Inequalities and Asymptotics for Riccati Matrix Difference Operators

AbstractIn the first part inequalities for solutions of Riccati matrix difference equations are obtained which correspond to the linear Hamiltonian difference system[formula]whereAk,Bk,Ck,Xk,Ukaren×n-matrices with symmetricBkandCk. If the matricesXkare invertible, then the matricesQk=UkX−1ksolve the Riccati matrix difference equation[formula]In contrast to some recent papers dealing with these equations we do not assume that the matricesBkare invertible. The second part of the paper deals with the asymptotic behaviour of solutionsQk(λ), as|λ|→∞, of the special Riccati matrix difference equation which corresponds to the Sturm–Liouville equation[formula]of even order 2nwith constant coefficientsr0,…,rn