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/

[1] A. B. Antonevich, Linear Functional Equations: Operator Approach, Universitetskoe, Minsk, 1988 (in Russian)

[2] A. B. Antonevich, “Coherent local hyperbolicity of a linear extension and essential spectra of the weighted shift operator on a segment”, Funct. Anal. Appl., 39:1 (2005), 52–69 (in Russian)

[3] A. B. Antonevich, “Right-sided invertibility of binomial functional operators and graded dichotomy”, Contemp. Math. Fundam. Directions, 67, no. 2, 2021, 208–236 (in Russian)

[4] A. B. Antonevich, A. A. Akhmatova, and Yu. Makovska, “Mappings with separable dynamics and spectral properties of the operators generated by them”, Math. Digest, 206:3 (2015), 3–34 (in Russian)

[5] A. B. Antonevich and E. V. Panteleeva, “Well-posed boundary-value problems, right-sided hyperbolicity and exponential dichotomy”, Math. Notes, 100:1 (2016), 13–29 (in Russian)

[6] O. A. Arkhipenko, “Boundary-value problems for difference equations”, Proc. BGTU. Ser. 3. Phys. Math. Sci. Inf., 2018, no. 1, 12–18 (in Russian)

[7] Yu. I. Karlovich and R. Mardiev, “On one-sided invertibility of functional operators with non-Carleman shift in Hölder spaces”, Bull. Higher Edu. Inst. Ser. Math., 3 (1987), 77–80 (in Russian)

[8] Ya. B. Lopatinskii, “On a method of reducing boundary-value problems for a system of differential equations of elliptic type to regular integral equations”, Ukr. Math. J., 5 (1953), 123–151 (in Russian)

[9] R. Mardiev, “A criterion for the semi-Noetherian property of a class of singular integral operators with non-Carleman shift”, Rep. Acad. Sci. UzSSR, 2 (1985), 5–7 (in Russian)

[10] A. Shukur Ali and O. A. Arkhipenko, “Resolvent of a boundary-value problem for a difference equation”, Probl. Phys. Math. Tech., 2016, no. 3(28), 70–75 (in Russian)

[11] Antonevich A., Makowska Yu., “On spectral properties of weighted shift operators generated by mappings with saddle points”, Complex Anal. Oper. Theory, 2 (2008), 215–240

[12] Karlovich A. Yu., Karlovich Yu. I., “One-sided invertibility of binomial functional operators with a shift in rearrangement-invariant spaces”, Integral Equ. Oper. Theory, 42 (2002), 201–228