Three-step iterative convergence theorems with errors for set-valued mappings in Banach spaces

Yen-Cherng Lin

Abstract

Let q > 1 and E be a real q-uniformly smooth Banach space, K be a nonempty closed convex subset of E and $T : K \rightarrow 2^K$ be a set-valued mapping. Let $\{u_n\}_{n=1}^\infty$, $\{v_n\}_{n=1}^\infty$ and $\{w_n\}_{n=1}^\infty$ be three sequences in K and $\{\alpha_n\}_{n=1}^\infty$, $\{\beta_n\}_{n=1}^\infty$ and $\{\gamma_n\}_{n=1}^\infty$ 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:
$x_{n+1}\in (1-\alpha_n)x_n+\alpha_n T y_n+u_n$, $y_n \in (1-\beta_n)x_n+\beta_n T z_n +v_n$, $z_n \in (1-\gamma_n)x_n+\gamma_n Tx_n+w_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of T are established. We also derive the corresponding new results on the strong convergence of the three-step iterative process with set-valued version and application to approximate the solution of the inclusion $f \in Tx$ in K.