A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings
Hongbing Zhou, YJ Cho, SM Kang · Fixed Point Theory and Applications · 2007
Abstract Suppose that "Equation missing" is a nonempty closed convex subset of a real uniformly convex and smooth Banach space "Equation missing" with "Equation missing" as a sunny nonexpansive retraction. Let "Equation missing" be two weakly inward and asymptotically nonexpansive mappings with respect to "Equation missing" with sequences "Equation missing", "Equation missing", respectively. Suppose that "Equation missing" is a sequence in "Equation missing" generated iteratively by "Equation missing", "Equation missing", for all "Equation missing", where "Equation missing", "Equation missing", and "Equation missing" are three real sequences in "Equation missing" for some "Equation missing" which satisfy condition "Equation missing". Then, we have the following. (1) If one of "Equation missing" and "Equation missing" is completely continuous or demicompact and "Equation missing", then the strong convergence of "Equation missing" to some "Equation missing" is established. (2) If "Equation missing" is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of "Equation missing" to some "Equation missing" is proved.