Dichotomic behavior of linear difference systems with advanced argument (Q1583941)

Consider the linear difference equation with advanced argument \[ y(n+1)= A(n)y(n)+ B(n)y(g(n)),\tag{\(*\)} \] where \((A(n))\) is a sequence of \(r\times r\) invertible matrices and \((B(n))\) is a sequence of matrices in general non-invertible. The sequence \((g(n))\) is such that \(1\leq g(n)- n\leq N\in\mathbb{N}\). We say that \((*)\) has an ordinary dichotomy in the sequential space \(S\) iff there exists a generating matrix \(\Psi(n)\) of \((*)\) in \(S\), a projection matrix \(P\) and a constant \(K\geq 1\) such that \[ |\Psi(n) P\Psi^{-1}(m+ 1)|\leq K,\quad n>m, \] \[ |\Psi(n)(I- P)\Psi^{-1}(m+ 1)|\leq K,\quad m\geq n. \] The authors establish the criterion of existence of the ordinary dichotomy to \((*)\). Some examples are included.