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This paper investigates new sufficient conditions for the stability, asymptotic stability, and global asymptotic stability of nonlinear autonomous systems, specifically in cases where the first derivative of the Lyapunov function candidate may have both positive and negative values on its domain. The main contribution of this approach is the introduction of a new auxiliary function that relaxes the stability conditions, allowing the first derivative of the Lyapunov function candidate to be less than or equal to a nonnegative function. The suggested auxiliary function should be integrable within our first theorem. Meanwhile, our first corollary presents a technique that simplifies the task by establishing specific conditions related to differential inequalities. This weaker condition in the proposed results enables the establishment of stability properties in cases where the Lyapunov function candidate is not well chosen or finding a Lyapunov function is not straightforward. Additionally, it is proven that the original Lyapunov method for autonomous systems is a special case of our first theorem. Furthermore, it is demonstrated that assumptions in previous studies, such as Matrosov's theorem or results on higher-order derivatives of the Lyapunov function, guarantee the existence of our auxiliary function. Finally, lemmas are provided to construct these auxiliary functions, and examples are presented to demonstrate the effectiveness of this approach. This work will contribute to the development of stability analysis techniques for nonlinear autonomous systems. [ABSTRACT FROM AUTHOR]

For time-invariant finite-dimensional systems, it is known that global asymptotic stability (GAS) is equivalent to uniform GAS (UGAS), in which the decay rate and transient overshoot of solutions are requested to be uniform on bounded sets of initial states. This paper investigates this relationship for time-invariant delay systems. We show that UGAS and GAS are equivalent for this class of systems under the assumption of robust forward completeness, i.e., under the assumption that the reachable set from any bounded set of initial states on any finite time horizon is bounded. We also show that, if the state space is a space in a particular family of Sobolev or H\"older spaces, then GAS is equivalent to UGAS and that robust forward completeness holds. Based on these equivalences, we provide a novel Lyapunov characterization of GAS (and UGAS) in the aforementioned spaces. [ABSTRACT FROM AUTHOR]

This paper presents criteria for the convergence of trajectories of time-varying nonlinear systems in terms of Lyapunov densities. The results are provided without assuming local stability and forward completeness of trajectories. As well as general time-varying nonlinear systems, periodically time-varying systems are also considered in this paper, where a weaker criterion is proposed for periodically time-varying systems. Also the existence of Lyapunov densities is proved for general and periodic time-varying nonlinear systems under the asymptotic stability of the equilibrium. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of standing waves for the FitzHugh--Nagumo equations, a well-known reaction-diffusion model of activator-inhibitor type for exhibiting Turing patterns. Similar to the Allen--Cahn equation, a balanced condition for the potential induced from the reaction terms is imposed in studying the existence of planar standing wavefronts. Furthermore, the diffusion rates of activator and inhibitor must be in an appropriate range to ensure the existence of such waves. For the standing front with a symmetry property, an application of the comparison argument yields a uniqueness result. Moreover, the asymptotic stability of wavefronts up to a phase shift is analyzed. [ABSTRACT FROM AUTHOR]