Two-point boundary value problems associated with first order nonlinear difference system -- existence and uniqueness (Q1294260)

This paper examines first-order matrix difference equations of the form \[ T(n+1) = A(n)T(n)B(n) + F(n,T(n)) \] in which \(A\), \(B\), \(F\) and \(T\) are square matrices, with \(F\) Lipschitz. A variation of parameters formula is obtained for the solutions, in terms of the fundamental matrices of the associated systems \(\Phi(n+1) = A(n) \Phi(n)\) and \(\Psi(n+1) = B(n)^*\Psi(n)B(n)\). The authors prove that certain associated two-point boundary value problems are uniquely solvable, by using a modified QR algorithm of \textit{R. H. Bartels} and \textit{G. W. Stewart} [Commun. ACM 15, No. 9, 820--826 (1972; Zbl 1372.65121)].