A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces

Abstract In this paper, we introduce an explicit iterative process for a finite family of asymptotically k -strict pseudo-contractions. By using the iterative process, we get a weak convergence theorem for a finite family of asymptotically k -strict pseudo-contractions and then we modify these algorithms to have strong convergence in the framework of Hilbert spaces. Our results improve and extend the recent ones announced by [G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 67 (2007) 2258–2271; T.H. Kim, H.K. Xu, Strong convergence of modified mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140–1152; C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400–2411; K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379; G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert space, J. Math. Anal. Appl. 329 (2007) 336–346; T.H. Kim, H.K. Xu, Convergence of the modified Manns iteration method for asymptotically strict pseudo-contractions, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.02.029 ] and many others.