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.

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