Stability of non-autonomous difference equations: simple ideas leading to useful results

We address the stability properties in a non-autonomous difference equation where f is continuous, and the zero solution is assumed to be the unique equilibrium. We focus our discussion on two techniques motivated by stability results for functional differential equations (FDEs) that proved recently to be useful in the frame of difference equations too. The first one involves the use of discrete inequalities and monotonicity arguments, and it is inspired by the so-called Halanay inequality; the second one is based on the well-known 3/2 stability results for FDEs. We give further insight into the simple ideas that are behind these methods, prove some new results and show applications and open problems.