The paper endeavours to solve the problem of the necessary and sufficientconditions for testing asymptotic stability of the equilibrium state withoutusing a positive definite or semi-definite Lyapunov function for time-invariantnonlinear discrete-time dynamical systems. The solution is based on the conceptof the $\bf G$-functions introduced in this paper. As a result, new necessaryand sufficient conditions for asymptotic stability of such systems and anestimation or the exact determination of the asymptotic stability domain of thestate $x=0$ are established. Examples are worked out to illustrate the results.
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机译：本文努力解决时不变非线性离散时间动力系统不使用正定或半定Lyapunov函数的情况，以解决测试平衡态渐近稳定性的充要条件的问题。该解决方案基于本文介绍的$ \ bf G $函数的概念。结果，建立了此类系统的渐近稳定性以及状态x = 0的渐近稳定域的估计或精确确定的新的充要条件。举例说明结果。
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