Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces

Gurucharan Singh Saluja

Suppose $K$ is a nonempty closed convex subset of a real uniformly convex Banach space $E$. Let $S,T: K \to K$ be two asymptotically quasi-nonexpansive mappings in the intermediate sense such that $F = F(S) \cap F(T) = \{x\in K : Sx = Tx = x\} \neq \emptyset$. Suppose $\{x_{n}\}$ is generated iteratively by $x_{1}\in K$, $x_{n+1} = (1 - \alpha_{n})T^{n}x_{n} + \alpha_{n}S^{n}y_{n}$, $y_{n} = (1-\beta_{n})x_{n} + \beta_{n}T^{n}x_{n}$, $n \geq 1$, where $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are real sequences in $[a,b]$ for some $a,b\in (0,1)$. If $S$ and $T$ satisfy condition (B) or either $S$ or $T$ is semi-compact, then the sequence $\{x_{n}\}$ converges strongly to some $q\in F$ and if $E$ satisfying the Opial's condition, then the sequence $\{x_{n}\}$ converges weakly to some $q\in F.$