Convergence theorems for two finite families of uniformly \(L\)-Lipschitzian mappings
Let \(E\) be a real Banach space and \(C\) be a nonempty closed convex subset of \(E\). A mapping \(T:C\rightarrow C\) is said to be uniformly \(L\)-Lipschitzian if there exists \(L>0\) such that, for any \(x,y\in C\), \[ \left\|T^n x-T^n y\right\| \leq L \left\|x-y\right\|\;\forall n\geq 1. \] For two given finite families of self mappings on \(C\), a family \(\{T_i\}_{i=1}^{N}\) of uniformly \(L_i\)-Lipschitzian mappings and a family \(\{S_i\}_{i=1}^{N}\) of uniformly \(K_i\)-Lipschitzian mappings with the property that \[ F=\bigcap_{i=1}^{N} \operatorname{Fix}(T_i) \cap \bigcap_{i=1}^{N} \operatorname{Fix}(S_i)\neq \emptyset, \] a modified Ishikawa iterative method \(\{x_n\}\) defined by \(x_0\in C\) and \[ \begin{aligned} x_{n+1}&= (1-\alpha_n) x_{n}+\alpha_n S_{r_{n}}^{n} y_n,\\ y_{n}&=(1-\beta_n) x_n+\beta_n T_{r_n}^{n},\quad n\geq 0, \end{aligned} \] where \(r_n=n \mod N\), is used in order to approximate a common fixed point \(x^*\in F\). The main result of the paper (Theorem 2.1) is a strong convergence theorem for the above iterative process in the case when the sequences \(\{\alpha_n\}\) and \(\{\beta_n\}\) in \((0,1)\) satisfy some technical conditions.
scientific article; zbMATH DE number 5556423