Solutions of generalized constrained discrete KP hierarchy

Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k{)}_{\ge m}+{\sum }_{i=1}^l{q}_i{\mathrm{\Delta }}^{-1}{\mathrm{\Lambda }}^m{r}_i$, are invesitigated by Darboux transformations $T_D(f)=f^{[1]}\cdot \mathrm{\Delta }\cdot f^{-1}$ and $T_I(g)=(g^{[-1]}{)}^{-1}\cdot {\mathrm{\Delta }}^{-1}\cdot g$. Due to this special constraint on Lax operator, it is showed that the generating functions $f$ and $g$ of the corresponding Darboux transformations, can only be chosen from (adjoint) wave functions or $(L^k{)}_{<m}={\sum }_{i=1}^l{q}_i{\mathrm{\Delta }}^{-1}{\mathrm{\Lambda }}^m{r}_i$. Then successive applications of Darboux transformations for gcdKP hierarchy are discussed. Finally based upon above, solutions of gcdKP hierarchy are obtained from $L^{\{0\}}=\mathrm{\Lambda }$ by Darboux transformations.