# Banach Fixed Point Theorem, System Has Solution

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.