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

Abstract

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 ?-hemicontractive mapping (with sequence kn ? [1,?), lim n?? kn = 1) such that F(T) = {p?K:Tp=p}. Let {?n}n?0, {?kn}n?0 be real sequences in [0,1] satisfying the conditions: (i) ?n?0 ?n = 1 (ii) limn?? ?n, ?kn = 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??n)xn + ?nTny1n, n?0, ykn = (1 ? ?kn )xn + ?kn Tnyk+1n, k = 1,2,..., p?2 (0.1), yp?1n=(1? ?p?1n)xn + ?p?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].