| Abstract: |
| This work is initially motivated motivated by the work of B\`{a}tkai and Piazzera \cite{BP2005,BP2001}. They studied partial differential equations with finite delay using an operator theoretical approach by means of $C_0$-semigroups. The theory of strongly continuous one-parameter operator semigroups is well-developed. However, also this theory has its limitations as one deals with operators on a given Banach space.
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A possible generalization of the concept of $C_0$-semigroups on Banach spsaces are operator semigroups on locally convex spaces. There has been a lot of research regarding the formulation of operator semigroup theory on those spaces. The class of locally equicontinuous semigroups has been investigated for example by Dembard \cite{D1974}, Babalola \cite{Ba1974}, Ouchi \cite{O1973} and Komura \cite{Ko1968}. Here, one has to be a little bit more careful when it comes to resolvents as one only can work with so-called asymptotic resolvents as the Laplace transform does not need to converge.
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We show that one can combine equations with both finite and infinite delay in one theory. Evolution equations with infinite delay have been explored on their own by several authors not only on Banach spaces but also on Frechet spaces. However, it is worth mentioning, that Picard, Trostorff and Waurick showed \cite{PTW2014} that they do not need different treatment.
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This is joint work with C.~Seifert (Technical University Hamburg, Germany).
\begin{thebibliography}{10}
\bibitem{Ba1974}
V.~A. Babalola.
\newblock Semigroups of operators on locally convex spaces.
\newblock {\em Trans. Am. Math. Soc.}, 199:163--179, 1974.
\bibitem{BP2001}
A.~B{\`a}tkai and S.~Piazzera.
\newblock Semigroups and linear partial differential equations with delay.
\newblock {\em J. Math. Anal. Appl.}, 264(1):1--20, 2001.
\bibitem{BP2005}
A.~B{\`a}tkai and S.~Piazzera.
\newblock {\em Semigroups for delay equations}, volume~10 of {\em Res. Notes
Math.}
\newblock Wellesley, MA: A K Peters, 2005.
\bibitem{D1974}
B.~Dembart.
\newblock On the theory of semigroups of operators on locally convex spaces.
\newblock {\em J. Funct. Anal.}, 16:123--160, 1974.
\bibitem{Ko1968}
T.~Komura.
\newblock Semigroups of operators in locally convex spaces.
\newblock {\em J. Funct. Anal.}, 2:258--296, 1968.
\bibitem{O1973}
S.~Ouchi.
\newblock Semi-groups of operators in locally convex spaces.
\newblock {\em J. Math. Soc. Japan}, 25:265--276, 1973.
\bibitem{PTW2014}
R.~Picard, S.~Trostorff, and M.~Waurick.
\newblock A functional analytic perspective to delay differential equations.
\newblock {\em Oper. Matrices}, 8(1):217--236, 2014.
\end{thebibliography} |
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