Generalization of the Banach fixed point theorem
Answer
Suppose $T^m(x)=x$, and assume $x$ is the only fixed point of $T^m$. Then
$$T^m(T(x))=T(x),$$
and hence $T(x)$ is also a fixed point of $T^m$. Since the fixed point of $T^m$ is unique, we must have
\[T(x)=x,\]
so $x$ is also a fixed point of $T$. It remains to show that $T$ does not have any other fixed point. Assume $T(y)=y$, for some $y \in M$. Then
\[T^{m}(y)=T^{m-1}(y)= \dots =T(x)=y,\]
so $y$ is a fixed point of $T^m$, and therefore $y=x$, since $T^m$ has only one fixed point.
$T$ also has the unique fixed point $x$.

4.8K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1522 views
- $25.00
Related Questions
- Let $f(x,y,z)=(x^2\cos (yz), \sin (x^2y)-x, e^{y \sin z})$. Compute the derivative matrix $Df$.
- Convergence and Integrability of Function Series in Measure Spaces and Applications to Series Expansion Integrals
- Integrate $\int x^2(1-x^2)^{-\frac{3}{2}}dx$
- Calculus on Submanifolds Challenge
- Prove Holder-continuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$
- Knot Theory, 3-colourbility of knots
- Find $n$ such that $\lim _{x \rightarrow \infty} \frac{1}{x} \ln (\frac{e^{x}+e^{2x}+\dots e^{nx}}{n})=9$
- Find the equation of the tangent line through the function f(x)=3x$e^{5x-5} $ at the point on the curve where x=1