Linear perturbations of a nonoscillatory second order differential equation II

Trench, William

doi:10.1090/S0002-9939-02-06682-0

Linear perturbations of a nonoscillatory second order differential equation II
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by William F. Trench PDF

Proc. Amer. Math. Soc. 131 (2003), 1415-1422 Request permission

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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