Abstract
Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Yao et al. [16] modification of Halpern iterative algorithm for nonexpansive mappings in uniformly smooth Banach spaces. Unfortunately, some deficiencies are found in the Yao et al. [16] control conditions imposed on the modified iteration to obtain strong convergence. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Yao et al. [16] are not sufficient and it is also proved that with some additional control conditions on the parameters strong convergence of the iteration is obtained.
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