Abstract
We are concerned with the nonexistence of L 2 -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 L 2 ( t 0 ,∞) 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.
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