Bulletin of the
Korean Mathematical Society BKMS

In this paper, we prove that the equivalence between the convergence of Mann and Ishikawa iterations for the generalized Lipschitzian and $\Phi$-strongly pseudocontractive mappings in real uniformly smooth Banach spaces. Our results significantly generalize the recent known results of [B. E. Rhoades and S. M. Soltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitz operators, Int. J. Math. Math. Sci. $\bf 42$ (2003), 2645--2651].

Keywords: Ishikawa iteration, Mann iteration, $\Phi$-strongly pseudocontractive maps, generalized Lipschitz maps, uniformly smooth Banach spaces

MSC numbers: 47H09, 47H10, 47H17