info prev up next book cdrom email home

Fixed Point Theorem

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 noting that

\begin{displaymath}
g(a)\geq a \qquad g(b)\leq b
\end{displaymath}


\begin{displaymath}
g(a)-a\geq 0 \qquad g(b)-b\leq 0.
\end{displaymath}

Since $g$ is continuous, the Intermediate Value Theorem guarantees that there exists a $c\in [a,b]$ such that

\begin{displaymath}
g(c)-c=0,
\end{displaymath}

so there must exist a $c$ such that

\begin{displaymath}
g(c)=c,
\end{displaymath}

so there must exist a Fixed Point $\in [a,b]$.

See also Banach Fixed Point Theorem, Brouwer Fixed Point Theorem, Kakutani's Fixed Point Theorem, Lefshetz Fixed Point Formula, Lefshetz Trace Formula, Poincaré-Birkhoff Fixed Point Theorem, Schauder Fixed Point Theorem




© 1996-9 Eric W. Weisstein
1999-05-26