非扩张映象带误差的Ishikawa迭代过程的收敛性(英文)

Convergence of the Ishikawa Iteration Processes with Errors for Nonexpansive Mappings

设D是赋范空间X的一子集,T:DX是一非扩张映射.给定D中序列{xn}和两个实数序列{tn}和{sn}满足: 0≤tn≤t<1和∑∞n=1tn=∞; 0≤sn≤1和∑∞n=1sn<∞; xn+1=tnT(snTxn+(1-sn)xn+vn)+(1-tn)xn+un,n=1,2,3,…,其中{un}和{vn}是两个在X中的可合序列,且limn→∞t-1n‖un‖=0.证明了若{xn}有界,则limn→∞‖Txn-xn‖=0.并给出了保证{xn}弱和强收敛到T的不动点时,关于D,X和T的条件.

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≤t<1 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.