Birkhoff–Kellogg invariant-direction theorem: Difference between revisions
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Revision as of 19:47, 11 October 2012
In functional analysis, the Birkhoff–Kellogg invariant-direction theorem, named after G. D. Birkhoff and O. D. Kellogg,[1] is a generalization of the Brouwer fixed-point theorem. The theorem states that:
Let U be a bounded open neighborhood of 0 in an infinite-dimensional normed linear space V, and let F:∂U —> V be a compact map satisfying ||F(x)|| ≥ α for some α > 0 for all x in ∂U. Then F has an invariant direction, i.e., there exist some xo and some λ > 0 satisfying xo = λ F(xo).
The Birkhoff–Kellogg theorem and its generalizations by Schauder and Leray have appllications to partial differential differential equations.
References
- ^ Birkhoff, G. D.; Kellogg, O. D. "Invariant points in function space" (PDF). Trans. Amer. Math. Soc. 23: 96–115.