Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-Type Conditions

Abstract

We consider a linear Hamiltonian Difference System for the so-called singular case so that discrete Sturm–Liouville Equations of higher order are included in our theory. We introduce the concepts of focal points for matrix-valued and generalized zeros for vector-valued solutions of the system and define disconjugacy for linear Hamiltonian Difference Systems. We prove a Reid Roundabout Theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional, among them the strengthened Jacobi's Condition and a condition on a certain Riccati Difference Equation. The key to this theorem is a discrete version of Picone's Identity. Furthermore, for the sake of generalization of our theorem, we introduce controllability for linear Hamiltonian Difference Systems and prove a Reid Roundabout Theorem for a more general functional and more general boundary conditions.