Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-Type Conditions

✍ Scribed by Martin Bohner

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 229 KB
• Volume: 199
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1996.0177

✦ Synopsis

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.