article copyright: © Copyright 1991 American Mathematical Society