Hybrid iteration method for common fixed points of a finite family of nonexpansive mappings in Banach spaces
Let \(E\) be a real uniformly convex Banach space, and let \(\{T_i:i\in I\}\) be \(N\) nonexpansive mappings from \(E\) into itself with \(F=\{x\in E:T_ix=x\), \(i\in I\}\neq\varnothing\), where \(I=\{1,2,\dots,N\}\). For an arbitrary initial point \(x_1\in E\), the hybrid iteration scheme \(\{x_n\}\) is defined as follows: \(x_{n+1}=\alpha_nx_n+(1-\alpha_n)(T_nx_n-\lambda_{n+1}\mu A(T_nx_n))\), \(n\geq 1\), where \(A:E\to E\) is an \(L\)-Lipschitzian mapping, \(T_n=T_i\), \(i=n\pmod N\), \(1\leq i\leq N\), \(\mu >0\), \(\{\lambda_n\}\subset [0,1)\), and \(\{\alpha_n\}\subset [a,b]\) for some \(a,b\in (0,1)\). Under some suitable conditions, strong and weak convergence theorems of \(\{x_n\}\) to a common fixed point of the mappings \(\{T_i:i\to I\}\) are obtained. The results presented in this paper extend and improve the results of \textit{L.\,Wang} [Fixed Point Theory Appl.\ 2007, Article ID 28619 (2007; Zbl 1159.47052)] and partially improve the results of \textit{M.\,O.\thinspace Osilike, F.\,O.\thinspace Isiogugu} and \textit{P.\,U.\thinspace Nwokoro} [ibid., Article ID 64306 (2007; Zbl 1156.47311)].
scientific article; zbMATH DE number 5632521
scientific article