The authors generalize the Krasnoselskii-Mann theorem and present several algorithms to solve the split feasibility problem (SFP)

${x}^{k+1}={P}_{C}({x}^{k}-y{A}^{T}(I-{P}_{Q})A{x}^{k})$ in case the projections

${P}_{C}$ and

${P}_{Q}$ of the algorithm proposed by

*Ch. Byrne* [Inverse Probl. 18, No. 2, 441-453 (2002;

Zbl 0996.65048)], are difficult or even impossible to compute. A perturbed projection method based on the

$CQ$ algorithm and an inverse method based on Mosco-convergence of sets are presented to solve the SFP and the convergence of these algorithms is established. An new efficient conjugate gradient method is used to make the algorithms more practical and easier to implement.