When a solution of a non-linear system exist?

Question

I was trying to solve a problem and end up with the following non-linear system $$\left\{\begin{array}{lll} a_{11}e^{x_1}+ \ldots+ a_{1n}e^{x_n}=f_1(x_1,\ldots,x_n)\\ \vdots\\ a_{n1}e^{x_1}+\ldots+a_{nn}e^{x_n}=f_n(x_1,\ldots,x_n) \end{array}\right.$$ where $f_i:\mathbb{R}^n \to \mathbb{R}$ is smooth, for all $i=1,\ldots,n$. I know that the real matrix $A:=(a_{ij})$ is invertible and I was wandering if this system has real solution. I am searching for methods of solving non-linear system, if somebody could give me some tip about this particular problem I would appreciate very much!

Answer 1

Your system has the form

$$\left\{\begin{array}{lll} g_1(x_1,\ldots,x_n)=0\\ \vdots\\ g_n(x_1,\ldots,x_n)=0 \end{array}\right.$$ and is just a general nonlinear system, for which there is no general method. Unless your $f_n$ have something very special (smoothness tells little).