Global attractors for difference equations dominated by one-dimensional maps

The global attractivity character of nonlinear higher order difference equations of the form x(n+1) = g(x(n), x(n-1), ..., x(n-k)), n >= 0 is investigated when g is dominated by an interval scalar map. Some basic properties of the interval map are obtained and used to prove new global attractivity criteria for the above equation with no monotonicity restrictions on g. Our results are applied to many models from mathematical biology and economy. The derived global attractivity criteria of these models are either new or improve substantially known ones.