Perturbation properties of fractional strongly continuous cosine and sine family operators

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 $A$ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $X$, and $B$ is a bounded linear operator in $X$, then $A+B$ is also the infinitesimal generator of a fractional strongly continuous cosine family in $X$. Our results coincide with the classical ones when $\alpha =2$. Furthermore, depending on commutativity condition of linear bounded operators, we propose the elegant closed-form formulas for uniformly continuous perturbed fractional operator cosine and sine functions. Finally, we present an example in the context of one-dimensional perturbed fractional wave equation to demonstrate the applicability of our theoretical results and we give some comparisons with the existing literature.