The Wronskian solution of the constrained discrete Kadomtsev–Petviashvili hierarchy

•By means of the gauge transformation, the Wronskian solutions of the DNLS equations have been given.•We have got the bright soliton and the dark-bright soliton solutions of the cdKP hierarchy.

From the constrained discrete Kadomtsev–Petviashvili (cdKP) hierarchy, the discrete nonlinear Schrödinger (DNLS) equations have been derived. By means of the gauge transformation, the Wronskian solution of DNLS equations have been given. The u1 of the cdKP hierarchy is a Y-type soliton solution for odd times of the gauge transformation, but it becomes a dark–bright soliton solution for even times of the gauge transformation. The role of the discrete variable n in the profile of the u1 is discussed.