Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47882120160101Periodicity and Stability in Nonlinear Neutral Dynamic Equations with Infinite Delay on a Time Scale51621671110.22034/kjm.2016.16711ENAbdelouahebArdjouniDepartment of Mathematics and Informatics, University of Souk Ahras, P.O.Box 1553, Souk Ahras, 41000, Algeria.AhceneDjoudiDepartment of Mathematics, University of Annaba, P.O. Box 12, Annaba
23000, Algeria.Journal Article20160221Let $\mathbb{T}$ 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 \[ x^{\Delta}(t)=-a(t)x^{\sigma}(t)+\left( Q(t,x(t-g(t))))\right) ^{\Delta }+\int_{-\infty}^{t}D\left( t,u\right) f\left( x(u)\right) \Delta u,\ t\in\mathbb{T}, \] 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].http://www.kjm-math.org/article_16711_9ebf94916138dbc647391b26cc4d1c8d.pdf