In this study, the stability problem of discrete-time switched systems without stable subsystems is considered. Using the k-sample-like method, the authors construct a Lyapunov function whose value at the switching instant is less than the value at the last switching instant when the corresponding dwell time belongs to a special time span. According to whether the dwell time belongs to the time span, switchings are divided into two categories: the switching without divergence time and the switching with divergence time. Based on that, a new less conservative exponential stability theorem is established with the bounded maximum average dwell time. Furthermore, by jointly considering the dynamic characteristics of the subsystems before and after switching instants, they also give the stability result via the dwell time with the floating lower and upper bounds. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.