## 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
Full Text: