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

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


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
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