This paper investigates the Lyapunov characterizations of input-to-output stability (IOS) and input-to-state stability (ISS) for nonlinear switched systems. The restriction on the negativity of derivatives of Lyapunov functions is relaxed; additionally. subsystems are not restricted to be ISS/IOS throughout the state space. These problems are overcome by a newly proposed method. https://www.adaptsurvival.com/product-category/rings/