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Comparison of the rate of convergence of various iterative methods for the class of weak contractions in Banach spaces. (English) Zbl 1294.47090
Let \(C\) be a nonempty convex subset of a Banach space \(X\) and let \(T:C \rightarrow C\) be a self map. In the iterative approximation of fixed points of \(T\), there exist several methods that can be incorporated in the general fixed point iterative process defined by \(u_1\in C\) and \[ h_n=(1-\gamma_n) w_n + \gamma_n T w_n, \]
\[ w_{n+1}=(1-\alpha_n) w_n+\alpha_n T\left((1-\beta_n)w_n+\beta_n T h_n\right), \;n\geq 1, \] where \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{\gamma_n\}\) are sequences in \([0,1]\).
If \(\gamma_n=0\), then this iterative sequence (generally known as Noor iteration) reduces to Ishikawa iteration; if \(\gamma_n=0\) and \(\beta_n=0\), then this iterative sequence reduces to Mann iteration, and so on.
By using the class of weak contractions (also called almost contractions), introduced by the reviewer [Nonlinear Anal. Forum 9, No. 1, 43–53 (2004; Zbl 1078.47042)], the authors establish convergence theorems for Mann, Ishikawa and Noor iterations and a new iterative process defined by \(x_1\in C\) and \[ \begin{gathered} z_n=(1-\gamma_n) x_n + \gamma_n T x_n, \\ y_n=(1-\beta_n) z_n + \beta_n T z_n, \\ x_{n+1}=(1-\alpha_n-\lambda_n) y_n+\alpha_n T y_n+\lambda_n T z_n, \;n\geq 1,\end{gathered} \] where \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{\gamma_n\}\), \(\{\lambda_n\}\) and \(\{\alpha_n+\lambda_n\}\) are sequences in \([0,1]\).
Then, by using the concept of rate of convergence introduced by the reviewer [Fixed Point Theory Appl. 2004, No. 2, 97–105 (2004; Zbl 1090.47053)], the authors of the paper under review prove that their new iterative process converges faster than Noor, Ishikawa and Mann iterations for the class of weak contractions with a unique fixed point.
Numerical examples are also given to illustrate the theoretical results.

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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