In this paper, we consider the method for solving a kind of special nonsmooth equations with max-type function and present a smoothing cautious DPRP conjugate gradient method based on the transformation by the smoothing function of the max-type function and the absolute value functions. The global  convergence of the smoothing cautious DPRP conjugate gradient method is given under the general assumptions. Finally, the effectiveness of the given method is shown by the related numerical experiments.