Schauder theorem

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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

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.