I am trying find bibliography on existence of solutions for difference equations but it seems that there is not much on the web.
I need existence and bound of a certain nonlinear difference equation of the form $$x(t+1)=a\, x(t)+f(x(t))$$ With $f$ analytic without constant or linear terms. I am searching for an analytic solution by the way.
So I have a couple of questions. Can I rearrange it somehow to get existence by fixed point theorem? Basically is there an analogue of Picard's theorem?
If I can majorate $f$ with some $F$ and I can solve $$X(t+1)=a\, X(t)+F(X(t))$$ then do I get any information about the solution $x(t)$?
Also with which classes of $F$ can I hope to actually solve the above equation?
Any ideas are kindly welcomed.