Vol. 90, No. 1, 1980

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Comparison and oscillation criteria for selfadjoint vector-matrix differential equations

Roger T. Lewis and Lynne C. Wright

Vol. 90 (1980), No. 1, 125–134


      ∑n     k      (k)   (k)
L(y) =   (− 1) (Pk(x)y  (x))

where the coefficients are real, continuous, m × m, symmetric matrices, y(x) is an m-dimensional vector-valued function, and Pn(x) is positive definite for all x I. We consider both the case for which the singularity is at , I = [1,), and the case for which the singularity is at 0, I = (0,1].

The main theorem is a comparison result in which the equation L(y) = 0 is compared with an associated scalar equation. Then, general theorems for the oscillation and nonoscillation of L(y) = 0 are presented which can be used when the comparison theorem does not apply. Some of the proofs indicate how scalar oscillation and nonoscillation criteria can be extended to the vector-matrix case when the associated scalar theorem has been proved using the quadratic functional criteria for oscillation. In general, proofs using the associated Riccati equation and other familiar methods do not extend as easily.

Mathematical Subject Classification 2000
Primary: 34C10
Received: 3 January 1979
Revised: 29 November 1979
Published: 1 September 1980
Roger T. Lewis
Lynne C. Wright