TheSchauder fixed-point theorem is an extension of theBrouwer fixed-point theorem tolocally convex topological vector spaces, which may be of infinite dimension. It asserts that if${\displaystyle K}$ is a nonemptyconvex closed subset of aHausdorff locally convex topological vector space${\displaystyle V}$ and${\displaystyle f}$ is a continuous mapping of${\displaystyle K}$ into itself such that${\displaystyle f(K)}$ is contained in acompact subset of${\displaystyle K}$, then${\displaystyle f}$ has afixed point.

A consequence, calledSchaefer's fixed-point theorem, is particularly useful for proving existence of solutions tononlinearpartial differential equations.Schaefer's theorem is in fact a special case of the far reachingLeray–Schauder theorem which was proved earlier byJuliusz Schauder andJean Leray.The statement is as follows:

Let${\displaystyle f}$ be a continuous and compact mapping of aBanach space${\displaystyle X}$ into itself, such that the set

${\displaystyle \{x\in X:x=\lambda f(x)\text{ for some }0\le \lambda \le 1\}}$

is bounded. Then${\displaystyle f}$ has a fixed point. (Acompact mapping in this context is one for which the image of every bounded set isrelatively compact.)

The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in theScottish book. In 1934,Tychonoff proved the theorem for the case whenK is a compact convex subset of alocally convex space. This version is known as theSchauder–Tychonoff fixed-point theorem. B. V. Singbal proved the theorem for the more general case whereK may be non-compact; the proof can be found in the appendix ofBonsall's book (see references).

• Fixed-point theorems
• Banach fixed-point theorem
• Kakutani fixed-point theorem

• F. F. Bonsall,Lectures on some fixed point theorems of functional analysis, Bombay 1962
• D. Gilbarg,N. Trudinger,Elliptic Partial Differential Equations of Second Order.ISBN 3-540-41160-7.
• H. Schaefer,Über die Methode der a priori-Schranken, Math. Ann. 129, 415–416 (1955),doi:10.1007/BF01362380
• J. Schauder,Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180
• A. Tychonoff,Ein Fixpunktsatz, Mathematische Annalen 111 (1935), 767–776
• E. Zeidler,Nonlinear Functional Analysis and its Applications,I - Fixed-Point Theorems

• "Schauder theorem",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• "Schauder fixed point theorem".PlanetMath.
• "proof of Schauder Fixed Point Theorem".PlanetMath..

• Fixed-point theorems
• Theorems in functional analysis
• Topological vector spaces

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