Limit behavior of solutions of singular difference equations

Necessary and sufficient conditions for a solution {z_k} of the difference equation $\text{A}\text{z}{}_{\mathrm{k+1}}\text{+}\text{B}\text{z}{}_{\mathrm{k}}\text{= b, k ?0}$, with $\text{A}$ singular, to be a convergent sequence of vectors are given under a variety of assumptions. Theoretical results on iterative schemes for solving Ax = b by singular splittings, $\text{A =}\text{A}\text{+}\text{B}$, are given first. In particular, the case when $\text{A = A}{}^1$ and $\text{A}$ is positive semi-definite is considered. Then applications to discrete control problems and backwards population projection are discussed.