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New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings. (English. Russian original) Zbl 1368.47089

Math. Notes 89, No. 3, 397-407 (2011); translation from Mat. Zametki 89, No. 3, 410-423 (2011).
Summary: 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 \(T_1, T_2: C \to X\) 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 \(\sum_{n=1}^{\infty}(k_n^{(i)}-1) < \infty\), \(\sum_{n=1}^{(i)} \delta_n^{(i)} < \infty\), and \(F = F(T_1)\cap F(T_2)\neq \emptyset\). For an arbitrary \(x_1\in C\), let \(\{x_n\}\) be the sequence in \(C\) defined by \[ \begin{gathered} y_n = P\left(\left(1-\beta_n-\gamma_n\right)x_n + \beta _n T_2 \left(PT_2\right)^{n-1}x_n + \gamma_n v_n\right), \\ x_{n+1} = P\left(\left(1-\alpha_n - \lambda_n \right)y_n + \alpha_n T_1 \left( PT_1\right)^{n-1}x_n + \lambda_n u_n \right), \quad n\geq 1, \end{gathered} \] 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.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators
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