Weak and strong convergence of Mann's-type iterations for a countable family of nonexpansive mappings
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Let K be a nonempty closed convex subset of a Banach space E. Suppose {Tn} (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ∩∞n=1 F(Tn) ≠0. For x0 ∈ K, define xn+1 = λn+1xn + (1 − λn+1)Tn+1xn, n ≥ 0. If λn ⊂ [0, 1] satisfies limn→∞ λn = 0, we proved that {xn} weakly converges to some z ∈ F as n → ∞ in the framework of reflexive Banach space E which satisfies the Opial’s condition or has Fr´echet differentiable norm or its dual E* has the Kadec-Klee property. We also obtain that {xn} strongly converges to some z ∈ F in Banach space E if K is a compact subset of E or there exists one map T ∈ {Tn; n = 1, 2, . . .} satisfy some compact conditions such as T is semicompact or satisfy Condition A or limn→∞ d(xn, F(T)) = 0 and so on.
Abstract
Let K be a nonempty closed convex subset of a Banach space E. Suppose {Tn} (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ∩∞n=1 F(Tn) ≠0. For x0 ∈ K, define xn+1 = λn+1xn + (1 − λn+1)Tn+1xn, n ≥ 0. If λn ⊂ [0, 1] satisfies limn→∞ λn = 0, we proved that {xn} weakly converges to some z ∈ F as n → ∞ in the framework of reflexive Banach space E which satisfies the Opial’s condition or has Fr´echet differentiable norm or its dual E* has the Kadec-Klee property. We also obtain that {xn} strongly converges to some z ∈ F in Banach space E if K is a compact subset of E or there exists one map T ∈ {Tn; n = 1, 2, . . .} satisfy some compact conditions such as T is semicompact or satisfy Condition A or limn→∞ d(xn, F(T)) = 0 and so on.
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