For a numerical optimization algorithm that I'm currently developing for my research, I am facing the following challenge for which any help is very much appreciated!
Let $w_{ij} \in \mathbb{R}_+$ be constants and consider the following system.
$$x_i - 1 = \sum_{j=1}^n w_{ij} x_j e^{-x_j}, \hspace{1cm} i=1,\dots,n,$$
with $x_i > 0$. Assume that the $w_{ij}$'s are such that a unique solution exists and let this solution be $x^*$.
Now, if we perturb the $w_{ij}$'s, e.g., $w'_{ij} = w_{ij} + \varepsilon$, can we bound the region in which the new solution $x'$ must lie? That is, can we find an $\eta(\varepsilon)$ such that:
$$|x^*_i - x'_i| \leq \eta(\varepsilon).$$
Informally stated, given that we change the system's parameters a little bit, can we say something about the 'distance' from the old to the new solution?
More generally, what if we perturb the system such that $|w'_{ij} - w_{ij}| < \varepsilon$ for all $i,j$?
Thanks once more.