Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces. (7th May 2008)

• Record Type: Journal Article
• Authors: Jung, Jong Soo
• Other Names: Khamsi Mohammed Academic Editor.
• Is Part Of: Fixed point theory and applications. Volume 2008(2008)
• Publication Date: 2008-05-07
• Subjects: Fixed point theory -- Periodicals; Mappings (Mathematics) -- Periodicals; 515.7248
• DOI: 10.1155/2008/167535
• Languages: English
• ISSNs: 1687-1812

Abstract

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.