Selfadjoint nonoscillatory second order linear 𝐵*-algebra differential equations

Abstract

ABSTRACT. The main result in this paper states that the second order linear B*-algebra differential equation {p(t)y') + q{t)y = 0, where p{t) is positive and q(t) is Hermitian for each t, is nonoscillatory on [io,co) if the scalar equation (IIP^Mir1^")' + ll<?(i)!|W = 0 is nonoscillatory on [to,oo).Consequently, every criterion on nonoscillation in the scalar case automatically produces another one in the B* -algebra case. Introduction.There is considerable literature concerning the oscillation of solutions of linear matrix differential equations (see [1,2,6, 7] and the references therein).Properties of determinants and the trace are used to obtain some of these results, thereby precluding a straightforward generalization to more general algebras. Hille's book [3] is devoted to a large extent to generalizing classical results to equations where the dependent variable takes values in a Banach algebra.In this paper, a method based on an integral inequality of the variational type is used to establish the main result (Theorem 3), which basically says that the oscillatory properties of second order linear B*-algebra differential equations are closely related to the oscillatory properties of a suitable selfadjoint linear scalar second order differential equation.Corollaries 1 and 2 show how to apply Theorems 2 and 3, respectively, to obtain nonoscillation criteria for second order linear ¿?*-algebra differential equations. Corollary 1, in particular, generalizes Reid's Theorem 5.1 in [6j.Theorem 4 was established in the matrix case by Barrett [1] and Reid [6].