Schur type comparison theorems for affine curves

Hille–Wintner type comparison theorems are developed in which two second order self-adjoint linear systems are compared. In particular, classes of $n \times n$ Hermitian matrix functions are defined, using Opial-type inequalities, which can be used for comparison purposes in the Hille theorems. Additional theorems are presented in which a system and a scalar differential equation are compared.

In this paper, we present a LEIGHTON type oscillation criterion and a STURM type comparison theorem. These results are obtained by using a generalized RICCATI transformation and can be applied in some cases in which classical oscillation results are not applicable.