Estimation of the solutions of the Sturm-Liouville equation

Abstract

Exact estimates are presented for the solutions of the problem $\ddot y + \lambda ^2 p(t)y = 0, y(0) = 0, \dot y(0) = 1$ with $p(t)$ satisfying one of the following conditions: $$(i) |p(t)| \leqslant M< \infty ; (ii) 0< \omega _1 \leqslant p(t) \leqslant \omega _2< \infty ; (iii) \mathop {sup}\limits_x \int_x^{x + T} {p(t)dt = P_T /T.}$$ The extremal solutions are found.