Successive approximations for common fixed points of a family of $$\alpha $$-nonexpansive mappings

In this paper, we consider the Halpern iteration for \(\alpha \)-nonexpansive mapping semigroups; such semigroups contain classical nonexpansive mapping semigroups as special cases. The strong convergence theorems of Halpern iteration sequence are shown for finding a common fixed point of an \(\alpha \)-nonexpansive semigroup, which exactly is a metric projection point of the anchor point u to the common fixed point set of such semigroups. Furthermore, the strong convergence of the Halpern iteration for an \(\alpha \)-nonexpansive mapping sequence is given also.