Abstract This paper is concerned with the asymptotic behaviors of nonlinear semi-group of Lipschitz operators. A series of basic properties similar to those of linear semigroups are characterized. It is shown that the Dahlquist constant, which is a non-linear extension of logarithmic norm of a bounded linear operator, can be applied to characterize the asymptotic behaviours of nonlinear semigroup of Lipschitz operators. Moreover, in order to characterize the asymptotic behaviours of nonlinear semigroups much better, a novel quantity of nonlinear operator, named the generalized Dahlquist constant, is defined. Unlike Dahlqust constant only defined for Lipschitz operator, this new quantity can be defined well for nonlinear operator. As an application, a new result on global exponential stability of Hopfield-type neural networks is proved.