Example of stable fixed point equation

Question

Schauder's fixed point theorem says that any continuous function $f:K\to{K}$, where $K$ is a nonempty convex and compact subset of a normed linear space $Y$ admits a fixed point. I came across this theorem: "Let $f:K\to K$ a continuous function. Then, for any $\varepsilon>0$, there exists $\delta>0$ such that for any $x ∈ K$ which it is satisfying the property $\Vert f(x)−x\Vert<\delta$ , there exists a fixed point of $f$ such $\Vert x−u\Vert <\varepsilon$." I want to see an example of a continuous function which satisfies both of this theorems. Is $f:[0,1]\to [0,1],\ f(x)=x^2-x+1$ a good one? How can I prove it?

Answer 1

This is a good example: the only fixed point of $f$ is $x =1$ because

$$x = f(x) = x^2 - x + 1 \implies (x-1)^2 = x^2 - 2x + 1 = 0 \implies x - 1$$

so for any $\varepsilon > 0$ we can set $\delta = \varepsilon^2$ and notice that $(x - 1)^2 = |f(x) - x| < \delta$ implies that $|1-x| < \sqrt{\delta} = \varepsilon$.

So the fixed point $1$ is within an $\varepsilon$-neighbourhood of $x$.