Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces
Jong Soo Jung · Fixed Point Theory and Applications · 2008
Abstract Let "Equation missing" be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of "Equation missing" has the fixed point property for nonexpansive mappings. Let "Equation missing" be a nonempty closed convex subset of "Equation missing", "Equation missing" a contractive mapping (or a weakly contractive mapping), and "Equation missing" nonexpansive mapping with the fixed point set "Equation missing". Let "Equation missing" be generated by a new composite iterative scheme: "Equation missing", "Equation missing", "Equation missing". It is proved that "Equation missing" converges strongly to a point in "Equation missing", which is a solution of certain variational inequality provided that the sequence "Equation missing" satisfies "Equation missing" and "Equation missing", "Equation missing" for some "Equation missing" and the sequence "Equation missing" is asymptotically regular.