Iterative Methods for Pseudocontractive Mappings in Banach Spaces. (27th March 2013) - Jung, Jong Soo
Abstract : 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 α n and β n be sequences in ( 0, 1 ) satisfying suitable conditions and for arbitrary initial value x 0 ∈ C, let the sequence x n be generated by x n = α n A x n + β n x n - 1 + ( 1 - α n - β n ) T x n, 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 x n converges strongly to a fixed point of T, which solves a certain variational inequality related to A .