Please give me some examples of contraction mapping on $(C[0,1]), \lvert \lvert \cdot \rvert \rvert_\infty)$ and $(C[0,1],\lvert \lvert \cdot \rvert \rvert_1) $.
Note that : 1. $\lvert \lvert f \rvert \rvert_\infty = \sup_{x\in [0,1]} f(x)$ and $\lvert \lvert f \rvert \rvert_1 = \int_0^1 \lvert f(x) \rvert dx $.
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$C[0,1]$ is a space of all continuous function on $[0,1]$
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Let $T$ be a function from $(X,\lvert \lvert \cdot \rvert \rvert)$ into itself. We call $T$ a contraction map if there exists $0<k<1$ such that $\lvert \lvert T(x) – T(y) \rvert \rvert \leq k \lvert \lvert x-y \rvert \rvert$ for all $x,y \in X$.
Best Answer
A very important one: $$\Phi: \mathcal C([a,b],\Bbb R)\to \Bbb R $$
defined by $$\Phi(v)(t) = x_0 + \int_0^tF(v(s))\, ds$$
where $F:\Bbb R\to \Bbb R$ is Lipschitz continuous.