%0 Journal Article
%T Periodicity and Stability in Nonlinear Neutral Dynamic Equations with Infinite Delay on a Time Scale
%J Khayyam Journal of Mathematics
%I 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)
%Z 2423-4788
%A Ardjouni, Abdelouaheb
%A Djoudi, Ahcene
%D 2016
%\ 01/01/2016
%V 2
%N 1
%P 51-62
%! Periodicity and Stability in Nonlinear Neutral Dynamic Equations with Infinite Delay on a Time Scale
%K fixed point
%K infinite delay
%K time scales
%K periodic solution
%K Stability
%R 10.22034/kjm.2016.16711
%X 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 \[ x^{\Delta}(t)=-a(t)x^{\sigma}(t)+\left( Q(t,x(t-g(t))))\right) ^{\Delta }+\int_{-\infty}^{t}D\left( t,u\right) f\left( x(u)\right) \Delta u,\ t\in\mathbb{T}, \] 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].
%U http://www.kjm-math.org/article_16711_9ebf94916138dbc647391b26cc4d1c8d.pdf