Strong Convergence of Modified Iteration Processes for Relatively Asymptotically Nonexpansive Mappings

Abstract

Ishikawa and Halpern’s iterations are modified to prove the strong convergence problems of such iteration processes for uniformly Lipschitzian mappings which are relatively asymptotically nonexpansive in Banach spaces, which extend the result due to Matsushita and Takahashi [J. Approx. Theory, 134 (2005), 257–266] for relatively nonexpansive mappings, and also some recent results due to Martinez-Yanez and Xu [Nonlinear Anal., 64 (2006), 2400–2411], and Kim and Xu [Nonlinear Anal., 64 (2006), 1140–1152] for nonexpansive mappings and asymptotically nonexpansive mappings, respectively, which are considered in the Hilbert space frameworks.