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Translation semigroups in the space \(L^ 1([-r,0],X)\). (Italian. English summary) Zbl 0563.47029

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.
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 \(L^ 1([-r,0],X).\)
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 [G. Da Prato, Applications croissantes et équations d’evolutions dans les espaces de Banach (1976; Zbl 0352.47002)] and [R. Villella-Bressan, Functional differential systems and related topics, Proc. 2nd Int. Conf., Blazejewko/Pol. 1981, 328-333 (1981; Zbl 0536.47041)].

MSC:

47D03 Groups and semigroups of linear operators
34G10 Linear differential equations in abstract spaces
47B44 Linear accretive operators, dissipative operators, etc.
47E05 General theory of ordinary differential operators
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References:

[1] G. Da Prato , Applioation croissantes et équations d’evolutions dans les espaces de Banach, Istitutiones Matematicae , 2 Academic Press ( 1976 ). MR 500309 | Zbl 0352.47002 · Zbl 0352.47002
[2] J. Dyson - R. Villella Bressan , Semigroups of translations associated with functional and functional differential equations , Proc. Roy. Soc. Edinburgh , 82A ( 1978 ). MR 532900 | Zbl 0419.47032 · Zbl 0419.47032
[3] J. Hale , Theory of functional differential equations , Springer , Berlin ( 1977 ). MR 508721 | Zbl 0352.34001 · Zbl 0352.34001
[4] A.T. Plant , Non linear semigroups of translations in Banach space generated by functional differential equations , J. Math. Anal. Appl. , 60 ( 1977 ), pp. 67 - 74 . MR 447745 | Zbl 0366.47021 · Zbl 0366.47021
[5] R. Villella-Bressan , An abstract functional equations in spaces of continuous funotions , Proceedings of the Second International Conference of Functional Differential Systems and Related Topics II, Zelona Gora ( 1981 ), pp. 328 - 333 . Zbl 0536.47041 · Zbl 0536.47041
[6] R. Villella-Bressan , Functional equations of delay type in L1 spaces , da apparire. Zbl 0579.34056 · Zbl 0579.34056
[7] R. Villella-Bressan , Appunti di Lezione di Teoria delle funzioni, Anno Accademico 1981 -92, Università di Padova .
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