THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES

Abstract

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a single-valued mapping. Let $\{u_n\}^{\infty}_{n=1}$, $\{v_n\}^{\infty}_{n=1}$, $\{w_n\}^{\infty}_{n=1}$ be three sequences in $K$ and $\{\alpha_n\}^{\infty}_{n=1}$, $\{\beta_n\}^{\infty}_{n=1}$ and $\{\gamma_n\}^{\infty}_{n=1}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the three-step iteration process with errors: $x_{n+1} = (1 − \alpha_n) x_n + \alpha_n Ty_n + u_n$, $y_n = (1 − \beta_n) x_n + \beta_n Tz_n + v_n$, $z_n = (1 − \gamma_n) x_n + \gamma_n Tx_n + w_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established. We also derive the corresponding new results on the strong convergence of the three-step iterative process.