Comparison theorem for conjugate points of systems of 𝑛th order nonselfadjoint differential equations

Abstract

A comparison theorem for conjugate points of the two systems of linear differential equations x ( n ) − ( − 1 ) n − k p ( t ) x = 0 {x^{(n)}} - {( - 1)^{n - k}}p(t)x = 0 and y ( n ) − ( − 1 ) n − k q ( t ) y = 0 {y^{(n)}} - {( - 1)^{n - k}}q(t)y = 0 , where p ( t ) p(t) and q ( t ) q(t) are m × m m \times m matrices of continuous functions, is given. It is assumed that q ( t ) q(t) is positive with respect to a certain cone but no positivity conditions of any kind are imposed on p ( t ) p(t) . No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.