Iterative approximation of common fixed points of two nonself asymptotically nonexpansive mappings

Suppose that K is nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction and F: = F (T1) ∩ F T2 = { x ∈ K: T1 x = T 2x = x } ≠ ∅. Let T1, T2: K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences { kn(i)} ⊂ [ 1, ∞) satisfying n = 1∞ (k n (i) - 1) < ∞(i = 1,2), respectively. For any given x1 ∈ K, suppose that { xn } is a sequence generated iteratively by x n+1 = (1 - αn) (PT1) n y n+ αn (PT2) n y n, yn= (1 - βn) xn + βn (P T1) n xn, n ∈ ℕ, where { αn } and { βn } are sequences in [ a, 1 - a ] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of { xn } to a common fixed point of T 1 and T2 are obtained. Copyright © 2011 Esref Turkmen et al.