Strong convergence of an iterative method for non-expansive mappings

作者：Song Yisheng^*; Chen Rudong

来源：Mathematische Nachrichten, 2008, 281(8): 1196-1204.

DOI：10.1002/mana.200510670

• reflexive Banach space
• weakly continuous duality mapping
• uniformly Gateaux differentiable norm
• sunny non-expansive retraction

Let E be a real reflexive Banach space having a weakly continuous duality mapping J(phi) with a gauge function phi, and let K be a nonempty closed convex subset of E. Suppose that T is a non-expansive mapping from K into itself such that F(T) not equal empty set. For an arbitrary initial value x(0) is an element of K and fixed anchor It E K. define iteratively a sequence {x(n)} as follows:
x(n+1) = alpha(n)u + beta(n)x(n) + gamma(n)Tx(n), n >= 0,
where {alpha(n)}, {beta(n)}, {gamma(n)} subset of (0, 1) satisfies alpha(n) + beta(n) + gamma(n) = 1, (C1) lim(n ->infinity) alpha(n) = 0, (C2) Sigma(infinity)(n=1) alpha(n) = infinity (B) 0 < lim inf(n ->infinity) beta(n) <= lim sup(n ->infinity) beta(n) < 1. We prove that {x(n)} converges strongly to Pu as n -> infinity, where P is the unique sunny non-expansive retraction of K onto F(T). We also prove that the same conclusions still hold in a uniformly convex Banach space with a uniformly Gateaux differentiable norm or in a uniformly smooth Banach space. Our results extend and improve the corresponding ones by C. E. Chidume and C. O. Chidume [Iterative approximation of fixed points of non-expansive mappings, J. Math. Anal. Appl. 318, 288-295 (2006)], and develop and complement Theorem I of T. H. Kim and H. K. Xu [Strong convergence of modified Mann iterations, Nonlinear Anal. 61, 51-60 (2005)].

• 出版日期2008
• 单位河南师范大学; 天津工业大学