Mezocontinuous operators and solutions of difference equations

We attempt to unify and extend the theory of asymptotic properties of solutions to difference equations of various types. Usually in difference equations some functions are used which generate transformations of sequences. We replace these functions by abstract operators and investigate some properties of such operators. We are interested in properties of operators which correspond to continuity or boundedness or local boundedness of functions. Next we investigate asymptotic properties of the set of all solutions to `abstract' and `functional' difference equations. Our approach is based on using the iterated remainder operator and the asymptotic difference pair. Moreover, we use the regional topology on the space of all real sequences and the `regional' version of the Schauder fixed point theorem.