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In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983.

A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings