ﬁnding a ﬁxed-point ˆ x∈R for the function

Question

I don't know how to verify this function if I can't use the Banach-Caccioppoli Contraction Principle. Please help

Try solving the equation $g(x)=x$. – EBP Jun 03 '20 at 10:16 — EBP, Jun 03 '20 at 10:16

score 0 · Answer 1 · answered Jun 03 '20 at 10:21

$g$ has a fixed point $\hat x$ if and only if $g(\hat x) = \hat x$, hence $$ g(\hat x) = \hat x \iff 1 + \hat x - \left(\frac{\hat x}{2} \right)^3 = \hat x \iff 1-\left(\frac{\hat x}{2} \right)^3 = 0 $$

Can you continue from here? Does exist $\hat x$? Is it unique?

score 0 · Answer 2 · answered Jun 03 '20 at 10:23

0

A fixed point for $g(x)$ should be a point $\hat{x}$ with the property that $g(\hat{x})=x$. I.e., we want to solve $$1+x-\frac{1}{8}x^3=x$$ for $x$. We rewrite this to $$1-\frac{1}{8}x^3=0,$$ and finally to $$x^3=-8.$$ It is clear that this last equation has $-2$ as its only solution.

answered Jun 03 '20 at 10:23

EBP

1,388

thank you it is very clear.could you also please help me for part c – Amy12 Jun 03 '20 at 10:38
Unfortunately not. – EBP Jun 03 '20 at 10:46

ﬁnding a ﬁxed-point ˆ x∈R for the function

2 Answers2