Convergence Theorems for a Pair of Nonexpansive Mappings

Let $E$ be a real Banach space and let $C$ be a nonempty closed convex subset of $E$. Then a mapping $T$ of $C$ into itself is called nonexpansive if $\Vert Tx-Ty\Vert \leq \Vert x-y\Vert$ for all $x,y\in C$, and quasi-nonexpansive if the set $F(T)$ of all fixed points of $T$ is nonempty and $\Vert Tx-y\Vert \leq \Vert x-y\Vert$ for all $x\in C$ and $y\in F(T)$. For two mappings $S,T$ of $C$ into itself G. Das and J. P. Debata ["Fixed points of quasi-nonexpansive mappings", Indian J. Pure Appl. Math. 17 (1986) 1263–1269] considered the following iteration scheme: $$x_1\in C\ \ \text{and}\ \ x_{n+1} = \alpha_n S [\beta_n Tx_n + (1-\beta_n)x_n] + (1-\alpha_n)x_n\ \ \forall n\geq 1,$$ where $\{\alpha_n\}$ and $\{\beta_n\}$ are sequences in $[0,1]$.
We first consider the weak convergence of the iterates $\{x_n\}$ in such iteration schemes in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable. Further, we discuss the strong convergence of the iterates $\{x_n\}$ in a strictly convex Banach space. The theorems generalize results of W. Takahashi and G.-E. Kim ["Approximating fixed points of nonexpansive mappings in Banach spaces", Math. Jap. 48/1 (1998) 1–9].