A new sufficient condition for the strong convergence of Halpern type iterations

The aim of this work is to give a new sufficient condition of the strong convergence of the Halpern type iteration for a non-expansive self-mapping defined on a Banach space with a uniformly Gâteaux differentiable norm. Several examples satisfying our condition are presented. Our results not only remove the restriction of the space with the fixed point property for non-expansive self-mappings, but also get rid of the dependence on the convergence of the implicit anchor-like continuous path zt in the proof.