Limit-point type solutions of nonlinear differential equations

We are concerned with the nonexistence of L²-solutions of a nonlinear differential equation x″=a(t)x+f(t,x). By applying technique similar to that exploited by Hallam SIAM J. Appl. Math. 19 (1970) 430-439] for the study of asymptotic behavior of solutions of this equation, we establish nonexistence of solutions from the class L²(t₀,∞) under milder conditions on the function a(t) which, as the examples show, can be even square integrable. Therefore, the equation under consideration can be classified as of limit-point type at infinity in the sense of the definition introduced by Graef and Spikes Nonlinear Anal. 7 (1983) 851-871]. We compare our results to those reported in the literature and show how they can be extended to third order nonlinear differential equations.