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 {\large{\bf{INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES}}}
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E.Cuesta$^{(1)}$ and C.Palencia$^{(2)}$
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\noindent
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 {\small{(1)}} Dpto. Matem\'atica Aplicada a la T\'ecnica.\\
     Escuela Universitaria Polit\'ecnica.\\
     Universidad de Valladolid.\\
     47014 Valladolid, Spain.\\

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 {\small{(2)}} Dpto. Matem\'atica Aplicada.\\
     Facultad de Ciencias.\\
     Universidad de Valladolid.\\
     47005 Valladolid, Spain.\\


Integro-differential equations of the form
\begin{equation}
u(t)=\int_0^t
k(t-s,\alpha)Au(s)ds \label{ID}
\end{equation}
where $A$ is the infinitesimal generator of
a holomorphic semigroup $e^{tA}, t\geq 0$ of linear and bounded operators in
Banach spaces $X$.

A first order method for the semidiscretization in time of (\ref{ID}) is
proposed. Sufficient conditions guaranteeing contractivity or positivity are
provided.

 \begin{thebibliography}{2}

\bibitem{1}{\sc Y.Fuyita}, {\em Integro-Differential Equation
which Interpolates the Heat Equation and the Wave Equation}, Osaka J. Math.
27 (1990), pp. 319-327.

\bibitem{2}{\sc J.M.Sanz-Serna}, {\em A Numerical Method for a Partial
Integro-Differential Equation}, SIAM J. NUMER. ANAL.
(1988), pp. 319-327.

\end{thebibliography}
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