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

Abstract

For $ν\in[0,1]$ and a complex parameter $σ,$ $Re\, σ>0,$ we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane $z\in\mathbb{C}$: \[ (a_{1}σ+a_{2}σ^ν)\mathcal{Y}(z+β,σ)-Ω(z)\mathcal{Y}(z,σ)=\mathbb F(z,σ), \quadβ\in\mathbb{R},\, β\neq 0, \] where $Ω(z)$ 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 $|z|\to +\infty$. 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.

Figures (1)

• Figure 1: Typical configurations of the contour $\ell_{d_{0}}$, $d_{0}\in[0,1]$.

Theorems & Definitions (38)

• Example 2.1
• Example 2.2
• Theorem 2.1
• Corollary 2.1
• Remark 2.1
• Remark 2.2
• Example 2.3
• Theorem 2.2
• Corollary 2.2
• Remark 2.3