Birkhoff–Kellogg invariant-direction theorem explained

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]

Notes and References

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