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 Hu [8] and Yao et al [21] modification of Halpern iterative algorithm for nonexpansive mappings in Banach spaces. It is the purpose of this paper to thoroughly analyze this modification and its convergence conditions. Unfortunately, Hu [8] and Yao et al [21] control conditions imposed on the modified Halpern iterative algorithm to have strong convergence are found to be not sufficient. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Hu [8] and Yao et al [21] are not sufficient. It is also proved that with some additional conditions on the control parameters, strong convergence of the defined iterative algorithm is obtained in different Banach space settings.
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