INEQUALITIES AND ASYMPTOTICS FOR RICCATI MATRIX DIFFERENCE-OPERATORS

In the first part inequalities for solutions of Riccati matrix differe nce equations are obtained which correspond to the linear Hamiltonian difference system Delta X-k = A(k)X(k+1) + BkUk, Delta U-k = CkXk+1 - A(k)(T)U(k), where A(k), B-k, C-k, X-k, U-k are n X n-matrices with sy mmetric B-k and C-k. If the matrices X-k are invertible, then the matr ices Q(k) = UkXk-1 solve the Riccati matrix difference equation Q(k+1) = C-k + (I - A(k)(T))Q(k)(I + B(k)Q(k))(-1)(I - A(k)). In contrast to some recent papers dealing with these equations we do not assume that the matrices Bk are invertible. The second part of the paper deals wi th the asymptotic behaviour of solutions Q(k)(lambda), as \lambda\ --> infinity, of the special Riccati matrix difference equation which cor responds to the Sturm-Liouville equation [GRAPHICS] of even order 2n w ith constant coefficients r(0),...r(n). (C) 1998 Academic Press.