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Zbl 0535.34023
Graef, John R.; Spikes, Paul W.
On the nonlinear limit-point/limit-circle problem.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 7, 851-871 (1983). ISSN 0362-546X

A perturbed second order nonlinear equation $(a(t)x')'+q(t)f(x)=r(t,x)$ is defined to be of the limit circle type if, for any solution x(t), either $\int\sp{\infty}x(u)f(x(u))du<\infty$ or $\int\sp{\infty}F(x(u))du<\infty$, where $F(v)=\int\sp{v}\sb{0}f(u)du$ (this is a generalization of {\it H. Weyl}'s [Math. Ann. 68, 220-269 (1910)] classification of second order linear differential equations $(a(t)x')'+q(t)x=0)$. The authors give sufficient conditions that such equations are of the limit circle type. Moreover, they discuss the relationships between the above property and the boundedness, oscillation and convergence to zero of the solution of the above equation.
[M.Boudourides]
MSC 2000:
*34C05 Qualitative theory of some special solutions of ODE
34A34 Nonlinear ODE and systems, general
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: limit cycle; limit circle; limit point; boundedness; oscillation

Cited in: Zbl 1215.34093 Zbl 1057.34023

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