Sturmian theory for a class of nonselfadjoint differential systems

Abstract

This paper investigates certain Sturmian properties of the differential system Y″+ P(t) Y=0, where Y is a real n-dimensional vector and P is a real continuous ipossibily nonsymmetric) n × n matrix which is positive with respect to a solid cone K in Rn. The presence or absence of focal points and conjugate points on a given interval is characterized in terms of integral inequalities on a Banach lattice. These inequalities are then used to give specific criteria for the presence or absence of a focal point or a conjugate point on a half-line [α, ∞). An example is given to demonstrate that in general the Sturmian properties of the above equation depend on whether P is assumed symmetric or nonsymmetric.