Comparison theorems for the square integrability of solutions of (r(t)y')'+q(t)y=f(t, y)

Abstract

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.