On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

For $\nu \in [0,1]$ and a complex parameter $\sigma ,$ $Re\sigma >0,$ we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane $z\in \mathbb{C}$: $$(a_1\sigma +a_2{\sigma }^{\nu })\mathcal{Y}(z+\beta ,\sigma )-\mathrm{\Omega }(z)\mathcal{Y}(z,\sigma )=\mathbb{F}(z,\sigma ),\beta \in \mathbb{R},\beta \mathrm{\ne }0,$$ where $\mathrm{\Omega }(z)$ and $\mathbb{F}(z)$ are given complex functions, while $a_1$ and $a_2$ are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as $\mathrm{\mid }z\mathrm{\mid }\to +\mathrm{\infty }$. Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed.