Convergence of composite iterative methods for finding zeros of accretive operators

Abstract

Strong convergence theorems on composite iterative schemes by the viscosity approximation methods for finding a zero of an accretive operator are established in Banach spaces. The main results generalize the recent corresponding results of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350–2360], Ceng et al. [L.C. Ceng, A.R. Khan, Q.H. Ansari, J.C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal. 70 (2009) 1830–1840], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643] to viscosity methods in a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm. Our results also improve the corresponding results of [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248–249 (2003) 62–71; R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m -accretive operators, Fixed Point Theory Appl. 2006 (2006) 1–10; S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000) 226–240; P.E. Maingé, Viscosity methods for zeroes of accretive operators, J. Approx. Theory 140 (2) (2006) 127–140; K. Nakajo, Strong convergence to zeros of accretive operators in Banach spaces, J. Nonlinear Convex Anal. 7 (2006) 71–81].