Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-type Conditions
Aron, Richard M. and Chen, Goong and Krantz, Steven G.
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.
M. Bohner, "Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-type Conditions," Journal of Mathematical Analysis and Applications, Elsevier, Jan 1996.
The definitive version is available at https://doi.org/10.1006/jmaa.1996.0177
Mathematics and Statistics
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© 1996 Elsevier, All rights reserved.