# Fixed point syntax!

• Dec 20th 2010, 02:56 AM
cantona
Fixed point syntax!
Hi everyone!

I am a little bit confused how write this:

My map is: $\displaystyle F(f)(x)=\frac{1}{2}f(x)-\frac{3}{5}$. I must calculated a fixed point. I know how to do that and I now how much it is. But problem is in syntax. Is that the right one: $\displaystyle F(f(x_0))=f(x_0)$ and the result in that form $\displaystyle f(x_0)=-\frac{6}{5}$ ? Or is there $\displaystyle f_0(x)$ and not $\displaystyle f(x_0)$

It is $\displaystyle f_0(x)$. Your map, F, is acting on functions, not numbers. A "fixed point" of F is a function, $\displaystyle f_0(x)$ such that $\displaystyle F(f_0)= f_0$. Since $\displaystyle F(f)(x)$ is defined as $\displaystyle \frac{1}{2}f(x)- \frac{3}{5}$ a fixed point is a function $\displaystyle f_0(x)$ such that $\displaystyle \frac{1}{2}f(x)- \frac{3}{5}= f(x)$ for all x. Of course, from that $\displaystyle \frac{1}{2}f(x)= -\frac{3}{5}$ so that the function $\displaystyle f_0$ is defined by $\displaystyle f_0(x)= -\frac{6}{5}$ for all x.