This paper investigates the problem of stability analysis and output-feedback stabilization of discrete-time Lur'e systems where the nonlinearity is odd and slope bounded. Using the linear matrix inequality (LMI) conditions from the literature to handle the 1 norm and positive realness constraints, an iterative algorithm based on LMIs is constructed to assess stability through the existence of a Zames-Falb multiplier of any given order based on independent positive definite matrices for the 1 norm and positive realness. More important, the method can also deal with output-feedback stabilization. Numerical examples illustrate the performance of the proposed approach when compared with other methods.