In order to prove the existence of a solution to the equation $f_1(x)-c_1(x) = 0$, I have shown that: \begin{align*} \lim_{x \rightarrow 0} f_1(x) &> \lim_{x \rightarrow \bar{x}} f_1(x) \\ \lim_{x \rightarrow 0} c_1(x) &< \lim_{x \rightarrow \bar{x}} c_1(x) \\ \frac{d f_1(x)}{d x}&<0 \\ \frac{d c_1(x)}{d x}&>0 \end{align*} That is, some version of the intermediate value theorem.
Now, I have two endogenous variables (x,y) and I want to prove that a solution to \begin{align*} f_1(x,y)-c_1(x,y) = 0 \\ f_2(x,y)-c_2(x,y) = 0 \end{align*}
Is there a similar approach I can use? I have also heard about Kakutani's fixed point theorem, but I do not now how to use it.
Thanks for your help!!