Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space

Abstract

In this paper, we study strong convergence of common fixed points of two asymptotically quasi-nonexpansive mappings and prove that if $K$ is a nonempty closed convex subset of a real Banach space $E$ and let $ S, T\colon K\to K $ be two asymptotically quasi-nonexpansive mappings with sequences $ \{u_n\}$, $\{v_n\}\subset [0,\infty) $ such that $ \sum_{n=1}^{\infty}u_n<\infty $ and $ \sum_{n=1}^{\infty}v_n <\infty $, and $ F=F(S)\cap F(T)=\{x\in K:Sx=Tx=x\}\neq\phi $. Suppose $ \{x_n\}_{n=1}^{\infty} $ is generated iteratively by $ x_1\in K $, and

$$ \begin{cases} x_{n+1}& = (1-\alpha_n)x_n + \alpha_nS^ny_n + l_n \\ y_n &= (1-\beta_n)x_n + \beta_nT^nx_n + m_n, \quad \forall n\in N \end{cases} $$

where $ \{l_n\}_{n=1}^{\infty},\{m_n\}_{n=1}^{\infty} $ are sequences in $K$ satisfying $ \sum_{n=1}^{\infty}\Vert l_n\Vert<\infty $, $ \sum_{n=1}^{\infty}\Vert m_n\Vert< \infty $ and $ \{\alpha_n\}, \{\beta_n\} $ are real sequences in $ [0,1] $. It is proved that $ \{x_n\}_{n=1}^{\infty} $ converges strongly to some common fixed point of $S$ and $T$. Our result is significant generalization of corresponding result of [3], [7] and [8].