Weak convergence theorems for asymptotically nonexpansive nonself-mappings

Abstract

Suppose that K is a nonempty closed convex subset of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T 1 , T 2 : K → E be two asymptotically nonexpansive nonself-mappings with sequences { k n } , { l n } ⊂ [ 1 , ∞ ) such that ∑ n = 1 ∞ ( k n − 1 ) < ∞ and ∑ n = 1 ∞ ( l n − 1 ) < ∞ , respectively and F ( T 1 ) ∩ F ( T 2 ) = { x ∈ K : T 1 x = T 2 x = x } ≠ 0̸ . Suppose that { x n } is generated iteratively by { x 1 ∈ K x n + 1 = P ( ( 1 − α n ) x n + α n T 1 ( P T 1 ) n − 1 y n ) y n = P ( ( 1 − β n ) x n + β n T 2 ( P T 2 ) n − 1 x n ) , ∀ n ≥ 1 , where { α n } and { β n } are two real sequences in [ ϵ , 1 − ϵ ] for some ϵ > 0 . If E also has a Fréchet differentiable norm or its dual E ∗ has the Kadec–Klee property, then weak convergence of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) are obtained.