  When extending bifurcation theory of dynamical systems to nonautonomous
problems, it is a central observation that hyperbolic equilibria persist as
bounded entire solutions under small temporally varying perturbations. In this
paper, we abandon the smallness assumption and aim to investigate the global
structure of the entity of all such bounded entire solutions in the situation
of nonautonomous difference equations. Our tools are global implicit function
theorems based on an ambient degree theory for Fredholm operators due to
Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restrict to
so-called homoclinic solutions, whose limit is 0 in both time directions.
