I have a system of two equations(non linear). $\phi_1(x_1,x_2)=0$ and $\phi_2(x_1,x_2)=0$. ($x_1,x_2 \in R)$ Substitution is not possible since both the equations are highly non linear. I was looking for a way to prove that a solution exists.(Numerical and graphical analysis seem to suggest that a solution exist).But i was unable to find any theorem that I can apply for proving an existence of solution. I read that the Banach fixed point theorem is sometimes used to show that solution will exist. But i am unable to see how I can use it since my function is defined implicitly. Here is how i went about it.
Defining $B_1(x_2)$ and $B_2(x_1)$ as the best response function, the problem becomes equivalent to finding the an $x^*$ such that $(B_1(x^*_2),B_2(x^*_1))-(x^*_1,x^*_2)=0$.I tried to move forward by trying to prove that the best response functions are concave. Hence there would $\exists \lambda \in(0,1)$ such that ||B(y)-B(x)||<$\lambda||y-x||$ for all y and x $\in R^2$. I am not sure if this is a valid approach. Is there any other result that i can use to prove the existence of a solution for such a non linear system.
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