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 $z\in \mathbb{C}$: \[ (a_{1}σ+a_{2}σ^ν)\mathcal{Y}(z+β,σ)-Ω(z)\mathcal{Y}(z,σ)=\mathbb F(z,σ), \quadβ\in\mathbb{R},\, β\neq 0, \] where $\Omega (z)$ 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 $|z|\to +\infty$. 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.

References in corpus (1)

• A dynamical trichotomy for structured populations experiencing positive density-dependence in stochastic environments