An oscillation criterion for discrete trigonometric systems

Abstract

In this paper, we investigate oscillation properties of discrete trigonometric systems whose coefficients matrices are simultaneously symplectic and orthogonal. The main result generalizes a necessary and sufficient condition of non-oscillation of trigonometric systems proved by M. Bohner and O. Došlý (J. Differential Equations 163 (2000), pp. 113–129) in the case when the block in the upper right corner of the coefficient matrix is symmetric and positive definite. Now, we present this oscillation criterion for an arbitrary trigonometric system. The obtained results are applied to formulate a necessary and sufficient condition for non-oscillation of even-order Sturm–Liouville difference equations.

Keywords::

• trigonometric difference systems
• focal points
• symplectic SVD
• oscillation criteria
• comparative index

Keywords::

• 39A12
• 39A21
• 39A22

Acknowledgements

The authors are grateful to professor Hongguo Xu from Department of Mathematics, University of Kansas for the productive discussion on representations (2.8), (2.10), (2.11). The second author thanks to the Masaryk University of Brno for the hospitality provided when conducting a substantial part of this project.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the grant 201/11/0768 of the Czech Grant Agency and by Federal Programme of Ministry of Education and Science of the Russian Federation in the framework of the state order in the scope of scientific activity [grant number 2014/105, project 1441].