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Convoluted $$C$$-operator families and abstract Cauchy problems. (English) Zbl 1199.47182
Let $$A$$ be a closed operator and $$K\in L^1_{\text{loc}}([0,t))$$, $$0<t\leq\infty$$. The authors define when $$A$$ is a subgenerator of a $$K$$-convoluted $$C$$-cosine function $$(C_K(t))_t$$ and when a strongly continuous operator family $$(S_K(t))_t$$ is a (local) $$K$$-convoluted $$C$$-semigroup having $$A$$ as a subgenerator. They prove the following: Assume that for each $$x\in R(C)$$ there exists a unique $$K$$-convoluted mild solution of $$(ACP_2)$$ at $$(Cx,0)$$, $$0<t\leq\infty$$. Then $$A$$ is a subgenerator of a K-convoluted C-cosine function on $$[0,t)$$.
##### MSC:
 47D09 Operator sine and cosine functions and higher-order Cauchy problems 47D60 $$C$$-semigroups, regularized semigroups 47D62 Integrated semigroups 34G10 Linear differential equations in abstract spaces