  Estimation of solution norms and stability for time-dependent nonlinear
systems is ubiquitous in numerous engineering, natural science and control
problems. Yet, practically valuable results are rare in this area. This paper
develops a novel approach, which bounds the solution norms, derives the
corresponding stability criteria, and estimates the trapping/stability regions
for some nonautonomous and nonlinear systems, which arise in various
application domains. Our inferences rest on deriving a scalar differential
inequality for the norms of solutions to the initial systems. Utility of the
Lipschitz inequality linearizes the associated auxiliary differential equation
and yields both the upper bounds for the norms of solutions and the relevant
stability criteria. To refine these inferences, we introduce a nonlinear
extension of the Lipschitz inequality, which improves the developed bounds and
allows estimation of the stability basins and trapping regions for the
corresponding systems. Finally, we confirm the theoretical results in
representative simulations.
