  In this paper, inspired by Iemoto and Takahashi [S. Iemoto, W. Takahashi,
Nonlinear Analysis 71, (2009), 2082-2089], we study the Halpern's method to
approximate strongly fixed points of a nonexpansive mapping and of a
nonspreading mapping. A crucial tool in our results is the regularization with
the averaged type mappings [C. Byrne, Inverse Probl. 20, (2004), 103-120].
