Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

Abstract

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.

Keywords:

• Dominant solution at infinity
• recessive solution at infinity
• discrete symplectic system
• genus of conjoined bases
• nonoscillation
• order of abnormality
• controllability
• Moore–Penrose pseudoinverse

AMS Subject Classifications:

• 39A21
• 39A12

Notes

No potential conflict of interest was reported by the authors.

Dedicated to the memory of Professor Ondřej Došlý

Additional information

Funding

This research was supported by the Czech Science Foundation [grant number GA16–00611S].