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

Abstract

Let B be a Banach lattice with order continuous norm, t(B) the algebra of bounded linear operators. Let B + denote the positive cone induced by the lattice structure of B, and +(#) the corresponding positive cone in (/?). We consider second-order operator-valued differential equations of the form " + Q(x)Y = 0, where Q: [ , + oo) -> t(B) is continuous in the uniform topology and is such that / x Q{) d E + (B) for all x > a. Comparison theorems of Hille-Wintner type are obtained.