Translation semigroups in the space \(L^ 1([-r,0],X)\) (Q1057465)

The theory of functional equations can be led to the general theory of abstract differential equations in Banach spaces through the method of semigroups of operators. The connexion with the functional equation consists in conditions on the domain \(D_ A\) of the operator \(A: D_ A\subset Y\to Y\), \(Au=-u'\), where Y is the space of initial data. In this study arises the problem of proving that the opertor A, if it is m- accretive, generates a semigroup of translations. So semigroups of translations have an important role in this kind of questions. \textit{A. T. Plant} proved that in the space C([-r,0],X) the derivative operator, if it is m-accretive, always generates a semigroup of translations [J. Math. Anal. Appl. 60, 67-74 (1977; Zbl 0366.47021)]. In this work we prove that the same operator, if it is m-accretive in the space \(L^ 1([-r,0],X)\), generates a semigroup of translations in \(L^ 1([-r,0],X).\) \textit{R. Villella-Bressan} had already proved this results in a particular case of a functional equation with initial data in \(L^ 1([-r,0],X)\) [Functional equations of delay type in spaces, to appear]. She related the semigroup generated by A in \(L^ 1([-r,0],X)\) to that one generated in C([-r,0],X) by the operator \(\tilde A,\) \(D_{\tilde A}=D_ A\cap C^ 1([-r,0],X)\); in this paper we prove that this method can be extended to the general case. Such an extension has been possible thanks to a characterization for the accretivity in \(L^ 1([-r,0],X)\) of the derivative operator, obtained using some results in [\textit{G. Da Prato}, Applications croissantes et équations d'evolutions dans les espaces de Banach (1976; Zbl 0352.47002)] and [\textit{R. Villella-Bressan}, Functional differential systems and related topics, Proc. 2nd Int. Conf., Blazejewko/Pol. 1981, 328-333 (1981; Zbl 0536.47041)].