I have two systems of non-linear equations. The equations are continuously differentiable in the domain that I am interested in.
Let $x_1, y_1$ solve the following system of equations.
Let $x_2,y_2$ solve
Were $c>0$ is a constant. For simplicity and for starters we can assume that these solutions are unique. I have the functional forms of $f$ and $g$ and they are very long, painful and non-linear. I have some non-math intuition that the following are true and I would love some help and suggestions on the things I can try to do to prove these statements:
$x_2 > x_1 \\ y_2 > y_1 \\ x_2-x_1 > y_2-y_1 $
Thanks a lot,
Please feel free to make simplifying assumptions. For example, we can take $c>0$ to arbitrarily small for starters and see where that takes us.
I'm not sure if I can show this but suppose $f(x,y)$ and $g(x,y)$ satisfied the implicit function theorem then perhaps we could get some traction on this.