Iterative Methods for Pseudocontractive Mappings in Banach Spaces

Let E a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let C be a nonempty closed convex subset of E, T : C → C a continuous pseudocontractive mapping with F(T) ≠ ∅, and A : C → C a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant k ∈ (0,1). Let {αn} and {βn} be sequences in (0,1) satisfying suitable conditions and for arbitrary initial value x0 ∈ C, let the sequence {xn} be generated by xn = αnAxn + βnxn−1 + (1 − αn − βn)Txn, n ≥ 1. If either every weakly compact convex subset of E has the fixed point property for nonexpansive mappings or E is strictly convex, then {xn} converges strongly to a fixed point of T, which solves a certain variational inequality related to A.