Schauder fixed-point theorem

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

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

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

History

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 the

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

References

• J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180
• A. Tychonoff, Ein Fixpunktsatz, Mathematische Annalen 111 (1935), 767–776
• F. F. Bonsall, Lectures on some fixed point theorems of functional analysis, Bombay 1962
• D. Gilbarg,
  ISBN 3-540-41160-7
  .
• 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..