How do i show that this system of nonlinear equations can be solved on $D:=\{(x,y)\in \mathbb R^2 : |x|,|y| \leq \frac{1}{2} $} $$\frac{1}{2}x^2-x- \frac{1}{2}y^2+\frac{3}{8}$$ $$\frac{1}{2}x^2-y+ \frac{1}{2}y^2-\frac{1}{2}$$
I was thinking on using Banach fixed-point theorem but im not really sure on how to deal with this.
The difference of both equations, $$ 0=-x+y+\frac38-\frac12 = y-x-\frac18 $$ allows you to eliminate one variable and reduce the problem to a univariate quadratic equation.
See the problem on WolframAlpha for a quick confirmation of the result. You will see that one of the solutions is at infinity, that is, the substitution eliminates all quadratic terms.