The nonlinear limit-point/limit-circle problem for higher-order equations. (English) Zbl 0914.34023

Bartušek, Miroslav; Došlá, Zuzana; Graef, John R.

Arch. Math., Brno 34, No. 1, 13-22 (1998).

The authors consider \(n\)th-order nonlinear differential equations \[ y^{(n)}+r(t)f(y,y',\dots ,y^{(n-1)}) = 0, \tag{E} \] where \(r\in L_{loc}[0, \infty)\) does not change the sign on \([t_0, \infty), t_0\geq 0, f:\mathbb{R} ^n \to \mathbb{R}\) is continuous, and \[ x_1 f(x_1, \dots , x_n)\geq 0 \;\;\;\text{on } \;\;\mathbb{R}^n. \] Only those solutions are considered that are continuable to all of \(\mathbb{R}_+ = [0, \infty)\) and are not eventually identically zero.
The equation (E) is of nonlinear limit-circle type if every solution \(y\) satisfies \[ \int _0^\infty y(t)f(y(t),y'(t),\dots ,y^{(n-1)}(t))dt < \infty. \] If \[ \int _0^\infty y(t)f(y(t),y'(t),\dots ,y^{(n-1)}(t))dt = \infty \] for at least one solution \(y\) to (E), then (E) is of nonlinear limit-point type.
The authors describe what is known for the higher-order nonlinear limit-point/limit-circle problem and indicate a number of open questions for further research.

Reviewer: F.Neuman (Brno)

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

higher-order equations; nonlinear limit-point; nonlinear limit-circle