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\title[Nonlinear neutral integro-dynamic equations]{\large Periodicity and
positivity in nonlinear neutral integro-dynamic equations with variable delay}



\author[M. Belaid]{MALIK BELAID}


\author[A. Ardjouni]{ABDELOUAHEB ARDJOUNI}


\author[A. Djoudi]{AHCENE DJOUDI}



\begin{abstract}

Let $\mathbb{T}$ be a periodic time scale. We use Krasnoselskii's fixed
point theorem for a sum of two operators to show new results on the
existence of periodic and positive periodic solutions of the nonlinear
neutral integro-dynamic equation with variable delay of the form%
\begin{align*}
x^{\triangle }\left( t\right) & =-\int_{t-\tau \left( t\right) }^{t}a\left(
t,s\right) x\left( s\right) \Delta s+Q\left( t,x\left( t-\tau \left(
t\right) \right) \right) ^{\triangle } \\
& +G\left( t,x\left( t\right) ,x\left( t-\tau \left( t\right) \right)
\right) ,\text{ }t%
\begin{array}{c}
\in%
\end{array}%
\mathbb{T}.
\end{align*}%
We invert this equation to construct a sum of a contraction and a completely
continuous map which is suitable for applying Krasnoselskii's theorem. The
uniqueness results of this equation are studied by the contraction mapping
principle. The results obtained here extend the work of Mesmouli, Ardjouni
and Djoudi \cite{mm}.

\end{abstract}



\begin{amssubject}

34K13, 34K30, 34L30, 34K40.

\end{amssubject}



\begin{keyword}

Fixed points, integro-dynamic equations, periodic solutions, positive solutions, time scales.

\end{keyword}



\thanks{The authors would like to thank the anonymous referee for his valuable comments.}



\maketitle

\let\thefootnote\relax\footnote{ \ }\hspace{-5mm}\let\thefootnote\relax\footnote{\hspace{-13pt}DOI: 10.24193/mathcluj.2020.2.02}




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\end{thebibliography}



\vspace{10pt}

\hspace{-4mm}{\small{Received December 20, 2018}}\\
\hspace{-4mm}{\small{Accepted March 19, 2019}}



% ADDRESS
\vspace{-25.3pt}
\ \hfill
\begin{tabular}{c}
{\small\em University of Annaba}\\
{\small\em Department of Mathematics}\\
{\small\em P.O. Box 12, Annaba, Algeria}\\
{\small\em E-mail: {\tt malikbelaid51@gmail.com}} \\\\
{\small\em University of Souk Ahras}\\
{\small\em Department of Mathematics and Informatics}\\
{\small\em P.O. Box 1553, Souk Ahras, Algeria}\\
{\small\em E-mail: {\tt malikbelaid51@gmail.com}} \\\\
{\small\em University of Annaba}\\
{\small\em Department of Mathematics}\\
{\small\em P.O. Box 12, Annaba, Algeria}\\
{\small\em E-mail: {\tt adjoudi@yahoo.com}}
\end{tabular}



\end{document} 
