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.

16.—Square Integrable Solutions of Perturbed Linear Differential Equations

“…In case of a = 1, Zettl [9] showed that if all solutions of (1.3) are in L 2 (0, oo) and A(0e£°°(0, oo), then every solution of (1.4) belongs to L 2 (0, co). This generalises a similar result of Bradley [3] in the second-order case. Our main results are the following two theorems: THEOREM 1.…”

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

“…As a final example, we consider the perturbed equation (tx')' + t 10 x 3 = t 2 x cos x + x' (sin x')/t+t sin t. Observe that so we may take h(t) = t 2 , g(t) = l/t, and k(t) = t(t+l) in condition (3). By taking B = 2, c = 1/2, H{t) = t 1+ ", 0</?< 1, and replacing condition (10) with condition (4) and condition (16) with [fc(s)H(s)/(g(s)) 1/2 ]rfs<oo, as indicated in the remark at rith Jto the end of the proof of Lemma 3, we see that this example also satisfies the hypotheses of Theorem 7.…”

Immunological competence of patients in remission from acute leukaemia: apparently normal T cell function but defective pokeweed mitogen‐driven immunoglobulin synthesis

“…was involved. This group consisted of 25 males and 16 females, median age 11 (range [4][5][6][7][8][9][10][11][12][13][14], whereas the patient group consisted of 19 males and 14 females, median age 6 (range 2-16).…”