In this paper, the convergent conditions in sequence or parallel update mode and the sufficient condition with only one global stable state for Hopfield network model with discrete time and continuous states when its neurons’ activation function is non-decreasing (not being strictly monotone increasing) are discussed. With the definition of a new energy function and the research on the properties of monotonously increasing function, the sufficient conditions is presented to converge in parallel or sequential update mode when neuron’s activation function is monotonously increasing (not be necessary to strictly increase). After obtained the condition for energy function to be convex with respect to the network states variables, it follows that a sufficient condition for network to have only one stable point with the minimum energy by regarding the operation of Hopfield network as solving a constrained convex optimal problem. When auto-connection weight value of each neuron in network is greater than the reciprocal of derivation of its activation function, the network will be convergent in sequence update mode. When the minimal eigenvalue of connection weights matrix is greater than the reciprocal of derivation of its neuron activation function, the network will be convergent in parallel update mode. If the energy function of network is convex, the network will have only one global stable point. These results extend the choice range of activation function of neuron when using Hopfield net to solution of optimization problem or to associative memory.