  Estimation of solution norms and stability for time-dependent nonlinear
systems is ubiquitous in numerous applied 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 a broad class of the
corresponding systems. 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
estimates the stability basins and trapping regions for the corresponding
systems. Finally, we conform the theoretical results in representative
simulations.
