Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Abstract

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let T 1 , T 2 : K → E be two nonself asymptotically nonexpansive mappings with sequences { k n } , { l n } ⊂ [ 1 , ∞ ) , lim n → ∞ k n = 1 , lim n → ∞ l n = 1 , F ( T 1 ) ∩ F ( T 2 ) = { x ∈ K : T 1 x = T 2 x = x } ≠ ∅ , respectively. Suppose { 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 . (1) Strong convergence theorems of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) are obtained under conditions that one of T 1 and T 2 is completely continuous or demicompact and ∑ n = 1 ∞ ( k n − 1 ) < ∞ , ∑ n = 1 ∞ ( l n − 1 ) < ∞ . (2) If E is real uniformly convex Banach space satisfying Opial's condition, then weak convergence of { x n } to some q ∈ F ( T 1 ) ∩ F ( T 2 ) is obtained.