A simple model predictive control (MPC) concept for nonlinear systems under input constraints is considered. The presented algorithm takes advantage of an MPC formulation without terminal constraints in order to solve the optimality conditions by a fixed-point iteration scheme that is easy to implement and of algorithmic simplicity. Sufficient conditions for the contraction of the fixed-point iterations are derived. To allow for a real-time implementation within an MPC scheme, a constant number of fixed-point iterations is used in each sampling step and sufficient conditions for asymptotic stability and incremental reduction of the suboptimality are presented. (C) 2012 Elsevier Ltd. All rights reserved.