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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
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