Weak and strong convergence of Mann's-type iterations for a countable family of nonexpansive mappings

作者：Song Yisheng; Chen Rudong

来源：Journal of the Korean Mathematical Society, 2008, 45(5): 1393-1404.

DOI：10.4134/jkms.2008.45.5.1393

• uniformly asymptotically regular sequence
• a countable family of nonexpansive mappings
• weak and strong convergence
• Mann's type iteration

Let K be a nonempty closed convex subset of a Banach space E. Suppose {T-n} (n = 1, 2....) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that boolean AND(infinity)(n=1) F(T-n) not equal 0. For x(0) is an element of K, define x(n+1) = gimel(n+1)x(n) + (1 - gimel(n+1))T(n+1)x(n,) n >= 0. If gimel(n) subset of [0, 1] satisfies lim(n ->infinity) , gimel(n) = 0, we proved that {x(n)} weakly converges to some z is an element of F as n -> infinity in the framework of reflexive Banach space E which satisfies the Opial's condition or has Frechet differentiable norm or its dual E* has the Kadec-Klee property. We also obtain that {x(n)} strongly converges to some z G F in Banach space E if K is a compact subset of E or there exists one map T is an element of {T-n; n = 1, 2, . . .} satisfy some compact conditions such as T is semicompact or satisfy Condition A or lim(n ->infinity) d(x(n), F(T)) = 0 and so on.

• 出版日期2008-9
• 单位河南师范大学