WEAK AND STRONG CONVERGENCES OF ISHIKAWA ITERATIONS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN THE INTERMEDIATE SENSE

摘要

Let C be a closed convex subset of a Banach space which satisfies Opial's condition. We first prove that if T : C → C is asymptotically nonexpansive in the intermediate sense, the Ishikawa iteration process with errors defined by x_1 ∈ C, x_(n+1) = α_nx_n + β_nT~ny_n + γ_nu_n, and y_n = α'_nx_n + β'_nT~nx_n + γ'_nv_n converges weakly to some fixed point of T, which generalizes the result due to Tan and Xu. Further, we show that if 5 and T are both comact and asymptotically nonexpansive in the intermediate sense, the iterations {x_n} and {y_n} defined by x_1 ∈ C, x_(n+1) = α_nx_n + β_nS~ny_n + γ_nu_n, and y_n = α'_nx_n + β'_nT~nx_n + γ'_nv_n converge strongly to the same common fixed point of S and T, which generalizes the result due to Rhoades.