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Fixed Point Theorem

The fixed point theorem states that if g is a continuous function g(x) in [a,b] for all x in [a,b], then g has a fixed point in [a,b]. This can be proven by supposing that

g(a)>=a g(b)<=b
(1)
g(a)-a>=0 g(b)-b<=0.
(2)

Since g is continuous, the intermediate value theorem guarantees that there exists a c in [a,b] such that

g(c)-c=0,
(3)

so there must exist a c such that

g(c)=c,
(4)

so there must exist a fixed point in [a,b].

See also

Banach Fixed Point Theorem, Brouwer Fixed Point Theorem, Hairy Ball Theorem, Kakutani's Fixed Point Theorem, Lefschetz Fixed Point Theorem, Lefschetz Trace Formula, Map Fixed Point, Poincaré-Birkhoff Fixed Point Theorem, Schauder Fixed Point Theorem

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References

Rosenlicht, M. Introduction to Analysis. New York: Dover, p. 170, 1968.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, p. 80, 1991.

Referenced on Wolfram|Alpha

Fixed Point Theorem

Cite this as:

Weisstein, Eric W. "Fixed Point Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FixedPointTheorem.html

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