Comparison and oscillation criteria for selfadjoint vector-matrix differential equations
Roger T. Lewis, Lynne C. Wright · Pacific Journal of Mathematics · 1980
L(y) = Σ (-l)*C fc0 where the coefficients are real, continuous, mxm, symmetric matrices, y(x) is an ^-dimensional vector-valued function, and P n (x) is positive definite for all xel.We consider both the case for which the singularity is at oo, /=[l,oo), and the case for which the singularity is at 0, / = (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.