You might want to check out the book "Fixed Point Theorems with Applications to Economics and Game Theory" by Kim C. Border. It's not new but it might give you some basic background to start.
As to papers: "Applications of fixed point theory to extended Nash equilibriums of nonmonetized noncooperative games on posets" by L Xie, J Li, CF Wen
Hope these help a bit to at least get you started.
You also have the full-text http://cupid.economics.uq.edu.au/mclennan/Advanced/advanced_fp.pdf) by Andrew McLennan. I have not checked the quality; it is quite recent (November 2012).
Tonu Puu, Nonlinear economic dynamics, 4th edition 1997, Springer (first edition 1989). The first three chapters are introductory to discrete and continuous dynamical systems, while the following chapters (4 to 8 ) present applications to economics (monopoly, duopoly and oligopoly, business cycles)
A very good book by Christopher-Oliver Ewald of school of Economics and finance, University of st. Andrew , titled " Game Theory, Fixed point and Mathematical Economics" . This will really assist you. Good luck.