On perturbation of convoluted \(C\)-regularized operator families. (English) Zbl 1286.47027
A strongly continuous exponentially bounded family of linear operators \(\{T_K(t): t \geq 0\}\) in a Banach space \(E\) is called a \(K\)-convoluted \(C\)-semigroup with 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.
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.
Reviewer: Hector O. Fattorini (Los Angeles)
MSC:
| 47D60 | \(C\)-semigroups, regularized semigroups |
| 34G10 | Linear differential equations in abstract spaces |
Keywords:
convoluted \(C\)-semigroups; convoluted \(C\)-cosine functions; subgenerators; incomplete Cauchy problems; multiplicative perturbations; additive perturbationsReferences:
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