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Maximal B-regular integro-differential equation. (English) Zbl 1187.45008

By using the Fourier transform, the authors establish maximal regularity of the convolution differential operator equation

$\sum _{k=0}^{i}{a}_{k}*\frac{{d}^{k}u}{d{x}^{k}}+A*u=f\left(x\right),$

in an $E$-valued Besov space. Here, $E$ is an arbitrary Banach space, $A=A\left(x\right)$ is an operator on $E$, ${a}_{k}={a}_{k}\left(x\right)$ are complex-valued functions. It is shown that the differential operator generated by this equation is the generator of an analytic semigroup. The result is used to establish maximal regularity for infinite systems of integro-differential equations.

##### MSC:
 45J05 Integro-ordinary differential equations 45F05 Systems of nonsingular linear integral equations
##### References:
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