Convergence of the Ishikawa Iteration Processes with Errors for Nonexpansive Mappings
XU Cheng-zhang · Journal of Southwest China Normal University · 2003
Let D be a subset of a normed space X and T: DX be a nonexpansive mapping. Given a sequence {x_n} in D and two real sequences {t_n} and {s_n} satisfying ? 0≤t_n≤t1 and ∑∞n=1t_n=∞; ? 0≤s_n≤1 and ∑∞n=1s_n∞; ? x_(n+1)=t_nT(s_nTx_n+(1-s_n)x_n+v_n)+(1-t_n)x_n+u_n,n=1,2,3,..., where {u_n} and {v_n} are two summable sequences in X and (lim)n→∞ t~(-1)_n‖u_n‖=0. We prove that if {x_n} is bounded, then (lim)n→∞‖Tx_n-x_n‖=0. The conditions on D,X and T are shown which guarantee the weak and strong convergence of the Ishikawa iteration processes with errors to a fixed point of T.