  Nonlinear higher order difference equations with linear arguments (containing
linear forms within nonlinear maps of the space) are well-defined on Banach
algebras. The scalar forms of these equations (i.e., with real variables and
parameters) have appeared frequently in the literature. By generalizing
existing results from real numbers to algebras and using a new result on
reduction of order, new sufficient conditions are obtained for the convergence
to zero of all solutions of nonlinear difference equations with linear
arguments. Where reduction of order is possible, these conditions extend the
ranges of parameters for which the origin is a global attractor even when all
variables and parameters are real numbers.
