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\begin{document}

\title{On the integro-differential equations with reflection}



\author[1]{El Hadi Ait Dads}%
\author[2]{Safoua Khelifi}%
\author[3]{Mohsen MIRAOUI}%
\affil[1]{Université Cadi Ayyad, Faculté des Sciences Semlalia B.P. 2390}%
\affil[2]{Université de Sfax Faculté des Sciences de Sfax}%
\affil[3]{IPEI of Kairouan}%


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  \date{March 17, 2025}


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\begin{abstract}
There are a few purely periodic phenomena in nature, which allows one to
consider several other generalizations, such as almost automorphic and
measure pseudo almost automorphic oscillations. In this paper, by
developing important properties on the composition of functions with
reflection, using some exponential dichotomy properties and an
application of the fixed-point theorem, several new sufficient
conditions for the existence and the uniqueness of an pseudo almost
automorphic solutions with measure for some general type reflection
integro-differential equations. We suppose that the nonlinear part is
measure pseudo almost automorphic and in which we distinguish the two
constant and variable cases for the lipschitz coefficients of the
functions associated with this part. It is assumed that the linear part
of the equation considered admits an exponential dichotomy. Finally, an
application is given on the very interesting model of Markus and Yamabe.%
\end{abstract}%



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