Iterative Approximation to Common Fixed Points of Two Nonexpansive Mappings in Banach Spaces

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E∗, and K be a nonempty closed convex subset of E. Suppose that T, S : K → K are two nonexpansive mappings such that F := F (ST ) = F (T ) ∩ F (S) = ∅. For arbitrary initial value x0 ∈ K and fixed anchor u ∈ K, define iteratively a sequence {xn} as follows: { yn = βnxn + (1− βn)Txn xn+1 = αnu+ (1− αn)Syn, n ≥ 0, where {αn}, {βn} ⊂ [0, 1] satisfies proper conditions. We prove that {xn} converges strongly to PFu as n → ∞, where PF is a unique sunny nonexpansive retraction of K onto F . Also we prove that the same conclusions still hold in a uniformly convex Banach space with uniformly Géteaux differentiable norm or uniformly smooth Banach spaces. Our results extend and improve the corresponding ones by Tae-Hwa Kim and Hong-Kun Xu [Strong convergence of modified Mann iterations, Nonlinear Anal. 61(2005) 51-60]. Mathematics Subject Classification: 47H05, 47H10, 47H17