h-Stability for Difference Systems

Pinto introduced the notion of h-stability. This notion is very useful because, when we study the asymptotic stability of difference systems it is not easy to work with non-exponential types of stability. Medina and Pinto applied the h-stability to obtain a uniform treatment for the various stability notions in difference systems and gave new insights about stability for weakly stable difference systems. In this paper, first, we introduce the background, the current situation of studies for h-stability. Second, we give some definition about h-stability and some lemmas. Finally, we prove that the two concepts of h-stability and h-stability in variation are equivalent via n_∞ - imilarity and Lyapunov function.