Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces

作者：Song Yisheng^*; Chen Rudong

来源：Nonlinear Analysis-Theory Methods & Applications, 2007, 66(3): 591-603.

DOI：10.1016/j.na.2005.12.004

• uniformly asymptotically regular mapping sequence
• viscosity approximation methods
• a family of infinitely many nonexpansive maps
• weakly sequentially continuous duality mapping

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*, and K be a nonempty closed convex subset of E. Suppose that {Tn} (n = 1, 2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K into itself such that F := boolean AND(infinity)(n=1) F(T-n) not equal theta. For arbitrary initial value x(0) epsilon K and fixed contractive mapping f : K -> K, define iteratively a sequence {x(n)} as follows:
x(n+1) = lambda(n+1) f (x(n)) + (1 - lambda(n+1))T(n+1)x(n), n >= 0,
where {lambda(n)} subset of (0, 1) satisfies lim(n ->infinity) lambda(n) = 0 and Sigma(infinity)(n=1) lambda(n) = infinity. We prove that {x(n)} converges strongly to p epsilon F, as n -> infinity, where p is the unique solution in F to the following variational inequality:\
<(I - f) p, j (p - u)> <= 0 for all u epsilon F (T).
Our results extend and improve the corresponding ones given by O'Hara et al. [J.G. O'Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417-1426], J.S. Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520], H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291] and O'Hara et al. [J.G. O'Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problem in Banach space, Nonlinear Anal. Available online 20 October 2005. doi: 10. 10 16/j.na.2005.07.36].

• 出版日期2007-2-1
• 单位天津工业大学