On perturbation of convoluted \(C\)-regularized operator families (Q1951076)

A strongly continuous exponentially bounded family of linear operators \(\{T_K(t): t \geq 0\}\) in a Banach space \(E\) is called a \textit{\(K\)-convoluted \(C\)-semigroup} with \textit{subgenerator \(A\)} if \[ A\int_0^t T_K(s) u \,ds = T_K(t) u - \left(\int_0^t K(s)\,ds\right) Cu, \quad t \geq 0, \;u \in E, \] where \(C\) is a bounded injective operator in \(E\) and \(K(t)\) is an exponentially bounded scalar function; it is also assumed that \(T_K(t)C = CT_K(t)\) and that \(T_K(t)A \subset AT_K(t)\) for \(t \geq 0.\) The semigroup \(T_K(t)\) is associated with the equation \(u'(t) = Au(t) + K(t)Cu.\) There is a corresponding definition for cosine functions. These objects are used in studying incomplete first and second order Cauchy problems. The objective of this paper is to prove perturbation results, both multiplicative and additive, for \(K\)-convoluted \(C\)-semigroups and cosine functions.