Exponential forward splitting for noninvertible difference equations (Q1600425)

The authors consider difference equations of the form: \[ x(k+1)= A(k)x(k), \tag{1} \] \(k\in Z_{\kappa_0}: =\mathbb{Z}\cap [\kappa_0,\infty)\), where for each \(k\in Z_{\kappa_0}\) the mapping \(A(k)\) belongs to the space \({\mathcal L}(\chi)\) of bounded linear operators on a Banach space \((\chi,\|\cdot\|)\), with the forward evolution operator \[ \varphi(k,n): =\begin{cases} A(k-1) A(k)\dots A(n), \quad & \text{for all }k<n \geq\kappa_0\\ \text{id}, \quad & \text{for all }k=n\geq \kappa_0. \end{cases} \] In the particular case when all operators \(A(k)\), \(k\in Z_{\kappa_0}\), are invertible, there is also considered \[ \varphi(k,n) =A(k)^{-1} A(k+1)^{-1} \dots A(n-1)^{-1}, \quad\text{for all }n>k \geq\kappa_0. \] Using \(\varphi (k,n)\), the notion of exponential forward splitting and the notion of exponential forward dichotomy are defined. The problem of how these notions can be used for difference equations whose right-hand sides are not supposed to be invertible, are discussed. The notion of regular exponential forward splitting is also defined and studied.