*(English)*Zbl 1213.65085

The author discusses iterative methods for solving the split feasibility problem (SFP), i.e., to find a point ${x}^{*}$ in a infinite-dimensional Hilbert space ${\mathscr{H}}_{1}$ such that ${x}^{*}\in C$ and $A{x}^{*}\in Q$. Here $C$ and $Q$ are nonempty closed convex subsets of ${\mathscr{H}}_{1}$, and $A$ is a bounded linear operator from ${\mathscr{H}}_{1}$ into another infinite-dimensional Hilbert space ${\mathscr{H}}_{2}$. Such problems have been considered as models in phase retrievals, medical image reconstruction and recently also in the intensity-modulated radiation therapy.

In this paper the SFP problem is reformulated as a minimization problem to enable approximations via gradient-projection methods. Equivalently, the SFP problem is also considered as a fixed point equation so that iterative methods with adaptive relaxation factors can be used to approximate the solution as a fixed point. Both weak convergence to the original solution and strong convergence to the regularized solution are established.

##### MSC:

65J22 | Inverse problems (numerical methods in abstract spaces) |

47J25 | Iterative procedures (nonlinear operator equations) |

47J06 | Nonlinear ill-posed problems |

49N45 | Inverse problems in calculus of variations |

15A29 | Inverse problems in matrix theory |