一致广义Lipschitz连续的逐次渐近Φ-强伪压缩型算子修正的具误差的Mann迭代和多步Noor迭代间的等价性

The Equivalence between Modified Mann and Multi-step Noor Iterations with Errors for Uniformly Generalized Lipschitz Continuous and Successively Asymptotically Φ-Strongly Pseudo-contractive Type Operators

研究了一致光滑Banach空间中一致广义Lipschitz连续的逐次渐近Φ-强伪压缩型算子修正的具误差的Mann迭代和多步Noor迭代间的等价性问题.本文的研究结果是对2006年Zhenyu Huang、Fanwei Bu和M.A.Noor(Zhenyu Huang,FanweiBu,M.A.Noor,On the equivalence of the convergence criteria between modified Mann-Ishikawa and multi-step iterations with errors for successively strongly pseudo-contractive operators,Applied Mathematics and Computation,2006,180:641-647.)在一致光滑Banach空间中所建立逼近具有有界值域的强伪压缩算子的不动点强假定下,修正的具误差的Mann迭代和带误差Ishikawa迭代两者的收敛是等价的这一问题更本质的和更一般的推广,且所用的方法不同于Zhenyu Huang、Fanwei Bu和M.A.Noor文,从而从更一般的意义上肯定地回答了Rhoades和Soltuz于2003年所提出的猜想.

In this paper, the equivalence between modified Mann and multi - step Noor iterations with errors is investigated for uniformly generalized Lipschitz continuous and successively asymptotically Ф - strongly pseudo - contractive type operators in uniformly smooth Banaeh spaces. Zhenyu Huang, Fanwei Bu and M. A. Noor （Zhenyu Huang, Fanwei Bu, M. A. Noor, On the equivalence of the convergence criteria between modified Mann - Ishikawa and multi - step iterations with errors for successively strongly pseudo - contractive operators, Applied Mathematics and Computation, 2006,180：641 -647. ） have established that the equivalence of the convergence criteria between modified Mann- Ishikawa and multi - step iterations with errors for successively strongly pseudo - contractive operators with bounded range. Our results extend to the more generalized multi - step Noor iterations with errors and the more generalized successively asymptotically Ф- strongly pseudocontractive type operators with uniformly generalized Lipschitz continuous. So, a more generalized affirmative answer is given to the conjecture raised by Rhoades and Soltuz in 2003.