The paper proposes and advocates a novel technique of Lyapunov analysis that is based on the concept of averaging functions and Steklov's averaging method. This approach relaxes quadratic criteria for incremental stability of nonlinear systems. The stability criteria rest on time-dependent quadratic or non-quadratic Lyapunov functions and offer better account for the role of the external excitation by providing input-dependent conditions. In particular, the proposed method works even if the system fails to satisfy the incremental version of the circle criterion and, for example, proves frequency entrainment for a piece-wise linear analog of the van der Pol oscillator whenever the input has sufficiently large amplitude.