The author discusses linear Hamiltonian difference systems
where , , , , , are -matrices, , symmetric, such that exist. For the controllable system (H) the extended Reid Roundabout Theorem is proved; that is equivalence of: a) positivity of some quadratic functional, b) disconjugacy, c) absence of focal points in the principal solution, d) Riccati condition. A particular case of this result, with boundary conditions , is considered separately.
To get the main result, a discrete version of Picone’s identity is proved, also several definitions like focal points or generalized zeros of vector-valued functions are introduced. Without assumption of nonsingularity of the matrix the presented theory includes discrete Sturm-Liouville equations of higher order. Various interconnections inside the theory and relations with earlier results are widely discussed. See also e.g. C. D. Ahlbrandt [J. Math. Anal. Appl. 180, No. 2, 498-517 (1993; Zbl 0802.39005)], L. H. Erbe and P. Yan [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)].