A convergence theorem of multi-step iterative scheme for nonlinear maps

Let K be a nonempty closed convex subset of a real Banach space X,T:K → K a nearly uniformly L-Lipschitzian (with sequence {rn}) asymptotically generalized Φ-hemicontractive mapping (with sequence kn [1,∞), lim n→∞ kn = 1) such that F(T) = {pK:Tp=p}. Let {αn}n≥0, {βkn}n≥0 be real sequences in [0,1] satisfying the conditions: (i) Σn≥0 αn = 1 (ii) limn→∞ αn, βkn = 0, k = 1, 2,..., p−1. For arbitrary x0 K, let {xn}n≥0 be a multi-step sequence iteratively defined by xn+1=(1−αn)xn + αnTny1n, n≥0, ykn = (1 − βkn )xn + βkn Tnyk+1n, k = 1,2,..., p−2 (0.1), yp−1n=(1− βp−1n)xn + βp−1n Tnxn, n ≥ 0, p ≥ 2. Then, {xn}n≥0 converges strongly to p F(T). The result proved in this note significantly improve the results of Kim et al. [2].