Applications of the \(S\)-iteration process to constrained minimization problems and split feasibility problems.

*(English)*Zbl 1281.47053Summary: In this paper, the \(S\)-iteration process introduced by R. P. Agarwal et al. [J. Nonlinear Convex Anal. 8, No. 1, 61–79 (2007; Zbl 1134.47047)] is further analyzed for contraction and nonexpansive mappings. It is shown, theoretically as well as numerically, that the \(S\)-iteration process is faster than the Picard and Krasnoselskij-Mann iteration processes for contraction operators. We also propose a new iterative algorithm and prove a strong convergence theorem for computing fixed points of nonexpansive operators in a Banach space. Our results are applied for finding solutions of constrained minimization problems and split feasibility problems. Our iteration methods are of independent interest.

##### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47H06 | Nonlinear accretive operators, dissipative operators, etc. |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |

65J15 | Numerical solutions to equations with nonlinear operators |

90C30 | Nonlinear programming |