Transformations and the number of focal points for conjoined bases of symplectic difference systems

Abstract

This paper studies symplectic transformations for conjoined bases of the symplectic difference systems

where the matrix W _i is symplectic for We derive new relations between the number of focal points of Y _i and P _i Y _i , where P _i is an arbitrary symplectic transformation. We formulate conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems. In particular, for the case and , we establish duality between eventual disconjugacy of general symplectic systems and their reciprocals.

Keywords:

• symplectic difference systems
• focal points
• symplectic transformations
• reciprocity principle

AMS Subject Classification::

• 39A10
• 39A11
• 39A12