# Schauder theorem

One of the fixed point theorems: If a completely-continuous operator maps a bounded closed convex set of a Banach space into itself, then there exists at least one point such that . 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 |

[2] | L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) |

[3] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |

[4] | R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) |

[5] | L. Nirenberg, "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974) |

#### Comments

The Tikhonov fixed-point theorem (also spelled Tychonoff's fixed-point theorem) states the following. Let be a locally convex topological space whose topology is defined by a family of continuous semi-norms. Let be compact and convex and a continuous mapping. Then has a fixed point in ([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 be a Banach space and a non-empty weakly compact subset. Let be a semi-group of mappings from to which is non-contracting, then there is a fixed point of . Here, a family of mappings is said to have a fixed point if for every , , [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) |

[a2] | A.N. [A.N. Tikhonov] Tychonoff, "Ein Fixpunktsatz" Math. Ann. , 111 (1935) pp. 767–776 |

[a3] | V.I. Istrăţescu, "Fixed point theory" , Reidel (1981) |

[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 |

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Schauder theorem.

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