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Inequalities and asymptotics for Riccati matrix difference operators. (English) Zbl 0914.39012
Inequalities for the solutions of the Riccati matrix difference equation \[ Q_{k+1}= C_k+(I-A_k^T) Q_k(I+B_kQ_k)^{-1} (I-A_k) \] which correspond to the linear Hamiltonian difference system are developed. Finally the asymptotic behaviour of solutions \(Q_k(\lambda)\), as \(|\lambda|\to \infty\), of the special Riccati matrix difference equation is established which corresponds to the Sturm-Liouville equation \[ \sum_{\mu=0}^n (-1)^\mu r_\mu \Delta^{2\mu} y_{k+1-\mu}= \lambda y_{k+1} \] of even order \(2n\) with constant coefficients \(r_0,\dots, r_n\).

MSC:
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
39A70 Difference operators
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