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

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−∞(σp_i=1di (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.