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

Abstract

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 \(z\in {{\mathbb {C}}}\):

$$\begin{aligned} (a_{1}\sigma +a_{2}\sigma ^{\nu })\mathcal {Y}(z+\beta ,\sigma )-\Omega (z)\mathcal {Y}(z,\sigma )={\mathbb {F}}(z,\sigma ), \quad \beta \in {\mathbb {R}},\, \beta \ne 0, \end{aligned}$$

where \(\Omega (z)\) 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 \(|z|\rightarrow +\infty \). 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.