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Linear perturbations of a nonoscillatory second order equation. (English) Zbl 0594.34057

Summary: It is shown that the equation \((r(t)x')'+g(t)x=0\) has solutions which behave asymptotically like those of a nonoscillatory equation \((r(t)y')'+f(t)y=0,\) provided that a certain integral involving f-g converges (perhaps conditionally) and satisfies a second condition which has to do with its order of convergence. The result improves upon a theorem of Hartman and Wintner.

MSC:

34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
34C99 Qualitative theory for ordinary differential equations
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References:

[1] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. · Zbl 0123.21502
[2] William F. Trench, Functional perturbations of second order differential equations, SIAM J. Math. Anal. 16 (1985), no. 4, 741 – 756. · Zbl 0577.34060 · doi:10.1137/0516056
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