Perturbation of fractional strongly continuous cosine family operators

TL;DR

This paper establishes perturbation results for infinitesimal generators of fractional strongly continuous cosine families, extending classical results to fractional orders in Banach spaces.

Contribution

It provides new sufficient conditions under which the sum of a generator and a bounded operator remains a generator of a fractional cosine family.

Findings

Perturbation results for fractional cosine families are proven.

Results generalize classical cases when fractional order alpha equals 2.

Conditions ensure stability of fractional cosine families under perturbations.

Abstract

Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that $\mathscr{A}$ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $\mathcal{X}$, and $\mathscr{B}$ is a bounded linear operator in $\mathcal{X}$, then $\mathscr{A}+\mathscr{B}$ is also the infinitesimal generator of a fractional strongly continuous cosine family in $\mathcal{X}$. Our results coincide with the classical ones when $\alpha=2$.