## Taiwanese Journal of Mathematics

### THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES

Yen-Cherng Lin

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 1 (2006), 75-86.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500403800

Digital Object Identifier
doi:10.11650/twjm/1500403800

Mathematical Reviews number (MathSciNet)
MR2186163

Zentralblatt MATH identifier
1106.47058

#### Citation

Lin, Yen-Cherng. THREE-STEP ITERATIVE CONVERGENCE THEOREMS WITH ERRORS IN BANACH SPACES. Taiwanese J. Math. 10 (2006), no. 1, 75--86. doi:10.11650/twjm/1500403800. https://projecteuclid.org/euclid.twjm/1500403800