ABSTRACT
In this chapter, we establish several results on the existence and uniqueness of solutions for a class of semilinear systems of difference equations with initial and boundary conditions. The approach is based on fixed point theory in vector-valued Banach spaces. Several aspects of the theory of semilinear difference equations can be understood as a proper generalization of the theory of ordinary difference equations. However, the fact that the state space for functional difference equations is infinite dimensional requires the development of methods and techniques coming from functional analysis (e.g., theory of semigroups of operators on Banach spaces, spectral theory, fixed point theory etc.). Some important contributions to the study of the mathematical aspects of such equations have been undertaken in [3,13,55, 84, 166].
