Iterative Methods for Pseudocontractive Mappings in Banach Spaces - Jong Soo Jung (Abstract and Applied Analysis)
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.