Vol. 72, No. 1, 1977

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A comparison theorem and oscillation criteria for second order differential systems

Garret J. Etgen and James Pawlowski

Vol. 72 (1977), No. 1, 59–69

Let be a Hilbert space and let = (,) be the B-algebra of bounded linear operators from to with the uniform operator topology. Let 𝒮 be the subset of consisting of the selfadjoint operators. This paper is concemed with second order, selfadjoint differential equations of the form

[P(x)Y′]′ + Q (x)Y = 0

on + = [0,), where P and Q are continuous mappings of + into 𝒮 with P(x) positive definite for all x ∈ℛ+. Let 𝒢 be the set of positive linear functionals on . Positive functionals are used in deriving a generalization of Sturm’s comparison theorem, and, in turn, the comparison theorem is used to obtain oscillation criteria for equation (1). These criteria are shown to include a large number of well-known oscillation criteria for (1) in the matrix and scalar case. Extensions of the results to nonlinear differential equations and differential inequalities are also discussed.

Mathematical Subject Classification 2000
Primary: 34C10
Received: 13 October 1976
Revised: 13 April 1977
Published: 1 September 1977
Garret J. Etgen
James Pawlowski