# Schauder theorem

One of the fixed point theorems: If a completely-continuous operator $A$ maps a bounded closed convex set $K$ of a Banach space $X$ into itself, then there exists at least one point $x\in K$ such that $Ax=x$. Proved by J. Schauder [1] as a generalization of the Brouwer theorem.

There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle, etc.

#### References

 [1] J. Schauder, "Der Fixpunktsatz in Funktionalräumen" Stud. Math. , 2 (1930) pp. 171–180 Zbl 56.0355.01 [2] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) MR0539144 MR0048693 Zbl 0141.11601 Zbl 0096.07802 [3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 [4] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) MR0221256 Zbl 0182.16101 [5] L. Nirenberg, "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974) MR0488102 Zbl 0286.47037