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On the asymptotic behaviour of solutions of x″ + a(t)f(x) = 0

Published online by Cambridge University Press:  24 October 2008

T. Burton  and
R. Grimmer
T. Burton
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
R. Grimmer
Affiliation:
Department of Mathematics, Southern Illinois University, Carbondale, Illinois 62901
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Extract

We consider the equation:

where a: [0, ∞) → R1, a(t) > 0, a'(t) is continuous,

f:( −∞, +∞) → R1, f is continuous, and xf(x) > 0 for x ≠ 0. The problem is to give conditions on a(t) and f(x) to ensure that all solutions of (1) tend to zero as t → ∞. First, however, we give some sufficient conditions and some necessary and sufficient conditions to ensure that all solutions of (1) are oscillatory or bounded.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 70 , Issue 1 , July 1971 , pp. 77 - 88
Copyright
Copyright © Cambridge Philosophical Society 1971

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References

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