Proof of Banach-Caccioppoli theorem

In summary, the uniqueness of the fixed point is easily seen by taking an arbitrary point and noting that the distance between any two points in the domain is always less than or equal to the given constant. Convergence of the series can be seen by noting that as the series approaches zero, its terms become arbitrarily small.
  • #1
ozkan12
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Let be a complete metric space and let be a mapping such that for each , there exists a constant such that

for all where

. 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 ?
 
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  • #2
ozkan12 said:
Let be a complete metric space and let be a mapping such that for each , there exists a constant such that

for all where

. 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 and are both fixed points then , which does not tend to zero.

For the existence, take an arbitrary point and let (for all ). Then It follows that Use that to show that the sequence is Cauchy and therefore converges to a limit point. Show that this limit point has to be a fixed point of .
 
  • #3
Dear professor,

How we get ?Also, how I can use to show that is Cauchy sequence ?

Thank you for your attention
 
  • #4
ozkan12 said:
How we get ?
Use

derived in post #2 and the given fact that .

ozkan12 said:
Also, how I can use to show that is Cauchy sequence ?
By triangle inequality, . Since the series converges, its tail becomes arbitrarily small.
 
  • #5
Dear Makarov,

What is the means of "its tail becomes arbitrarily small. " ?
 
  • #6
By a tail of a series I mean the series for some . And if the first series converges, then for every there exists an such that for all .
 
  • #7
Dear Makarov,

İf then series of is convergent...İs this true ?...Also, if , then ...İs this true ? I ask these questions because my knowledge of functional analysis is not well...Thank you for your attention...
 
  • #8
ozkan12 said:
İf then series of is convergent...İs this true ?
This is the definition of the notation , at least when all are nonnegative.
ozkan12 said:
Also, if , then ...İs this true ?
Yes.

ozkan12 said:
I ask these questions because my knowledge of functional analysis is not well.
These are questions from calculus, not functional analysis.
 
  • #9
Dear Makarov,

So, if first one is not true, how I can prove that is cauchy sequence ? Can you prove that {x_n} is cauchy sequence and uniqueness of fixed point ? Also, how series of convergent ? I didnt understand ? Thank you for your attention...
 
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  • #10
ozkan12 said:
how I can prove that is cauchy sequence ?
The proof is written in posts 2, 4 and 6.

ozkan12 said:
Also, how series of 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.
 
  • #11
Please, Can you talk on these hints ? Because, I have not any calculus or functional analysis book in my home
 
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