Exponential forward splitting for noninvertible difference equations

While the basic theory of exponential dichotomies deals with differential and difference equations with uniquely determined forward and backward solutions, nowadays applications require to have a corresponding theory for equations whose backward solutions are not guaranteed to exist or to be unique. In this paper, we study the problem of how to generalize the notion of exponential dichotomy to difference equations whose solutions are only assumed to exist in forward time. It turns out that, for this kind of equation, a straightforward generalization of the notion of exponential dichotomy does not provide satisfactory results but that, on the other hand, with an additional assumption that desired goal can be reached.