Authors’ abstract: This article briefly surveys some recent progress in the investigation of Halpern’s iterative algorithm for approximating fixed points of nonexpansive mappings. An implicit iterative method is first discussed, followed by details on the strong convergence of Halpern’s iteration in various cases, covering averaged mappings, perturbation techniques, common fixed points for finite families of nonexpansive mappings, and the viscosity approximation method. Extensions of Halpern’s iteration to nonexpansive semigroups and Hadamard manifolds are discussed. Applications to split feasibility and the multiple sets split feasibility problems, and to accretive operators are also included.
For the entire collection see [Zbl 1190.00027].