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.