Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces. (7th May 2008) - Jung, Jong Soo
Abstract : LetE be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset ofE has the fixed point property for nonexpansive mappings. LetC be a nonempty closed convex subset ofE, f : C     →     C a contractive mapping (or a weakly contractive mapping), andT : C     →     C nonexpansive mapping with the fixed point setF ( T )     ≠     ∅ . Let{ x n } be generated by a new composite iterative scheme:y n = λ n f ( x n ) + ( 1 − λ n ) T x n, x n + 1 = ( 1 − β n ) y n + β n T y n, ( n ≥ 0 ) . It is proved that{ x n } converges strongly to a point inF ( T ), which is a solution of certain variational inequality provided that the sequence{ λ n } ⊂ ( 0, 1 ) satisfieslim n → ∞ λ n = 0 and∑ n = 1 ∞ λ n = ∞, { β n } ⊂ [ 0, a ) for some0 < a < 1 and the sequence{ x n } is asymptotically regular.