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Regularized resolvent families. (English) Zbl 1157.45006

So called \(C\)-regularized resolvent families for the Volterra-type equation
\[ u(t) = f(t) + \int_0^t a(t-s)Au(s)\,ds, \quad t\geq 0 \]
with \(A\) a closed linear operator on a complex Banach space \(X\), \(a \in L_{\text{loc}}^1(\mathbb R^+)\) and \(f:\mathbb R^+\to X\) are introduced and studied. Analyticity criteria for such resolvent families are presented and examples are given.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45D05 Volterra integral equations
47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
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