Birkhoff–Kellogg invariant-direction theorem

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^[2] 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 x_o and some λ > 0 satisfying x_o = λF(x_o).

The Birkhoff–Kellogg theorem and its generalizations by Schauder and Leray have applications to partial differential equations.^[3]

References

1. Birkhoff, G. D.; Kellogg, O. D. (1922). "Invariant points in function space" (PDF). Trans. Amer. Math. Soc. 23: 96–115. doi:10.1090/s0002-9947-1922-1501192-9.
2. Granas, Andrzej; Dugundji, James (2003). Fixed Point Theory. New York: Springer-Verlag. pp. 125–126. ISBN 0-387-00173-5.
3. Morse, Marston (1946). "George David Birkhoff and his mathematical work, VI. MISCELLANEOUS WORKS, (a) Fixed points in function space, pages 385–386". Bull. Amer. Math. Soc. 52 (5, Part 1): 357–391. doi:10.1090/S0002-9904-1946-08553-5. MR 0016341.