Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces
An Ishikawa iteration process with errors is considered: Let \(K\) be a nonempty convex subset of a Banach space \(X\) and a mapping \(T: K\to K\). The sequences \(\{x_n\}\), \(\{y_n\}\) are defined by \(x_0\in K\), \[ x_{n+1}= (1- \alpha_n- \gamma_n) x_n+\alpha_n Ty_n+ \gamma_n u_n, \] \[ y+n= (1- \beta_n- \delta_n)x_n+ \beta_n Tx_n+ \delta_n, v_n,\qquad n\geq 0, \] where \(\{u_n\}\) and \(\{v_n\}\) are two sequences in \(K\) and \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{\gamma_n\}\), \(\{\delta_n\}\) are real sequences in \([0, 1]\) satisfying certain restrictions. The notions ``convex structure'' and ``convex metric space'' are introduced. An iterative scheme similar to the Ishikawa one for approximating a. common fixed point of two sequences of uniformly quasi-Lipschitzian mappings is given and steadied in convex metric spaces. Under certain assumptions convergence of the two sequences to a common fixed point is proved. The authors pointed out a gap in the proof published by \textit{Ch. Wang} and \textit{L.-W. Liu} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5 (A), 2067--2071 (2009; Zbl 1225.47114)] and they present a corrected version of this theorem.
scientific article; zbMATH DE number 5707068
scientific article