Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales

• Published in: Journal of difference equations and applications Vol. 7; no. 2; pp. 265 - 295
• Main Authors: Dořlý, Ondřrej, Hilscher, Roman
• Format: Journal Article
• Language: English
• Published: Gordon and Breach Science Publishers 01.01.2001
• Subjects: 1991 Mathematics Subject Classification: 34C10,39A 10,93C70; Disconjugacy; Focal point; Jacobi condition; Linear Hamiltonian system; Principal solution; Quadratic functional; Riccati equation; Symplectic system; Time scale
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236190108808273

In this paper we study qualitative properties of the socalled symplectic dynamic system (S) z δ A =Stz on an arbitrary time scale T, providing a unified theory for discrete symplectic systems and differential linear Hamiltonian systems . We define dis-conjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equival ent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also the solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales.