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arXiv · 1812.07957

A Hilbert space approach to fractional difference equations

Pham The Anh·Artur Babiarz·Adam Czornik·Konrad Kitzing·Michał Niezabitowski·Stefan Siegmund·Sascha Trostorff·Hoang The Tuan
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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 $1 - \tau^{-1}$ with the right shift $\tau^{-1}$ on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.

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Pham The Anh, Artur Babiarz, Adam Czornik, Konrad Kitzing, Michał Niezabitowski, Stefan Siegmund, Sascha Trostorff, Hoang The Tuan. 2018-12-19. A Hilbert space approach to fractional difference equations. https://arxiv.org/abs/1812.07957

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