A Hilbert space approach to fractional difference equations

2018/12/19 by Phạm Thế Anh, Anh, Pham The, Artur Babiarz +13
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Dynamical Systems (math.DS) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.1812.07957

openalex publication_date 2018/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We formulate fractional difference equations of Riemann-Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator 1 - τ^-1 with the right shift τ^-1 on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.

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