Ishikawa Iterative Process for a Pair of Single-valued and Multivalued Nonexpansive Mappings in Banach Spaces

Let [inline-graphic not available: see fulltext] be a nonempty compact convex subset of a uniformly convex Banach space [inline-graphic not available: see fulltext], and let [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] for all [inline-graphic not available: see fulltext]. We prove that the sequence of the modified Ishikawa iteration method generated from an arbitrary [inline-graphic not available: see fulltext] by [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext], where [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext] are sequences of positive numbers satisfying [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext], converges strongly to a common fixed point of [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext]; that is, there exists [inline-graphic not available: see fulltext] such that [inline-graphic not available: see fulltext].