On the formulation of boundary-value problems for binomial functional equations

In a number of previous works it was found that for binomial functional equations of the form
$$ \hspace{-1.5cm} a(x)u(\alpha(x)) - \lambda u(x) = v(x), x \in X, $$
where $\alpha:X \to X$ is an invertible mapping of the set $X$ into itself, a situation typical for differential equations is possible: the equation is solvable for any right-hand side and there is no uniqueness of the solution. As in the case of differential equations, the question arises of formulating well-posed boundary value problems, i.e., of specifying additional conditions under which the solution exists and is unique. In this paper, we discuss the question of what kind of additional conditions lead to well-posed boundary-value problems for the equations under consideration.