Banach空间中关于增生算子方程解带误差的Ishikawa迭代序列

Ishikawa Iteration Process with Errors for Solutions to Equations Involving Accretive Operators in Banach Space

设X是任意实Banach空间,T∶X→X是L ipsch itz连续的增生算子,在没有假设∞∑n=0αnnβ<∞之下,证明了由xn+1=(1-αn)xn+αn(f-Tyn)+un及yn=(1-nβ)xn+βn(f-Txn)+vn,n≥0生成的、带误差的Ish ikawa迭代序列强收敛到方程x+Tx=f的唯一解,并给出了更为一般的收敛率估计:若un=vn=0,n≥0,则有xn+1-x*≤(1-γn)xn-x*≤…≤n∏j=0(1-γj)x0-x*,其中{γn}是(0,1)中的序列,满足nγ≥12max{η,1-η}-14m in{η,1-η}αn,n≥0。

Let X be an arbitrary real Banach space and T：X→X be a Lipschitz continuous accretive operator. Under the lack of the assumption of ∞∑n=0αnβn〈∞ ,it is shown that the Ishikawa iterative sequence with errors engendered by xa＋1=（1-αn）xn＋αn（f-Tyα）＋un and yn=（1-βn）xn＋βn（f-Txn）＋vn,A↓n≥0. converges strongly to the unique solution of the equation x ＋ Tx =f. Moreover, this result provides a general convergence rate estimation for such a sequence： if un= vn = 0 for n all n≥0,then we have ‖xa＋1-x^＊‖≤（1-γn）‖x-x^＊‖≤…nПf=0（1-γ1）‖x0-x^＊‖,where ｜γn｜ is a sequence in （0, 1） ,such that for all γn≥[1/2max{η,1-η}-1/4min{η,1-η}]αn,A↓n≥0.