Convergence Theorems for Two Asymptotically Nonexpansive Non-self Mappings in Uniformly Convex Banach Spaces
GURUCHARAN SINGH SALUJA · Journal of the Indian Mathematical Society · 2014
Let K be a nonempty closed convex non expansive retract of a uniformly convex Banach space E with P as a non expansive retraction. Let T 1 , T 2 : K → E be two asymptotically non expansive non-self mappings with sequences {k n }, {h n } ⊂[1,(∞) such that Σ ∞ n=1 (k n h n -1) < ∞ and F = F(T 1 ) ∩ F(T 2 ) = {x E K : T 1 x = T 2 x = x}≠ Φ . Let {x n } ∞ n=1 be the sequence generated iteratively by x l ∈ K and x n+1 = P(a n x n + b n T 1 (PT 1 ) n-1 y n + c n l n ) ∀ n ≥1 y n = P(ā n x n + b n T n (PT 2 ) n-1 x n + c n m n ),∀ n ≥1 where {l n }, {m n } are bounded sequences, a n +b n +c n = 1 = ā n +b n +c n ,0 ≤ a n +b n +c n , ā n +b n +c n ≤ 1, ∀ n ∈ N, Σ ∞ n=1 c n < ∞ and Σ ∞ n=1 b n c n <∞ . If T 1 is completely continuous or T 1 and T 2 satisfy condition (A'), then {x n } converges strongly to a point in F = F(T 1 ) ∩ F(T 2 ). Also if E satisfies Opial's condition or the dual E* of E has the Kedec-Klee property, then {x n } converges weakly to a point in F.