Geodesic
Comparison theorems for the square integrability of solutions of (r(t)y')'+q(t)y=f(t, y)
Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 75-79

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Bellman [1], [2, p. 116] proved that, if all solutions of the equationare in L2, ∞) and b(t) is bounded, then all solutions ofare also in L2(a, ∞). The purpose of this paper is to present conditions on the function f that guarantee that all solutions ofbe in the class L2(a, ∞) whenever all solutions of the equationhave this property. It is assumed that r(t) >0, r and qare continuous on a half line (a, ∞) and f is continuous. Actually the continuity assumptions may be weakened to local integrability and L2 (a, ∞) may be replaced by Lp(a, ∞) for any p > 1. 
Bradley, John S. Comparison theorems for the square integrability of solutions of (r(t)y')'+q(t)y=f(t, y). Glasgow mathematical journal, Tome 13 (1972) no. 1, pp. 75-79. doi: 10.1017/S0017089500001415
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[1] 1.Bellman, R., A stability property of solutions of linear differential equations, Duke Math. J. 11 (1944), 513–516. Google Scholar | DOI

[2] 2.Bellman, R., Stability theory of differential equations (New York, 1953). Google Scholar

[3] 3.Gollwitzer, H. E., A note on a functional inequality, Proc. Amer. Math. Soc. 23 (1969), 642–647. Google Scholar | DOI

[4] 4.Halvorsen, S., On the quadratic integrability of solutions of d'x/dt2+f(t)x =0, Math. Scand. 14 (1964), 111–119. Google Scholar | DOI

[5] 5.Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (New York, 1955). Google Scholar

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