# Existence and uniqueness of solutions for neutral periodic integro-differential equations with infinite delay on time scale

Document Type : Research Articles

Authors

1 Department of mathematic, Faculty of Science, University of Benghazi, Benghazi, Libya

2 Department of Mathematics, Faculty of Science, University of Benghazi, Benghazi, Libya

3 College of Electrical and Electronic Technology, Benghazi, Libya

10.22080/cjms.2021.19819.1552

Abstract

In this article, we will shed the light on the following nonlinear neutral dynamic equation with infinite delay
\begin{eqnarray*}\label{e1}
{x(t)}^{\Delta }=&& G(t,\ x\left(t\right),x\left(t-\tau \left(t\right)\right))+{Q(t,x(t-\tau (t)))}^{\Delta }\\
&&+\int^t_{-\infty }{\left(\sum^p_{i=1}{D_i\left(t,s\right)}\right)f\left(x\left(s\right)\right)}\Delta s,
\end{eqnarray*}
where $\mathbb{T}$ is a periodic time scale. Using the fixed-point method by Krasnoselskii, we will show that equation has a periodic solution. In addition, we will prove this solution is unique by using the contraction mapping principle.

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