In short, I had this problem that required me to solve a system of simultaneous linear equations which turned out to be of the form $$w_j = \frac{1}{n_j}\sum_{k=1}^N \frac{n_{jk}c_{jk}}{1-c_{jk}}w_{k},$$
where $w_j$ is the only unknown for $j = 1,\dots, N$. We can formulate it as an eigenvalue problem by defining $A_{jk} = \frac{n_{jk}c_{jk}}{1-c_{jk}}$ which means $A \boldsymbol w = \boldsymbol w$. Nice and clean.
But then I altered my mathematical model. So this is a new problem which is independent of the first. I only included the first to illustrate the analogy to an easy linear eigenvector problem. This is the system of nonlinear equations that the new model needs to satisfy:
$$\begin{align} w_j = \frac{1}{n_j}\sum_k \frac{n_{jk}w_k c_{jk}}{2w_k c_{jk}-w_k-c_{jk}}. \end{align} $$
So, $w_k$ shows up in both numerator and denominator, which means I can't view this sum as a scalar product. But it sort of has a similar flavor to the easy problem above. Is this just a case of "standard" satisfyability of nonlinear systems problem, or does this possess more structure? Is it an instance of a well-studied problem? Any references appreciated.
references: a book on scientific computing, many techniques, clear examples... scientific computing an introductory survey by micheal heath
you must linearize your termz. produce final eqn that looks like Ax=b solve using likely approximate method of your choice. also if all of right hand side is known... then like whatever? your eqn here is just a constraint. you must read more to understand and write your question better.