Halpern iteration for firmly type nonexpansive mappings

The main aim of this paper is to study the strong convergence of Halpern iteration for firmly type nonexpansive mappings defined on a Banach space with a uniformly Gâteaux differentiable norm, and the proof techniques are simpler and different from much existing literature also, which gives the affirmative answer of the Halpern open problem for this class of mapping.