Perturbation theory for fractional evolution equations in a Banach space

TL;DR

This paper develops a perturbation theory for fractional evolution equations of order between 1 and 2 in Banach spaces, demonstrating their stability under bounded time-dependent perturbations and extending the mathematical framework for modeling physical processes.

Contribution

It introduces a novel perturbation theory for fractional evolution equations with order in (1,2], showing stability of solutions under bounded time-dependent perturbations.

Findings

Fractional evolution equations of order in (1,2] are well-posed under bounded perturbations.

The study extends the theory of fractional cosine functions in Banach spaces.

Results applicable to modeling complex physical systems with fractional dynamics.

Abstract

A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order $\alpha\in (1,2]$ associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator $A$ of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of $A$. We also provide some necessary special cases.