Consider a system of $n$ nonlinear equations in $n$ variables, given by
\begin{align*} f_1(x_1, x_2, \dots, x_n) &= 0 \ f_2(x_1, x_2, \dots, x_n) &= 0 \ &\vdots \ f_n(x_1, x_2, \dots, x_n) &= 0. \end{align*}
Under what conditions can we prove the existence and uniqueness of a solution to this system?
I am aware of the implicit function theorem, which can be used to prove the existence and uniqueness of a solution in the case where the Jacobian matrix of the system is non-singular. However, this requires the functions $f_i$ to be continuously differentiable, which may not always be the case. Is there a more general result that holds for arbitrary (possibly non-differentiable) functions $f_i$?
Best Answer
if the system is from $R^n\rightarrow R^n$ and if it is lipschitz with $d<1$ then there exists a unique fixed point.
let $f_i(x_1,...,x_n)+x_i=x_i,\;i\in\{1,...,n\}$
Then if
$\left(|f_i(x_1-y_1,...,x_n-y_n)|\leq |d|||\mathbf{x}-\mathbf{y}||\right) \land d<1$
the system has a unique fixed point by Banach fixed point theorem.