Periodicity and Stability in Nonlinear Neutral Dynamic Equations with Infinite Delay on a Time Scale

Abstract

Let $mathbb{T}$ be a periodic time scale. We use a fixed point theorem due to Krasnoselskii to show that the nonlinear neutral dynamic equation with infinite delay [ x^{Delta}(t)=-a(t)x^{sigma}(t)+left(  Q(t,x(t-g(t))))right)  ^{Delta }+int_{-infty}^{t}Dleft(  t,uright)  fleft(  x(u)right)  Delta u, tinmathbb{T}, ] has a periodic solution. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle. Also, by the aid of the contraction mapping principle we study the asymptotic stability of the zero solution provided that $Q(t,0)=f(0)=0$. The results obtained here extend the work of Althubiti, Makhzoum and Raffoul [1].