We present an iterative method for solving, eventually infinite systems of inequalities g(ω, x)≤0, a.e. (Ω), in which the functions g(ω,·) are defined, convex, lower semicontinuous and essentially uniformly bounded on a Banach space. We show that the proposed method produces sequences which accumulate weakly to solutions of the system, provided that such solutions exist. In the most usual Banach spaces, the size of the constraint violations along the sequences generated by the iterative procedure converges in mean to zero. We prove that this method can be implemented for solving consistent operator equations (like first kind Fredholm or Volterra equations, some potential equations, etc.) when they have solutions in Hilbert, Lebesgue or Sobolev spaces.