Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions
The author discusses linear Hamiltonian difference systems \[ \Delta x_k = A_k x_{k + 1} + B_k u_k,\quad \Delta u_k = C_k x_{k + 1} - A^T_k u_k, \quad k \in J : = \{0, 1, \dots, N\}, \tag{H} \] where \(x_k\), \(u_k \in \mathbb{R}^n\), \(k \in \overline J : = J \cup \{N + 1\}\), \(A_k\), \(B_k\), \(C_k\) are \(n \times n\)-matrices, \(B_k\), \(C_k\) symmetric, \(A_k\) such that \(\widetilde A_k = (I - A_k)^{-1}\) 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 \(x_0 = x_N = 0\), 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 \(B_k\) 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. \textit{C. D. Ahlbrandt} [J. Math. Anal. Appl. 180, No. 2, 498-517 (1993; Zbl 0802.39005)], \textit{L. H. Erbe} and \textit{P. Yan} [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)].
scientific article; zbMATH DE number 902576