On the nonlinear limit-point/limit-circle problem. (English) Zbl 0535.34023

Graef, John R.; Spikes, Paul W.

Nonlinear Anal., Theory Methods Appl. 7, 851-871 (1983).

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 \(\int^{\infty}x(u)f(x(u))du<\infty\) or \(\int^{\infty}F(x(u))du<\infty\), where \(F(v)=\int^{v}_{0}f(u)du\) (this is a generalization of 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.

Reviewer: M.Boudourides

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
34C11	Growth and boundedness of solutions to ordinary differential equations

Keywords:

limit cycle; limit circle; limit point; boundedness; oscillation

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