Limit circle type results for sublinear equations (Q791014)

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 \textit{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.