- #1

- 408

- 12

## Homework Statement

Hi everybody! Here is another problem about contraction and Banach fixed-point theorem that I don't get:

The function ƒ: C([0,½]) → C([0,½]) is defined by:

[tex]

[f(x)](t) := 1 + \int_{0}^{t} x(s) ds ∀ t∈[0,\frac{1}{2}].

[/tex]

Is ƒ a contraction with respect to the norm || ⋅ ||

_{∞}? If yes, which function is the fixed point of ƒ?

## Homework Equations

Contraction mapping theorem, Banach fixed-point theorem

## The Attempt at a Solution

Well I could not really get anywhere, because I don't understand really the definition of the function. Here is what I would do anyway:

[tex]

|| [f(x)](t) - [f(y)](t) ||_{\infty} = \mbox{max } | 1 + \int_{0}^{t} x(s) ds - 1 - \int_{0}^{t} y(s) ds | \\

= \mbox{max } | \int_{0}^{t} x(s) - y(s) ds |

[/tex]

Then I have no idea how to integrate that since x is a function of s... What is the primitive of x(s)? Also not sure if I did the right thing in the first place as well! Some help would be very appreciated. :)

Thank you in advance for your answers.

Julien.