Modified Halpern Iteration of Asymptotically Non-Expansive Mappings

For an asymptotically non-expansive self-mapping T, we will prove the strong convergence of {xn} defined by xn+1 = (1−αn−βn)xn+αnu+βnTnxn, xn+1 = αnu+(1−αn)Tnxn, whenever {βn}, {αn} С (0, 1) satisfy (C1) lim n rightarrow ∞ αn = 0, (C2) ∑∞n=0 αn = ∞, (C3) lim n rightarrow ∞ kn − 1/αn = 0 or (C4) ∑∞n=0 (kn − 1) +∞ . As an application, we also establish the strong convergence of the viscosity approximation schemes with a contraction f given by xn+1 = αnf(xn) + (1 − αn)Tnxn and xn+1 = (1 − αn − βn)xn + αnf(xn) + βnTnxn.