Suppose that we have, $\forall i \in N$, where $|N|<\infty$: \begin{gather*} u_i=x-\left[(1-\rho)\cdot\left(\frac{1}{|N|}\sum_{j\in N}^{j\neq i}u_j+\frac{1}{|N|}\left(x-\alpha_i\right)\right)+\rho\cdot\mu_i\right]\\ u_j=x-\left[(1-\rho)\cdot\left(\frac{1}{|N|}\sum_{i\in N}^{i\neq j}u_i+\frac{1}{|N|}\left(x-\alpha_j\right)\right)+\rho\cdot\mu_j\right]\\ ...\\ ...\\ ...\\ u_n=x-\left[(1-\rho)\cdot\left(\frac{1}{|N|}\sum_{n\in N}^{i\neq n}u_n+\frac{1}{|N|}\left(x-\alpha_n\right)\right)+\rho\cdot\mu_n\right] \end{gather*}
Further, $\rho\in(0,1]$; and everything except the vector $\mathbf{u}\equiv(u_i)_{i\in N}\equiv(u_i,u_j,...,u_n)$ is known. My goal is to identify each element in $\mathbf{u}$ as a function of the known parameters, but not as a function of other unknown elements of $\mathbf{u}$. Because we have $|N|$ equations and $|N|$ unknowns, I assume that the problem can be solved, but I just don't know how.
Thus, my question is two-fold: what is the solution to the problem above and how do we reach it? Additionally, could anybody please tell me what kind of problem is this and how one usually goes about facing it? Some external resources (for beginners) would also be really appreciated.
Thank you all very much in advance for your time and your effort.
You should write the system in matrix-vector form. That is you want to write it the form $Ax=b$ where $A$ is an $n\times n$ constant matrix, that is, every element of $A$ is known, $$x=\pmatrix{u_1\\u_2\\ \vdots\\u_n}$$ and $b$ is a column vector of $n$ elements.
Then if $A$ is invertible, $\boxed{x=A^{-1}b}$ If $A$ is not invertible, then either there is no solution, or there are infinitely many.
It is not usually possible to find a closed-form formula for $A^{-1}$ except in very special cases. If you are looking for a formula, I suggest you try to find one in the case $n=2$ to start and see if you get any insight. If you are simply interested in a numerical solution, then there are lots of computer systems that will compute the answer for you, even when n is rather large, say $n=1000,$ or probably much more these days.
I don't mean to imply that it's necessary to compute the inverse in order to solve the system. There are more efficient ways. I would expect that finding a closed-form formula for the solution would entail finding a formula for the inverse, but I suppose it might be possible without that.
It's hard to give you any more advice unless you give more context. What exactly do you want to do?