Existence and uniqueness of periodic solutions in neutral nonlinear summation-difference systems with infinite delay

Abstract

We use Krasnoselskii’s fixed point theorem to show that the neutral nonlinear summation-difference system with infinite delay $\mathrm{\Delta }x\left(n\right)=P\left(n\right)+A\left(n\right)x\left(n-\tau \left(n\right)\right)+\mathrm{\Delta }Q\left(n,x\left(n-g\left(n\right)\right)\right)+{\sum }_{k=-\infty }^{n}D\left(n,k\right)f\left(x\left(k\right)\right)$ has a periodic solution. We also use the contraction mapping principle to show that the periodic solution is unique. An example is given to illustrate our results.