Weak and strong convergence to common fixed points of non-self nonexpansive mappings

Abstract

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T1, T2 and T3 : K → E be nonexpansive mappings with nonempty common fixed points set. Let {αn}, {βn}, {γn}, {α′n}, {β′n}, {γ′n}, {α″n}, {β″n} and {γ″n} be real sequences in [0, 1] such that αn+βn+ γn=α′n + β′n + γ′n = α″n + β″n + γ″n = 1, starting from arbitrary X1 e K, define the sequence {xn} by {zn = P(α″nT1xn + β″nXn + γ″nwn) Yn = P(α′nT2zn + β′nxn + γ′nvn) xn+1 = PαnT3yn + βnxn + γnun) with the restrictions Σn=1∞ γn < ∞, Σn=1∞γ′n < ∞, Σn=1∞ γ″n < ∞. (i) If the dual E* of E has the Kadec-Klee property, then weak convergence of a {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is proved; (ii) If T1, T2 and T3 satisfy condition (A′), then strong convergence of {xn} to some x* ∈ F(T1) ∩ F(T2) ∩ (T3) is obtained.