Poisson Stable Solutions and Their Exponential Attractiveness for Difference Equations

We consider the Poisson stable solutions and their exponential attractiveness for the linear difference equation z(n+1)=Az(n)+g(n) and semi-linear difference equation z(n+1)=Az(n)+G(n,z(n)). Via Shcherbakov’s comparability principle, we show that if the forcing g (respectively, G) has some Poisson stable property, there is precisely one bounded solution that shares the same recurrence character as g(respectively, G) under appropriate assumptions. Moreover, the unique Poisson stable solution exponentially attracts every other solution.