MHB Proof of Banach-Caccioppoli theorem

[ozkan12]

Let $(X,d)$ be a complete metric space and let $f:X\to X$ be a mapping such that for each $n\ge1$, there exists a constant ${c}_{n}$ such that

$d\left({f}^{n}x,{f}^{n}y\right)$ $\le{c}_{n}d\left(x,y\right)$ for all $x,y\in X$ where

$\sum_{n=1}^{\infty}{c}_{n}<\infty$. Then f has a unique fixed point.

I didnt prove this theorem and I didnt find article which is related to this theorem...Please can you help me ?

 

[Opalg]

Let $(X,d)$ be a complete metric space and let $f:X\to X$ be a mapping such that for each $n\ge1$, there exists a constant ${c}_{n}$ such that

$d\left({f}^{n}x,{f}^{n}y\right)$ $\le{c}_{n}d\left(x,y\right)$ for all $x,y\in X$ where

$\sum_{n=1}^{\infty}{c}_{n}<\infty$. Then f has a unique fixed point.

I didnt prove this theorem and I didnt find article which is related to this theorem...Please can you help me ?

The uniqueness is more or less obvious: if $x$ and $y$ are both fixed points then $d(f^nx,f^ny) = d(x,y)$, which does not tend to zero.

For the existence, take an arbitrary point $x_0\in X$ and let $x_n = f^nx_0$ (for all $n\geqslant1$). Then $d(x_n,x_{n+1}) = d(f^nx_0,f^{n+1}x_0) \leqslant c_nd(x_0,x_1).$ It follows that $\sum_{n=0}^\infty d(x_n,x_{n+1}) < \infty.$ Use that to show that the sequence $(x_n)$ is Cauchy and therefore converges to a limit point. Show that this limit point has to be a fixed point of $f$.

 

[ozkan12]

Dear professor,

How we get $\sum_{n=0}^{\infty}d\left({x}_{n},{x}_{n+1}\right)\le\infty$ ?Also, how I can use $\sum_{n=0}^{\infty}d\left({x}_{n},{x}_{n+1}\right)\le\infty$ to show that $\left\{{x}_{n}\right\}$ is Cauchy sequence ?

Thank you for your attention

 

[Evgeny.Makarov]

How we get $\sum_{n=0}^{\infty}d\left({x}_{n},{x}_{n+1}\right)\le\infty$ ?

Use
\[
d(x_n,x_{n+1}) = d(f^nx_0,f^{n+1}x_0) \leqslant c_nd(x_0,x_1)
\]
derived in post #2 and the given fact that $\sum_{n=1}^{\infty}{c}_{n}<\infty$.

Also, how I can use $\sum_{n=0}^{\infty}d\left({x}_{n},{x}_{n+1}\right)\le\infty$ to show that $\left\{{x}_{n}\right\}$ is Cauchy sequence ?

By triangle inequality, $d(x_n,x_m)\le\sum_{k=n}^{m-1}d(x_k,x_{k+1})\le\sum_{k=n}^\infty d(x_k,x_{k+1})$. Since the series $\sum_{k=0}^\infty d(x_k,x_{k+1})$ converges, its tail becomes arbitrarily small.

 

[ozkan12]

Dear Makarov,

What is the means of "its tail becomes arbitrarily small. " ?

 

[Evgeny.Makarov]

By a tail of a series $\sum_{k=0}^\infty a_n$ I mean the series $\sum_{k=n}^\infty a_n$ for some $n$. And if the first series converges, then for every $\varepsilon>0$ there exists an $N>0$ such that $\left|\sum_{k=n}^\infty a_n\right|<\varepsilon$ for all $n>N$.

 

[ozkan12]

Dear Makarov,

İf $\sum_{n=1}^{\infty}{c}_{n}<\infty$ then series of $\sum_{n=1}^{\infty}{c}_{n}$ is convergent...İs this true ?...Also, if $\sum_{n=1}^{\infty}{c}_{n}<\infty$, then $\sum_{n=1}^{\infty}{c}_{n}d\left({x}_{0},{x}_{1}\right)<\infty$...İs this true ? I ask these questions because my knowledge of functional analysis is not well...Thank you for your attention...

 

[Evgeny.Makarov]

İf $\sum_{n=1}^{\infty}{c}_{n}<\infty$ then series of $\sum_{n=1}^{\infty}{c}_{n}$ is convergent...İs this true ?

This is the definition of the notation $\sum_{n=1}^{\infty}{c}_{n}<\infty$, at least when all $c_n$ are nonnegative.

Also, if $\sum_{n=1}^{\infty}{c}_{n}<\infty$, then $\sum_{n=1}^{\infty}{c}_{n}d\left({x}_{0},{x}_{1}\right)<\infty$...İs this true ?

Yes.

I ask these questions because my knowledge of functional analysis is not well.

These are questions from calculus, not functional analysis.

 

[ozkan12]

Dear Makarov,

So, if first one is not true, how I can prove that $\left\{{x}_{n}\right\}$ is cauchy sequence ? Can you prove that $\left\{{x}_{n}\right\}${x_n} is cauchy sequence and uniqueness of fixed point ? Also, how series of $\sum_{k=0}^{\infty}d\left({x}_{k},{x}_{k+1}\right)$ convergent ? I didnt understand ? Thank you for your attention...

 

[Evgeny.Makarov]

how I can prove that $\left\{{x}_{n}\right\}$ is cauchy sequence ?

The proof is written in posts 2, 4 and 6.

Also, how series of $\sum_{k=0}^{\infty}d\left({x}_{k},{x}_{k+1}\right)$ convergent ?

This is also answered in post #4.

In order to understand these hints, you need to review the definition and theory of Cauchy sequences and convergent series.

 

[ozkan12]

Please, Can you talk on these hints ? Because, I have not any calculus or functional analysis book in my home