- #1
JulienB
- 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.