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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear perturbations of a nonoscillatory second order differential equation II


Author: William F. Trench
Journal: Proc. Amer. Math. Soc. 131 (2003), 1415-1422
MSC (2000): Primary 34A30
Published electronically: September 5, 2002
MathSciNet review: 1949871
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Abstract: Let $y_1$ and $y_2$ be principal and nonprincipal solutions of the nonoscillatory differential equation $(r(t)y')'+f(t)y=0$. In an earlier paper we showed that if $\int^\infty(f-g)y_1y_2\,dt$ converges (perhaps conditionally), and a related improper integral converges absolutely and sufficently rapidly, then the differential equation $(r(t)x')'+g(t)x=0$ has solutions $x_1$ and $x_2$ that behave asymptotically like $y_1$ and $y_2$. Here we consider the case where $\int^\infty(f-g)y_2^2\,dt$ converges (perhaps conditionally) without any additional assumption requiring absolute convergence.


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Additional Information

William F. Trench
Affiliation: 95 Pine Lane, Woodland Park, Colorado 80863
Email: wtrench@trinity.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06682-0
PII: S 0002-9939(02)06682-0
Keywords: Asymptotic, nonoscillatory, principal, nonprincipal
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: December 6, 2001
Published electronically: September 5, 2002
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society