Iterative Methods for Pseudocontractive Mappings in Banach Spaces

Abstract EN

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.