This study focuses on state-feedback and output-feedback neural learning control problems for discrete-time nonlinear systems in the pure-feedback form.First,an extended result for the exponential stability for a class of discrete-time linear time-varying(LTV)systems with n-step delays is proposed to verify the exponential convergence of estimated weights.Subsequently,both state-feedback and output-feedback adaptive neural network(NN)controllers are constructed by combining the classical n-step and n-step input-output predictors.After ensuring convergence of the system output to a recurrent reference signal,the radial basis function subvector of NN is verified to satisfy the persistent exciting condition using the system state equation and the implicit function theorem.By combining the extended stability corollary of an LTV system,the estimated weights are verified to exponentially converge to their optimal values.By constructing“learning rules”and using a“mod”function,the estimated weights with a convergent sequence are synthetically represented and stored as the experience knowledge,which is reused to construct neural learning controllers.The proposed neural learning controllers not only accomplish similar control tasks but also reduce the burden of online computation compared with the conventional adaptive NN controllers.Finally,simulation results are presented to demonstrate the effectiveness of the presented schemes.