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

Abstract

Let C be a nonempty closed convex subset of a real Banach space X whose norm is uniformly Gâteaux differentiable and T : C → C be a continuous pseudo-contraction with a nonempty fixed point set F ( T ) . For arbitrary given element u ∈ C and for t ∈ ( 0 , 1 ) , let { y t } be the unique continuous path such that y t = ( 1 − t ) T y t + t u . Assume that y t → p ∈ F ( T ) as t → 0 . Let { α n } , { β n } and { γ n } be three real sequences in (0, 1) satisfying the following conditions: (i) α n + β n + γ n = 1 ; (ii) lim n → ∞ α n = lim n → ∞ β n = 0 ; (iii) lim n → ∞ β n 1 − γ n = 0 ; or (iii)′ ∑ n = 0 ∞ α n 1 − γ n = ∞ . Let { ϵ n } be a summable sequence of positive numbers. For arbitrary initial datum x 0 = x 0 0 ∈ C and a fixed n ≥ 0 , construct elements { x n m } as follows: x n m + 1 = α n u + β n x n + γ n T x n m , m = 0 , 1 , 2 , … . Suppose that there exists a least positive integer N ( n ) satisfying the following condition: ‖ T x n N ( n ) + 1 − T x n N ( n ) ‖ ≤ γ n − 1 ( 1 − γ n ) ϵ n . Define iteratively a sequence { x n } in an explicit manner as follows: x n + 1 = x n + 1 0 = x n N ( n ) + 1 = α n u + β n x n + γ n T x n N ( n ) , n ≥ 0 . Then { x n } converges strongly to a fixed point of T . For all the continuous pseudo-contractive mappings for which is possible to construct the sequence x n , 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].