The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Abstract

Let $E$ be an arbitrary uniformly smooth real Banach space, let $D$ be a nonempty closed convex subset of $E$, and let $T:D\to D$ be a uniformly generalized Lipschitz generalized asymptotically $\mathrm{\Phi }$-strongly pseudocontractive mapping with $q\in F(T)\ne \varnothing$. Let $\{a_n\},\{b_n\},\{c_n\},\{d_n\}$ be four real sequences in $[\mathrm{0,1}]$ and satisfy the conditions: (i) $a_n+c_n\le 1$, $b_n+d_n\le 1$; (ii) $a_n,b_n,d_n\to 0$ as $n\to \infty$ and $c_n=o(a_n)$; (iii) ${\Sigma }_{n=0}^{\infty }{a}_n=\infty$. For some $x_0,z_0\in D$, let $\{u_n\},\{v_n\},\{w_n\}$ be any bounded sequences in $D$, and let $\{x_n\},\{z_n\}$ be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of $\{x_n\}$ is equivalent to that of $\{z_n\}$.