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Disconjugacy and transformations for symplectic systems. (English) Zbl 0894.39005

Let I be the n×n identity matrix, and let J=0I-I0. The 2n×2n matrix S=ABCD is called symplectic if S T JS=J holds, and the system z k+1 =S k z k , 0kN, is symplectic if the matrix S is symplectic. A solution z=x u to the system has a generalized zero in [k,k+1] if x k 0, x k+1 ImB k and x k T B k + x k+1 =0, where B k + denotes the Moore-Penrose inverse of the matrix B k . The system is called disconjugate on 𝒥=[0,N] if no solution of the system has more than one generalized zero (if x 0 =0 no zero).

Theorem 1 shows a lot of equivalent facts about disconjugacy, possession of generalized zeros and other similar properties of the solution to the symplectic system. Similar results are obtained relative to the reciprocal symplectic system z k+1 =S k -1 z k . The problems of eventually disconjugate solutions and disconjugacy preserving transformations are considered as well.


MSC:
39A12Discrete version of topics in analysis
39A10Additive difference equations