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In fixed point theorem, If g is a continuous function $ g(x) \in [a,b]$ for all $ x \in [a,b]$, then g has a fixed point in $[a,b]$ i.e. $ c \in [a,b]$ such that $g(c)=c$

According to this theorem, We have exactly one fixed point or at least one fixed point?

pinaki nayak
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Anonymous
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1 Answers1

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Without loss of generality you can reparametrize $[a,b]$ with $[0,1]$.

Given $g$ define $f(t)=g(t)-t$.

$f$ is continuous. Since $g$ takes value in $[0,1]$, then and $f(0)\geq 0$ and $f(1)\leq 0$. By the Intermediate Value Theorem, there is at least one $t\in[0,1]$ so that $f(t)=0$, which implies $g(t)=t$.

user126154
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  • Reparametrization is just psicological. Even if you don't reparametrize, if you set $f(t)=g(t)-t$ you have $f(a)\geq 0$ and $f(b)\leq 0$, and the intermediate value theorema apply. – user126154 May 11 '18 at 13:44