The paper starts from the suggestion of R. E. Kalman that additional information on nonlinearity slope may improve the sufficient conditions for absolute stability. This leads to the so called systems with augmented dynamics. Motivated also by the problem of the PIO II aircraft oscillations-self sustained oscillations induced by the saturation nonlinearities, which are both sector and slope restricted-the paper considers a generalization of the Yakubovich criterion to the case of the systems with critical and unstable linear part. The same generalization concerns a quite well known stability criterion where only slope restrictions are taken into account: the published version is improved by using all advantages of the Liapunov method and of the frequency domain stability inequalities. The results are illustrated by several applications.