Strong convergence of modified Noor iterations (Q2468974)

Let \(C\) be a closed convex subset of a uniformly smooth Banach space \(E\) and \(T\) a nonexpansive self-mapping of \(C\) with \(Fix (T)\neq \emptyset\). The authors consider the following composite iteration scheme \[ \begin{alignedat}{2} w_n &= \delta_n x_n +(1-\delta_n) T_n, &\qquad z_n &= \gamma_n x_n+ (1-\gamma_n)Tw_n,\\ y_n &= \beta_n x_n+ (1-\beta_n) Tz_n, &\qquad x_{n+1} &= \alpha_n u+ (1-\alpha_n) y_n, \end{alignedat} \] where \(u\) and \(x_0\) are arbitrarily fixed and \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{\gamma_n\}\) and \(\{\delta_n\}\) are sequences in \((0,1)\). This iterative scheme contains, as particular cases, the modified Ishikawa iteration, the modified Mann iteration, and the Halpern iteration. The authors prove that, under certain appropriate assumptions on the sequences \(\{\alpha_n\}\), \(\{\beta_n\}\), \(\{\gamma_n\}\) and \(\{\delta_n\}\), the sequence \(\{x_n\}\) converges strongly to a fixed point of \(T\).