An approximation method for continuous pseudocontractive mappings

Let[inline-graphic not available: see fulltext] be a closed convex subset of a real Banach space[inline-graphic not available: see fulltext],[inline-graphic not available: see fulltext] is continuous pseudocontractive mapping, and[inline-graphic not available: see fulltext] is a fixed[inline-graphic not available: see fulltext]-Lipschitzian strongly pseudocontractive mapping. For any[inline-graphic not available: see fulltext], let[inline-graphic not available: see fulltext] be the unique fixed point of[inline-graphic not available: see fulltext]. We prove that if[inline-graphic not available: see fulltext] has a fixed point and[inline-graphic not available: see fulltext] has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of[inline-graphic not available: see fulltext] has the fixed point property for nonexpansive self-mappings, then[inline-graphic not available: see fulltext] converges to a fixed point of[inline-graphic not available: see fulltext] as[inline-graphic not available: see fulltext] approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).