Kim Border has lots of excellent lecture notes related to mathematical economics, including some on general equilibrium theory. The lecture notes on general equilibrium theory by Nicholas Yannelis are very extensive and introduce a lot of the related mathematics.

The original paper by Arrow and Debreu, probably the most famous paper on the existence of an general equilibrium, can be found here. It is based on a result by Debreu to be found here. Ted Bergstrom has a nice survey on how to weaken several assumptions used in many existence proofs. In particular, how to substantially weaken the rationality requirements.

Mathematicians might be interested that existence of equilibrium has also proven for some infinite dimensional models that are important for finance, growth theory and various other fields. This literature makes much use of functional analysis, especially Riesz spaces. A good survey by Mas-Colell and Zame can be found here.