非扩张映象带误差的Ishikawa迭代过程的收敛性

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.

Convergence of the Ishikawa Iteration Processes with Errors for Nonexpansive Mappings

Let D be a subset of a normed space X and T: DX be a nonexpansive mapping. Given a sequence {xn} in D and two real sequences ｛tn｝ and ｛sn｝ satisfying 0≤tn≤t＜1 and ∑∞ n=1tn=∞; 0≤sn≤1 and ∑∞n=1sn＜∞; xn+1=tnT(snTxn+(1-sn)xn+vn)+(1-tn)xn+un,n=1,2,3,…,where ｛un｝ and ｛vn｝ are two summable sequences in X and limn→∞ t-1n‖un‖=0. We prove that if {xn} is bounded, then limn→∞‖Txn-xn‖=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.