Using Banach's Fixed Point Theorem, show that the following system has at least one solution:

$$ x = 0.000001x^2 + 10\sin y + 1 $$ $$y = 0.000001y^3 - 0.01\cos x - 1 $$

Here is what I have tried:

Consider the function defined by
$$ f(x,y) = (0.000001x^2 + 10\sin y + 1, 0.000001y^3 - 0.01\cos x - 1) $$ and show that it has a fixed point. If it does, then that fixed point is clearly a solution to our system.

I used the standard Euclidean distance between two points and attempted to show that
$$ d(f(x_1, y_1), f(x_2, y_2)) ≤ d((x_1, y_1), (x_2, y_2)) $$

In doing so, I greatly struggle to manipulate the left-hand side so that it is clearly less than or equal to the right-hand side of the inequality above. I tremendously appreciate any advice, and if there is a smarter way to approach this, I appreciate any hints you are willing to offer. Thank you.


Idea: the mean value theorem shows that if $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f'(x)\leq M$ on $(a,b)$, then $|f(x)-f(y)|\leq M|x-y|$ on $(a,b)$. See if you can adapt that to a function of two variables.


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