Geodesic
On the formulation of boundary-value problems for binomial functional equations
Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 70 (2024) no. 3, pp. 343-355

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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.
Keywords: binomial functional equation, uniqueness of solution, well-posed boundary-value problem. 
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A. B. Antonevich; D. I. Kravtsov. On the formulation of boundary-value problems for binomial functional equations. Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 70 (2024) no. 3, pp. 343-355. http://geodesic.mathdoc.fr/item/CMFD_2024_70_3_a0/