Limit circle type results for sublinear equations. (English) Zbl 0535.34024

Graef, John R.

Pac. J. Math. 104, 85-94 (1983).

The author considers forced second order nonlinear equations of the type \((a(t)x')'+q(t)f(x)=r(t)\) and calls them of nonlinear limit circle type if every solution x(t) has \(\int^{\infty}_{t_ 0}x(u)f(x(u))du<\infty\) and of nonlinear limit point type otherwise (this definition generalizes H. Weyl’s [Math. Ann. 68, 220-269 (1910)] classification of second order linear differential equations \((a(t)x')'+q(t)x=0)\). The author considers the sublinear case \(f(x)=x^{\gamma}\), \(0<\gamma \leq 1\). Necessary and sufficient conditions are found that such a forced or unforced \((r=0)\) equation is of nonlinear limit circle type and also sufficient conditions that it is of nonlinear limit point type.

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
34A30	Linear ordinary differential equations and systems

Keywords:

limit cycle; limit circle; second order linear differential equations; nonlinear limit point