Ishikawa Iteration Convergence Theorems for Nonexpansive Mappings when Parameter Approximate End Point of [0,1]
Guanghui Gu · 2005
Let X be a uniformly convex Banach space and C be a nonempty colsed convex subset of X,and T:C→C be a nonexpansive mapping,for x_1∈C and {s_n},{t_n}[0,1],exsits subsequence{x_(n_k)}of x_(n+1)=(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n),t_n→1,s_n→0,∑∞n=1(1-t_n)=+∞, such that‖x_(n_k)-Tx_(n_k)‖→0(k→∞),if T is compact then Ishikawa iterative process {x_(n_k)}converges strongly to a fixed point,if X satisfies Opial's condition then Ishikawa iterative process {x_(n_k)}converges weakly to a fixed point.