On well-posedness of vector-valued fractional differential-difference equations

TL;DR

This paper introduces an operator-theoretical approach to analyze the well-posedness of vector-valued fractional differential-difference equations, establishing existence and uniqueness of solutions under mild conditions.

Contribution

It develops a novel method inspired by the Poisson distribution to handle fractional operators in difference equations, extending the analysis to Banach space-valued functions.

Findings

Proves existence and uniqueness of solutions.

Applies the method to weighted Lebesgue spaces.

Provides original examples illustrating the results.

Abstract

We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n) &= Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; u(0) &= u_0; u(1) &= u_1, \end{array} \right. \end{equation*} where $A$ is an closed linear operator defined on a Banach space $X$. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on strongly continuous sequences of bounded operators generated by $A,$ and natural restrictions on the nonlinearity $f$. Finally we present some original examples to illustrate our results.