I think it is worth pointing out that, from a purely mathematical point of view, economic general equilibrium theory is an exercise in fixed point theory.
The same may be said of the theory of non-cooperative games. John Nash invented the solution concept now known as the Nash equilibrium in his thesis. VonNeumann dismissed Nash's result as "just a fixed point theorem" but Nash eventually received the nobel prize in economics for this work.
The mathematical setup for economic general equilibrium theory focuses on constructing what is called the "excess demand correspondence". This is derived from assumptions about how consumers and producers formulate their plans to take best advantage of the prices they observe in the market place. The excess demand correspondence associates to any vector of market prices (one price for each commodity) the convex set of vectors of aggregate excess demands (one excess demand- possibly negative -for each commodity) that will arise when consumers and producers respond to that specified price vector.
The main idea of the proof is then to find another convex set of price vectors, each of which can be interpreted as a supporting hyperplane to the given convex set of excess demands, and each of which maximizes the market value of the excess demand.
This construction then then can be shown to yield an upper semi-continuous, convex set valued function from the convex set of allowable vectors of market prices to the space of is convex subsets. One then applies an appropriate fixed point theory to deduce the existence of a price vector which, because of the structure of the excess demand correspondence, has the property that the value of excess demand in each market is zero. This is the market equilibrium price vector.