We introduce iterative methods approximating fixed points for nonlinear operators defined on infinitedimensional spaces. The starting points are the Implicit and Explicit Midpoint Rules, which generate polygonal functions approximating a solution for an ordinary differential equation in finite-dimensional spaces. The purpose is to determine suitable conditions on the mapping and the underlying space, in order to get strong convergence of the generated sequence to a common solution of a fixed point problem and a variational inequality. The authors contributions appear in the papers [34], [60], [61].