Gérard Debreu in the context of "Sonnenschein–Mantel–Debreu theorem"

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function ${\displaystyle f}$ mapping a nonempty compact convex set to itself, there is a point ${\displaystyle x_{0}}$ such that ${\displaystyle f(x_{0})=x_{0}}$. The simplest forms of Brouwer's theorem are for continuous functions ${\displaystyle f}$ from a closed interval ${\displaystyle I}$ in the real numbers to itself or from a closed disk ${\displaystyle D}$ to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset ${\displaystyle K}$ of Euclidean space to itself.

Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.

Marie-Esprit-Léon Walras (French: [valʁas]; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist. He formulated the marginal theory of value (independently of William Stanley Jevons and Carl Menger) and pioneered the development of general equilibrium theory. Walras is best known for his book Éléments d'économie politique pure, a work that has contributed greatly to the mathematization of economics through the concept of general equilibrium.

For Walras, exchanges only take place after a Walrasian tâtonnement (French for "trial and error"), guided by the auctioneer, has made it possible to reach market equilibrium. It was the general equilibrium obtained from a single hypothesis, rarity, that led Joseph Schumpeter to consider him "the greatest of all economists". The notion of general equilibrium was very quickly adopted by major economists such as Vilfredo Pareto, Knut Wicksell and Gustav Cassel. John Hicks and Paul Samuelson used the Walrasian contribution in the elaboration of the neoclassical synthesis. For their part, Kenneth Arrow and Gérard Debreu, from the perspective of a logician and a mathematician, determined the conditions necessary for equilibrium.

In mathematical economics, the Arrow–Debreu model is a theoretical general equilibrium model. It posits that under certain economic assumptions (convex preferences, perfect competition, and demand independence), there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.

The model is central to the theory of general (economic) equilibrium, and it is used as a general reference for other microeconomic models. It was proposed by Kenneth Arrow, Gérard Debreu in 1954, and Lionel W. McKenzie independently in 1954, with later improvements in 1959.

In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general equilibrium. General equilibrium theory contrasts with the theory of partial equilibrium, which analyzes a specific part of an economy while its other factors are held constant.

General equilibrium theory both studies economies using the model of equilibrium pricing and seeks to determine in which circumstances the assumptions of general equilibrium will hold. The theory dates to the 1870s, particularly the work of French economist Léon Walras in his pioneering 1874 work Elements of Pure Economics. The theory reached its modern form with the work of Lionel W. McKenzie (Walrasian theory), Kenneth Arrow and Gérard Debreu (Hicksian theory) in the 1950s.