Maximum and anti-maximum principles for singular Sturm–Liouville problems

Abstract

The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problemswith boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.

On maximum principles in the class of oscillating functions

“…Our approach is related to the paper [4], where some maximum principles for solutions of the Sturm-Liouville problem −(r(x)u ′ ) ′ + p(x)u = λm(x)u + h(x) with h = 0 almost everywhere were obtained. We assume h ≡ 0 and our techniques and results are different.…”

Strongly Singular Sturm - Liouville Problems

“…we have considered in [6] boundary and eigenvalue problems of the form Lu = λm(t)u , u(a) = 0 , lim t→b u(t) exists (finite) .…”

“…The results obtained in [6] have been extended in [8] to some situations where p is not necessarily positive. The results obtained in [6] have been extended in [8] to some situations where p is not necessarily positive.…”

“…In the present paper, using the methods of [6], we investigate the case when 0 ≤ p ∈ A; this case may be called strongly singular. More precisely, we shall assume that, for example, 1/r ∈ L 1 [a) , 1/r ∈ L 1 (b] , p ≥ 0 , pr 1 ∈ L 1 [a) and pr 1 …”

“…As for the singular case studied in [6], we shall also give a (global) maximum principle and a (local) anti -maximum principle for the strongly singular case. We shall also show that under some assumptions on m stronger than m ∈ A ∩ L 1 (a, b) , the eigenvalue problem where H is a suitable Hilbert space of functions.…”