Comparison theorems for second-order operator-valued linear differential equations

Geoffrey J. Butler and Lynn Harry Erbe

Vol. 112 (1984), No. 1, 21–34

DOI: 10.2140/pjm.1984.112.21

Abstract

Let B be a Banach lattice with order continuous norm, L(B) the algebra of bounded linear operators. Let B₊ denote the positive cone induced by the lattice structure of B, and L₊(B) the corresponding positive cone in L(B). We consider second-order operator-valued differential equations of the form Y ′′ + Q(x)Y = 0, where Q : [a,+∞) → L(B) is continuous in the uniform topology and is such that ∫ _x^∞Q(t)dt ∈ L₊(B) for all x ≥ a. Comparison theorems of Hille-Wintner type are obtained.

Mathematical Subject Classification 2000

Primary: 34G10

Milestones

Received: 13 January 1981

Revised: 11 June 1982

Published: 1 May 1984