New Iterations with Errors for Approximating Common Fixed Points for two Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings

Let $X$ be a real uniformly convex Banach space and $C$ a nonempty closed convex nonexpansive retract of $X$ with $P$ as a nonexpansive retraction. Let $T_1,T_2\colon C \to X$ be two uniformly $L$-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of $C$ satisfying condition $A'$ with sequences $\{k_n^{(i)}\}$ and $\{\delta_n^{(i)}\} \subset [1,\infty)$, $i=1,2$, respectively such that $\sum_{n=1}^{\infty} (k_n^{(i)} -1) < \infty$, $\sum_{n=1}^{\infty} \delta_n^{(i)} < \infty$, and $F=F(T_1)\cap F(T_2)\ne \varnothing$. For an arbitrary $x_1 \in C$, let $\{x_n\}$ be the sequence in $C$ defined by
\begin{align*} y_n&=P((1-\beta_n-\gamma_n)x_n+\beta_nT_{2}(PT_{2})^{n-1}x_n+\gamma_n v_n), \\ x_{n+1}&=P((1-\alpha_n-\lambda_n )y_n+\alpha_nT_{1}(PT_{1})^{n-1}x_n+\lambda_n u_n),\qquad n \ge 1, \end{align*}
where $\{\alpha_n\}$, $\{\beta_n\}$, $\{\gamma_n\}$, and $\{\lambda_n\}$ are appropriate real sequences in $[0,1)$ such that $\sum_{n=1}^{\infty} \gamma_n < \infty$, $\sum_{n=1}^{\infty}\lambda_n < \infty$, and $\{u_n\}$, $\{v_n\}$ are bounded sequences in $C$. Then $\{x_n\}$ and $\{y_n\}$ converge strongly to a common fixed point of $T_1$ and $T_2$ under suitable conditions.