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We present a new approach to the theory of asymptotic properties of solutionsof difference equations. Usually, two sequences $x,y$ are called asymptoticallyequivalent if the sequence $x-y$ is convergent to zero i.e., $x-y\in c_0$,where $c_0$ denotes the space of all convergent to zero sequences. We replacethe space $c_0$ by various subspaces of $c_0$. Our approach is based on usingthe iterated remainder operator. Moreover, we use the regional topology on thespace of all real sequences and the `regional' version of the Schauder fixedpoint theorem.

Qualitative approximation of solutions to difference equations