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首页> 外文期刊>Journal of Mathematical Analysis and Applications >Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Phi-pseudocontractive mappings without Lipschitzian assumptions
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Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Phi-pseudocontractive mappings without Lipschitzian assumptions

机译：对于不带Lipschitzian假设的广义强连续Phi-pseudo压缩映象，Ishikawa和Mann迭代之间的收敛等价定理有误差

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摘要

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [YG. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91-101] and Lui [L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114-125] respectively is proven for the generalized strongly successively Phi-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, E Bu, The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption, J. Math. Anal. Appl. 325 (1) (2007) 586-594; B.E. Rhoades, S.M. Soltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocon tractive maps, J. Math. Anal. Appl. 289 (2004) 266-278; B.E. Rhoades, S.M. Soltuz, The equivalence between Mann-Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219-228] by extending to the most general class of the generalized strongly successively Phi-pseudocontractive mappings and hence improve the corresponding results of all the references given in this paper by providing the equivalence of convergence between all of these iteration schemes for any initial points u(1), x(1) in uniformly smooth Banach spaces. (c) 2006 Elsevier Inc. All rights reserved.
机译：在本文中，修正的Mann和Ishikawa迭代之间的强收敛的等价性在Xu的两种不同方案中是相同的。徐，石川和曼恩的迭代过程，带有非线性强积算算子方程的误差。肛门应用224（1998）91-101]和Lui [L.S. Liu，Ishikawa和Mann的Banach空间中非线性强积映射的具有误差的迭代过程。肛门应用194（1995）114-125]分别证明了没有Lipschitzian假设的广义强连续Phi-pseudo-contractive映射。我们的结果概括了论文的最新结果[Zhenyu Huang，E Bu，在没有Lipschitzian假设的情况下，对于强连续伪压缩映射，Ishikawa和Mann迭代的收敛性与错误之间的等价关系，J。Math。肛门应用325（1）（2007）586-594;是。罗德（S.M.） Soltuz，在中间意义上的渐近非扩张和强连续伪控制图的Ishikawa和Mann迭代的收敛之间的等价关系。肛门应用289（2004）266-278;是。罗德（S.M.） Soltuz，“ Mann-Ishikawa迭代与多步迭代之间的等价关系”，“非线性肛门”。 58（2004）219-228]，扩展到广义的强连续Phi-pseudocontractive映射的最一般类别，并通过提供所有这些迭代方案之间的收敛等价性，从而改进本文给出的所有参考的对应结果。对于均匀光滑Banach空间中的任何初始点u（1），x（1）。 （c）2006 Elsevier Inc.保留所有权利。

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