Perturbation of fractional strongly continuous cosine family operators

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 $\mathcal{A}$ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $\mathcal{X}$, and $\mathcal{B}$ is a bounded linear operator in $\mathcal{X}$, then $\mathcal{A}+\mathcal{B}$ is also the infinitesimal generator of a fractional strongly continuous cosine family in $\mathcal{X}$. Our results coincide with the classical ones when $\alpha =2$.