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Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces. (English) Zbl 1134.47049
Three-step iterative schemes with errors for two and three nonexpansive maps are introduced in the paper. Finding common fixed points of maps acting on a Banach space is a problem that often arises in applied mathematics. In fact, many algorithms have been introduced for different classes of maps with nonempty set of common fixed points [see, e.g., N. Shahzad, Nonlinear Anal. 61, No. 6 (A), 1031–1039 (2005; Zbl 1089.47058)]. Let \(C\) be a nonempty convex subset of a real Banach space \(E\) and let \(T_i: C \to C\) be nonexpansive maps \((i=1,2,3)\). The following three-step iterative scheme with errors is considered: \( x_1 \in C\), \(z_n=\alpha^{(3)}_n x_n + \beta^{(3)}_n T_3 x_n + \gamma^{(3)}_n u^{(3)}_n\), \(y_n=\alpha^{(2)}_n x_n + \beta^{(2)}_n T_2 z_n + \gamma^{(2)}_n u^{(2)}_n\), \(x_{n+1}=\alpha^{(1)}_n x_n + \beta^{(1)}_n T_1 y_n + \gamma^{(1)}_n u^{(1)}_n\), for all \(n \geq 1\), where \(\{ u^{(j)}_n \}\) is a bounded sequence in \(C\) and \(\{ \alpha^{(j)}_n \}\), \(\{ \beta^{(j)}_n \}\), \(\{ \gamma^{(j)}_n \}\) are sequences in \([0,1]\) satisfying \(\alpha^{(j)}_n + \beta^{(j)}_n + \gamma^{(j)}_n =1\), \(n \geq 1\), \(j=1,2,3\).
Under suitable conditions, the weak and strong convergence of the above scheme to a common fixed point of nonexpansive maps in a uniformly convex Banach space is proved. By modifying the iteration schemes, the corresponding results can be proved for asymptotically nonexpansive mappings with suitable changes. The convergence theorems improve and generalize some important results in the current literature.

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI
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