Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces

Let CC be a nonempty closed convex subset of a real Banach space XX whose norm is uniformly Gâteaux differentiable and T:C→CT:C→C be a continuous pseudo-contraction with a nonempty fixed point set F(T)F(T). For arbitrary given element u∈Cu∈C and for t∈(0,1)t∈(0,1), let {yt}{yt} be the unique continuous path such that yt=(1−t)Tyt+tu.yt=(1−t)Tyt+tu. Assume that yt→p∈F(T)yt→p∈F(T) as t→0t→0. Let {αn},{βn}{αn},{βn} and {γn}{γn} be three real sequences in (0, 1) satisfying the following conditions: (i)αn+βn+γn=1αn+βn+γn=1;(ii)limn→∞αn=limn→∞βn=0limn→∞αn=limn→∞βn=0;(iii)limn→∞βn1−γn=0; or(iii)′∑n=0∞αn1−γn=∞.Let {ϵn}{ϵn} be a summable sequence of positive numbers. For arbitrary initial datum x0=x00∈C and a fixed n≥0n≥0, construct elements {xnm} as follows: xnm+1=αnu+βnxn+γnTxnm,m=0,1,2,…. Suppose that there exists a least positive integer N(n)N(n) satisfying the following condition: ‖TxnN(n)+1−TxnN(n)‖≤γn−1(1−γn)ϵn.Define iteratively a sequence {xn}{xn} in an explicit manner as follows: xn+1=xn+10=xnN(n)+1=αnu+βnxn+γnTxnN(n),n≥0. Then {xn}{xn} converges strongly to a fixed point of TT. For all the continuous pseudo-contractive mappings for which is possible to construct the sequence xnxn, this result improves and extends a recent result of Yao et al. [Yonghong Yao, Yeong-Cheng Liou, Rudong Chen, Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces, Nonlinear Anal., 67 (2007) 3311–3317].