Comparative index for solutions of symplectic difference systems

Abstract

We introduce the comparative index of two conjoined bases of a symplectic difference system, which generalizes difference analogs of canonical systems of differential equations. We consider the main properties of the comparative index and its relation to the number of focal points of a conjoined basis of the symplectic system. We prove a formula relating the number of focal points (including multiplicities) of two bases in the interval (i, i + 1] and corollaries of this formula such as an estimate for the difference of the numbers of focal points of two conjoined bases in the interval (0, N + 1], the equality of the numbers of focal points of principal solutions for the primal and reciprocal systems, sufficient conditions for the solvability of the Riccati equation for a disconjugate symplectic system, etc.