Regarding $\displaystyle a_n=A_n + B_n \sum_{j=-\infty}^{+\infty} a_j a_{n-j}$, an infinite system of non-linear algebraic equations

Question

Solving a PDE, I ran into an infinite system of equations of the form

$$a_n=A_n + B_n \sum_{j=-\infty}^{+\infty} a_j a_{n-j}, \qquad n \in \mathbb{Z}$$

where $A_n$ and $B_n$ and known coefficients while the $a_n$'s are the unknowns.

I wonder if

• Conditions may be set to ensure that at least a solution $\{a_n\}$ exists?

and

• How to compute it?

In the literature for linear systems I couldn't find any hint for this type of equations. Thank you for any suggestion.
Dario