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

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^{{star operator}} continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping J_Φ with gauge φ{symbol}. 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].