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

Abstract

In this article, we will shed the light on the following nonlinear neutral dynamic equation with infinite delay
x(t)Δ=G(t, x(t),x(t−τ(t)))+Q(t,x(t−τ(t)))Δ+∫−∞t(∑i=1pDi(t,s))f(x(s))Δs,
where 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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