Fixed Point Iteration Convergence

[Scootertaj]

Homework Statement

Consider the system
x = $\frac{1}{\sqrt{2}}$ * $\sqrt{1+(x+y)^2}$ - 2/3
y = x = $\frac{1}{\sqrt{2}}$ * $\sqrt{1+(x-y)^2}$ - 2/3

Find a region D in the x,y-plane for which a fixed point iteration

x_n+1 = $\frac{1}{\sqrt{2}}$ * $\sqrt{1+(x_n + y_n)^2}$ - 2/3

y_n+1 = $\frac{1}{\sqrt{2}}$ * $\sqrt{1+(x_n - y_n)^2}$ - 2/3

is guaranteed to converge to a unique solution for any (x₀,y₀)$\in$D

a) State clearly what properties this region must have
b) find a region with these properties and show it has these properties

Homework Equations

Seen above

The Attempt at a Solution

Not really sure where to start.
I don't know, in general, what properties are required.

 

[Eynstone]

Try to find bounds on x_(n+1) in terms of x_n ,y_n ( for instance, when it's less than x_n).

 

[Scootertaj]

Try to find bounds on x_(n+1) in terms of x_n ,y_n ( for instance, when it's less than x_n).

I'm confused as to where that leads :(.

Also, I realized there is a typo. There shouldn't be an "x =" in the second line.

 

[Scootertaj]

Try to find bounds on x_(n+1) in terms of x_n ,y_n ( for instance, when it's less than x_n).

Here is what I tried:

Assume x_n<y_n.

Then,

y_n+1 = $\frac{1}{\sqrt{2}}$*$\sqrt{1+(x_n-y_n)^2}$
= $\frac{1}{\sqrt{2}}$ * 1 - 2/3
= $\frac{3-2\sqrt{2}}{3\sqrt{2}}$

So, y_n+1 bounded above by $\frac{3-2\sqrt{2}}{3\sqrt{2}}$ ?

 

[Scootertaj]

amath.colorado.edu/courses/5600/2005fall/Tests/.../final-96ans.pdf

This problem seems similar (#7) but I don't quite understand it.