부산대학교 도서관

Let $K$ be a nonempty closed convex subset of a uniformly convex Banach space $E$, let $S_1, S_2, S_3 \colon K \to K$ be nonexpansive mappings, let $T_1, T_2, T_3 \colon K \to K$ be asymptotically nonexpansive mappings, and suppose that the fixed point sets of all the $S_k$ and $T_k$ meet. If $\{u_n^{(i)}\}$, $i = 1, 2, 3$ are bounded sequences in $K$ and $\{a_n^{(i)}\}, \{b_n^{(i)}\}, \{c_n^{(i)}\} \subset [0,1]$ are such that $a_n^{(i)} + b_n^{(i)} + c_n^{(i)} = 1$, $i = 1, 2, 3$, consider the 3-step iterative scheme $$ \cases x_1\in K,\\ z_n=a_n^{(3)} T_3^n x_n + b_n^{(3)} S_3 x_n + c_n^{(3)}u_n^{(3)},\\ y_n=a_n^{(2)} T_2^n z_n + b_n^{(2)} S_2 x_n + c_n^{(2)}u_n^{(2)},\\ x_{n+1}=a_n^{(3)} T_1^n y_n + b_n^{(1)} S_1 x_n +c_n^{(1)}u_n^{(1)},\quad n\ge1.\endcases $$ It is assumed that there is a sequence $\{k_n\} \subset [1, \infty)$ such that $\lim_{n \to \infty}k_n = 1$ and $\|T_i^n x - T_i^n y\| \le k_n \|x - y\|$ for all $x, y \in K$, $n \ge 1$ and $i = 1, 2, 3$. The authors impose conditions on the various collections $\{k_n\}$, $\{a_n^{(i)}\}$, $\{c_n^{(i)}\}$, $\{T_i\}$, and $\{S_i\}$ in order to obtain convergence (weak or strong) to a common fixed point of the six maps $S_1$, $S_2$, $S_3$, $T_1$, $T_2$, and $T_3$. They assert, but do not prove, that similar statements are true for $N$-step iterative schemes defined in like manner.