Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Abstract

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak ⋆ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping J Φ with gauge ϕ. Let f be an α-contraction and { T n } a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes (1) x n + 1 = α n f ( x n ) + ( 1 − α n ) T n x n with a general theorem and then recover and improve some specific cases studied in the literature [K. Aoyoma, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed point of a countable family of nonexpansive mappings, Nonlinear Anal. 67 (8) (2007) 2350–2360; G. Lopez, V. Martin, H.-K. Xu, Perturbation techniques for nonexpansive mappings with applications, Nonlinear Anal. Real World Appl., in press, available online 4 May 2008; H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (1) (2004) 279–291; T.-H. Kim, H.-K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (1–2) (2005) 51–60; Y. Song, R. Chen, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput. 180 (2006) 275–287; Y. Song, R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (1) (2006) 316–326; J. Chen, L. Zhang, T. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl. 334 (2) (2007) 1450–1461; Y. Kimura, W. Takahashi, M. Toyoda, Convergence to common fixed points of a finite family of nonexpansive mappings, Arch. Math. 84 (2005) 350–363].