Continuous mapping and fixed point

In summary: In this context, $T^k y=\underbrace{TTT\dots T}_{k \, \text{times}}y,$ so that $T^k$ is shorthand for "compose $T$ with itself $k$ times". So, you are correct.
  • #1
ozkan12
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0
Let $T$ be a continuous mapping of a complete metric space $X$ into itself such that ${T}^{k}$ is a contraction mapping of $X$ for some positive integer $k$. Then $T$ has a unique fixed point in $X$.

Proof:

${T}^{k}$ has a unique fixed point $u$ in $X$ and $u=\lim_{{n}\to{\infty}}\left({T}^{k}\right)^n{x}_{0}$ ${x}_{0}\in X$ arbitrary.

Also $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^nT{x}_{0}=u$. Hence

$u=\lim_{{n}\to{\infty}}({T}^{k})^nT{x}_{0}$=$\lim_{{n}\to{\infty}}T\left({T}^{k}\right)^n{x}_{0}$ (2)
=$T\lim_{{n}\to{\infty}}\left({T}^{k}\right)^n{x}_{0}$
=$Tu$.İn this proof, I didnt understand, How (2) happened ? Please, can you explain ? Thank you for your attention...Best wishes...
 
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  • #2
ozkan12 said:
Let $T$ be a continuous mapping of a complete metric space $X$ into itself such that ${T}^{k}$ is a contraction mapping of $X$ for some positive integer $k$. Then $T$ has a unique fixed point in $X$.

Proof:

${T}^{k}$ has a unique fixed point $u$ in $X$ and $u=\lim_{{n}\to{\infty}}\left({T}^{k}\right)^n{x}_{0}$ ${x}_{0}\in X$ arbitrary.

Also $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^nT{x}_{0}=u$. Hence

$u=\lim_{{n}\to{\infty}}({T}^{k})^nT{x}_{0}$=$\lim_{{n}\to{\infty}}T\left({T}^{k}\right)^n{x}_{0}$ (2)
=$T\lim_{{n}\to{\infty}}\left({T}^{k}\right)^n{x}_{0}$
=$Tu$.İn this proof, I didnt understand, How (2) happened ? Please, can you explain ? Thank you for your attention...Best wishes...
Since $T^k$ is a contraction map, $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^ny = u$ for any $y\in X$. In particular, this holds for $y=Tx_0$, so that $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^nT{x}_{0}=u$. But powers of $T$ commute with each other, so $\left({T}^{k}\right)^nT = T\left({T}^{k}\right)^n$. Therefore $u = \lim_{{n}\to{\infty}}T\left({T}^{k}\right)^n{x}_{0}$, as claimed in (2).
 
  • #3
Dear professor,

Firstly, Thank you so much...But why powers of T is commute each other ?
 
  • #4
Opalg said:
Since $T^k$ is a contraction map, $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^ny = u$ for any $y\in X$. In particular, this holds for $y=Tx_0$, so that $\lim_{{n}\to{\infty}}\left({T}^{k}\right)^nT{x}_{0}=u$. But powers of $T$ commute with each other, so $\left({T}^{k}\right)^nT = T\left({T}^{k}\right)^n$. Therefore $u = \lim_{{n}\to{\infty}}T\left({T}^{k}\right)^n{x}_{0}$, as claimed in (2).

ozkan12 said:
But why powers of T is commute each other ?
Index laws: $\left({T}^{k}\right)^nT = T^{kn+1} = t^{1+kn} = T\left({T}^{k}\right)^n$.
 
  • #5
Dear professor,

İn there $({T}^{k})$ is composite function, isn't it ? That is, İn my opinion, we don't get exponentiate
 
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  • #6
In this context, $T^k y=\underbrace{TTT\dots T}_{k \, \text{times}}y,$ so that $T^k$ is shorthand for "compose $T$ with itself $k$ times". So, you are correct.
 
  • #7
ozkan12 said:
İn there $({T}^{k})$ is composite function, isn't it ? That is, İn my opinion, we don't get exponentiate
The operation of composition is associative, and therefore obeys the index laws.
 

FAQ: Continuous mapping and fixed point

What is continuous mapping?

Continuous mapping is a mathematical concept that describes the relationship between two topological spaces. It means that a small change in one space will result in a small change in the other space.

What is a fixed point?

A fixed point is a point in a function or mapping where the output or result is equal to the input. In other words, the point remains unchanged after applying the function or mapping.

Why is continuous mapping important?

Continuous mapping is important because it allows us to study the behavior of functions and mappings in a more precise and systematic way. It also helps us understand the properties and relationships between different topological spaces.

How do you determine if a mapping is continuous?

A mapping is continuous if the pre-image of an open set is an open set. In simpler terms, this means that if a small change is made to the input, the output will also change by a small amount.

What is an example of a continuous mapping?

An example of a continuous mapping is the function f(x) = x, where the input and output are both real numbers. No matter how small the change in the input, the output will also change by a small amount, showing continuity in the mapping.

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