  As local and nonlocal reductions of a discrete second-order
Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type
equations are considered. Through the bilinearization reduction method, we
construct double Casoratian solutions of the reduced discrete nonlinear
Schr\"odinger type equations, including soliton solutions and Jordan-block
solutions.Dynamics of the obtained one-soliton and two-soliton solutions are
analyzed and illustrated. Moreover,both semi-continuous limit and full
continuous limit, are applied to obtain solutions of the local and nonlocal
semi-discrete nonlinear Schr\"odinger type equations, as well as the local and
nonlocal continuous nonlinear Schr\"odinger type equations.
