In the paper we study the subject of stability of systems with -differences of Caputo-, Riemann-Liouville- and Grünwald-Letnikov-type with fractional orders. The equivalent descriptions of fractional -difference systems are presented. The sufficient conditions for asymptotic stability are given. Moreover, the Lyapunov direct method is used to analyze the stability of the considered systems with -orders.
In this paper, there are derived sufficient conditions for exponential and asymptotic stability of differential and difference systems.
The paper deals with a difference equation arising from the scalar pantograph equation via the backward Euler discretization. A case when the solution tends to zero but after reaching a certain index it loses this tendency is discussed. We analyse this problem and estimate the value of such an index. Furthermore, we show that the utilized proof technique enables us to investigate some other numerical formulae, too.