A modified Halpern-type iteration algorithm for quasi-φ{symbol} -asymptotically nonexpansive mappings and applications

The purpose of this article is to modify the Halpern-type iteration algorithm for quasi-φ{symbol}-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of Qin et al. [Convergence of a modified Halpern-type iterative algorithm for quasi-φ{symbol}-nonexpansive mappings, Appl. Math. Lett. 22 (2009), 1051-1055], Wang et al. [A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011), 2364-2371], Su et al. [Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010), 3890-3906], Nartinez-Yanes et al}. [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2400-2411], and others.