Strong convergence and control condition of modified Halpern iterations in Banach spaces (Q5971234)

Let \(X\) be a real Banach space, \(C\) a closed convex subset of \(X\) and denote by \(\Pi_C\) and \(\Gamma_C\) the collection of all contractions on \(C\) and, respectively, the collection of all nonexpansive mappings on \(C\). For a given \(f\in \Pi_C\) and \(T\in \Gamma_C\), the authors consider the following Halpern type ``viscosity'' iteration process \(\{x_n\}\) \[ x_n=\beta_n x_n+(1-\beta_n) [\alpha_n f(x_n)+(1-\alpha_n)T x_n],\quad n\geq 0 , \] to approximate fixed points of \(T\), for which some strong convergence theorems are given.