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

Comments

The Tikhonov fixed-point theorem (also spelled Tychonoff's fixed-point theorem) states the following. Let $ X $ be a locally convex topological space whose topology is defined by a family $ \{ p _{i} \} $ of continuous semi-norms. Let $ C \subset X $ be compact and convex and $ f : \ C \rightarrow C $ a continuous mapping. Then $ f $ has a fixed point in $ C $([a2]; [a3], p. 175). Both the Kakutani fixed-point theorem and the Markov fixed-point theorem are generalized in the Ryll-Nardzewski fixed-point theorem, which states: Let $ X $ be a Banach space and $ Q $ a non-empty weakly compact subset. Let $ S $ be a semi-group of mappings from $ Q $ to $ Q $ which is non-contracting, then there is a fixed point of $ S $. Here, a family $ S $ of mappings is said to have a fixed point $ p $ if for every $ f \in S $, $ f(p) = p $, [a4]; cf. [a3], Chapt. 9, for a discussion of the Ryll-Nardzewski fixed-point theorem in relation to the Kakutani and Markov ones and other fixed-point theorems for families of mappings.

References

[a1]	J. Dugundji, A. Granas, "Fixed-point theory" , I , PWN (1982) MR0660439 Zbl 0483.47038
[a2]	A.N. [A.N. Tikhonov] Tychonoff, "Ein Fixpunktsatz" Math. Ann. , 111 (1935) pp. 767–776 MR1513031 Zbl 0012.30803 Zbl 61.1195.01
[a3]	V.I. Istrăţescu, "Fixed point theory" , Reidel (1981) MR0620639 Zbl 0465.47035
[a4]	C. Ryll-Nardzewski, "On fixed points of semi-groups of endomorphisms of linear spaces" , Proc. 5-th Berkeley Symp. Probab. Math. Stat. , 2: 1 , Univ. California Press (1967) pp. 55–61