In functional analysis, the BirkhoffâKellogg invariant-direction theorem, named after G. D. Birkhoff and O. D. Kellogg, [1] is a generalization of the Brouwer fixed-point theorem. The theorem [2] states that:
Let U be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space V, and let F:âU â V be a compact map satisfying ||F(x)|| ℠α for some α > 0 for all x in âU. Then F has an invariant direction, i.e., there exist some xo and some λ > 0 satisfying xo = λF(xo).
The BirkhoffâKellogg theorem and its generalizations by Schauder and Leray have applications to partial differential equations. [3]