Asymptotic integration of nonoscillatory differential equations: a unified approach

  • S. Bodine
  • D. A. Lutz


We consider the equation [r(t)x′]′ + f(t)x = 0 as a perturbation of the equation [r(t)y′]′ + g(t)y = 0, where the latter is assumed to be nonoscillatory at infinity. The functions r and g are real-valued, r is positive, and f is complex-valued. The problem of the asymptotic integration of the perturbed equation in comparison with solutions of the unperturbed equation has been studied by many mathematicians, including Hartman and Wintner, Trench, ˇSimˇsa, Chen, and Chernyavskaya and Shuster. Here we apply a unified approach. Working in a matrix setting, we use preliminary and so-called conditioning transformations to bring the system in the form \( \vec{z} = \left[ {\Lambda (t) + R(t)} \right]\vec{z} \), where Λ is a certain diagonal matrix and R is an absolutely integrable perturbation. This allows us to use Levinson’s fundamental theorem to find the asymptotic behavior of solutions and, in addition, to estimate the error involved. This method allows us to derive these known results in a more unified setting and to weaken the hypotheses in some instances.

Key words and phrases

Differential equations second-order nonoscillatory perturbations asymptotic behavior 

2000 Mathematics Subject Classification



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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Puget SoundTacomaUSA
  2. 2.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA

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