On the integrability of solutions of perturbed non-linear differential equations

Abstract

SynopsisSufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

Limit circle type results for sublinear equations

“…In order to see that condition (10) is sharp, it is convenient to consider a special case of (3), namely (12) jt" + ί σ jt γ = O. Now (10) implies that σ>l + l/w=l + 2/(γ + 1) which is in agreement with what is known from asymptotic integrations of equation (12) (see, for example, Bellman [2; p. 163]).…”

“…This of course reduces to the square integrability of solutions in the case of equation (1). While some authors have discussed the nonlinear limit point-limit circle problem (see [1,3,5,7,9,12,13,14,16]), the majority of the results obtained have been of the nonlinear limit point type for unforced equations. In fact only the papers of Graef [7] and Spikes [12,13] contain limit circle results for equation (2).…”

“…While some authors have discussed the nonlinear limit point-limit circle problem (see [1,3,5,7,9,12,13,14,16]), the majority of the results obtained have been of the nonlinear limit point type for unforced equations. In fact only the papers of Graef [7] and Spikes [12,13] contain limit circle results for equation (2). Moreover, in the papers written on the nonlinear limit point-limit circle problem to date, the case of f(x) being sublinear has either been ignored completely or explicitly excluded by hypothesis from consideration.…”