- #1

mathmari

Gold Member

MHB

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For which values of $a\neq 0$ can we solve iteratively $ax^3-x-2=0$ by $x_{n+1}=ax_n^3-2, \ n=1,2, \ldots $ with appropriate $x_1$ ?

I have done the following:

$$ax^3-x-2=0 \Rightarrow x=ax^3-2$$ So we can consider a fix point problem with $\phi (x)=ax^3-2$.

So we have to check when the fix point iteration converges right?

For that $\phi$ has to be Lipshitz continuous, or not? :unsure: