-1
$\begingroup$

Let $g(x) =\frac{2}{3} (x+\frac{1}{x^2})$, show that $g$ has a unique fixed point $r\in [1,2]$. what is it?

If $x_0∈[1,2]$, show that the functional iterates converge to $r$, $x_n→r$, and give the rate of convergence?

help me understand this i am pretty sure it's a contractive mapping theorem question but i am having a hard time understanding the procedure.

$\endgroup$
  • $\begingroup$ Tell us how far you got in computing $\frac{g(y)-g(x)}{y-x}$. $\endgroup$ – LutzL Oct 29 '17 at 5:32
1
$\begingroup$

Contraction mapping theorem? Don't use bombs to kill ants. By definition of a fixed point, we have to solve $$ x = g(x) = \frac23 \left(x + \frac{1}{x^2} \right), $$ so we have to solve $$ x^3 = 2. $$ This equation a unique solution in $[1,2]$, namely $\sqrt[3]{2}$.

$\endgroup$
  • $\begingroup$ If x_0∈[1,2], show that the functional iterates converge tor, x_n→r, and give the rate of convergence? $\endgroup$ – August Haze Oct 28 '17 at 20:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.