Consider $N$ nonlinear equations for $N$-dimensional vector ${x}$: $F({x})=0$. $F:\mathbb{R}_{++}^{N}\rightarrow \mathbb{R}^{N}$ is continuous and differentiable, and specifically, following form:
\begin{equation} F_{i}(x) = C_{i}x_{i}^{-\alpha}\left(\sum_{n}D_{in}x_{n}^{-\beta}\right)-E_{i} \end{equation}
where $C_i$, $D_{in}$, $E_{i}$ are positive constants. $\alpha$ and $\beta$ are some parameters.
I am trying to show the sufficient conditions for the uniqueness of the solution to this system.
My idea is: the solution is unique if $\alpha>0$ and $\beta>0$, i.e.,
- $dF_{i}({x})/dx_{i} <0$ for any element $i$, and $dF_{i}({x})/dx_{j}<0$ for any $j\neq i$;
- $dF_{i}({x})/dx_{i} < dF_{i}({x})/dx_{j}<0$ for any pair of $i$ and $j$
because in the case of $N=2$, it leads that the Jacobian of $F$ is non-singular and the implicit function theorem can be applied to characterize the unique solution. I am considering these conditions can be generalized?
Any suggestions are very appreciated.
*** ADDED ***
Let us think the simplified version:
\begin{equation}
x_{i}^{\alpha} = \sum_{n}D_{in}x_{n}^{\beta}
\end{equation}
All elements of $D_{in}$ are strictly positive. Then, what conditions on $\alpha$ and $\beta$ are needed for the unique solution of $x$?