MHB Why is the contraction constant important in the Banach contraction principle?

[ozkan12]

It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist...But I didnt understand this case...how is contraction constant h important ? I don't understand...please talk about this

 

[Evgeny.Makarov]

It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist.

As far as I know, the term "contractive mapping", as distinct from "contraction", or "contraction mapping", does not have a universally accepted meaning in English. You'll have to give us its definition. Do you mean a "nonexpansive map"?

how is contraction constant h important ?

You said it yourself (I am guessing): if $h\ge 1$, a fixed point need not exist.

I don't understand...

What exactly?

 

[ozkan12]

no I say contractive mapping...the definition of contractive mapping: let f:X to X be self mapping on (X,d) metric space
if d(fx,fy)<d(x,y) then f is contractive mapping and note that in this concept 'h' contraction constant equal to 1 and inequality is strict inequality

 

[Evgeny.Makarov]

let f:X to X be self mapping on (X,d) metric space
if d(fx,fy)<d(x,y) then f is contractive mapping

OK. So, what is your question?

 

[ozkan12]

It was important in the proof of BCP that the contraction constant h be strictly less than 1. That gave us control over the rate of convergence of f^n (x0) to the fixed point since h^n goes to 0 as n goes to infinity...If we consider f is contractive mapping instead of a contraction, then we lose that control and indeed a fixed point need not exist...

how this happened ? that is why if we choose contractive map we lost control of rate of convergence of f^n(x0)

x0 is arbitrary point in (X,d) metric space