In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a number of
fixed-point theorems in infinite-dimensional spaces generalise the
Brouwer fixed-point theorem. They have applications, for example, to the proof of
existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
s for
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s.
The first result in the field was the
Schauder fixed-point theorem, proved in 1930 by
Juliusz Schauder (a previous result in a different vein, the
Banach fixed-point theorem for
contraction mappings in complete
metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of
algebraic topology, first proved for finite