On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval

In this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations. It is proved that the SP-iteration is equivalent to and converges faster than the others. Our results extend and improve the corresponding results of Borwein and Borwein [D. Borwein, J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991) 112-126], Qing and Qihou [Y. Qing, L. Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006) 1383-1386], Rhoades [B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741-750], and many others. Moreover, we also present numerical examples for the SP-iteration to compare with the Mann, Ishikawa and Noor iterations.     

关于广义Φ-压缩映射几种迭代序列收敛的等价性

在广义Φ-压缩映射条件下,分别得到了Picard迭代序列与Krasnoselskii迭代序列以及Mann迭代序列与Ishikawa迭代序列收敛的等价性.     

Convergence theorems for continuous functions on an arbitrary interval

In this paper, we introduce a new iterative scheme for finding a fixed points of continuous functions on an arbitrary interval. The convergence theorems are also established. Further, the numerical examples comparing with Mann, Ishikawa and Noor iterations are demonstrated. Main results generalize and unify the corresponding ones announced in the literature.     

具一致广义Lipschitz连续算子的带误差的多步迭代间的收敛等价性

研究了一致光滑Banach空间中具一致广义Lipschitz连续的逐次渐近Φ-强伪压缩型算子的具误差的修正Mann迭代和具误差的修正多步Noor迭代间的收敛等价性问题,所得结果是对2007年Zhenyu Huang在一致光滑Banach空间中所建立的逼近具有有界值域的逐次Φ-强伪压缩算子的不动点具误差的修正Mann迭代和具误差的修正Ishikawa迭代两者的收敛是等价的这一结论更本质的和更一般的推广,所用的方法不全同于ZhenyuHuang所使用的方法,因此,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.     

The equivalence between the convergence of Ishikawa and Mann iterations with errors for strongly successively pseudocontractive mappings without Lipschitzian assumption

In this paper we prove some new equivalences between convergence of the Ishikawa and Mann iteration sequences with errors in two schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91-101] and Liu [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, for strongly successively pseudocontractive mappings. Our main results improve and extend the corresponding results of the all references listed in this article.     

Equivalence theorems of the convergence between Ishikawa and Mann iterations with errors for generalized strongly successively Φ-pseudocontractive mappings without Lipschitzian assumptions

In this paper, the equivalence of the strong convergence between the modified Mann and Ishikawa iterations with errors in two different schemes by Xu [Y.G. Xu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91-101] and Liu [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 Φ-pseudocontractive mappings without Lipschitzian assumption. Our results generalize the recent results of the papers [Zhenyu Huang, F. 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 pseudocontractive 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 Φ-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 u₁, x₁ in uniformly smooth Banach spaces.     

Banach空间中修正的Mann迭代与Ishikawa迭代收敛的等价性

在实Banach空间框架下,证明了新修正的Mann迭代与修正的Ishikawa迭代关于一致Lipschitz映射不动点收敛的等价性.     

A reconsideration on equivalency of convergence between iterations for uniformly L-Lipschitzian operators

This paper is to illustrate that the conditions and proofs of three theorems of S.S. Chang, Y.J. Cho and J.K. Kim [The equivalence between the convergence of modified Picard, modified Mann and modified Ishikawa iterations, Math. Comput. Modelling 37 (2003) 985–991] concerning the equivalence between the convergence of modified Picard, modified Mann and modified Ishikawa iterations for uniformly Lipschitzian operators in arbitrary Banach spaces are incorrect.     

The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps

The convergence of modified Mann iteration is equivalent to the convergence of modified Ishikawa iterations, when T is an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive map.     

On the equivalence of the convergence criteria between modified Mann–Ishikawa and multi-step iterations with errors for successively strongly pseudo-contractive operators

In this paper, the equivalence of the convergence between the modified Mann–Ishikawa and multi-step Noor iterations with errors is proven for the successively strongly pseudo-contractive operators without Lipschitzian assumption. Our results generalize the recent results of the paper [B.E. Rhoades, S.M. Soltuz, The equivalence between Mann–Ishikawa iterations and multi-step iteration, Nonlinear Anal. 58 (2004) 219–228; 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 pseudo-contractive maps, J. Math. Anal. Appl. 289 (2004) 266–278] by extending to the more generalized multi-step iterations with errors and hence improve the corresponding results of all the references in bibliography by providing the equivalences of convergence between all of these up-to-date iteration schemes.     

一类具有广义Lipschitz条件的非线性映象的迭代过程

本文研究了广义Ｌｉｐｓｃｈｉｔｚ强增生映象的Ｉｓｈｉｋａｗａ型迭代和Ｍａｎｎ型迭代过程的收敛性．所得结果统一和扩展了近期相关结果     

Convergence theorems of fixed points for asymptotically hemi-pseudocontractive mappings

In this paper, the concept of asymptotically hemi-pseudocontractive mapping in Banach space, which is a proper generalization of asymptotically pseudocontractive mapping with nonempty fixed point set as shown by an example, is introduced. Some sufficient and necessary conditions on the strong convergence of the modified Ishikawa and the modified Mann iteration sequences with mean errors for uniformly L-Lipshitzian asymptotically hemi-pseudocontractive mappings are presented.     

关于非Lipschitz的渐近伪压缩映象的迭代法的强收敛性

本文在任意的实Banach空间中研究用带误差的修改的Ishikawa与Mann迭代程序来逼近非Lipschitz的渐近伪压缩映象的不动点的强收敛性问题．本文所得结果在多方面改进和推广了张石生教授的结果．     

The Equivalence of Ishikawa-Mann and Multistep Iterations in Banach Space

Let E be a real Banach space and T be a continuous Φ-strongly accretive operator. By using a new analytical method, it is proved that the convergence of Mann, Ishikawa and three-step iterations are equivalent to the convergence of multistep iteration. The results of this paper extend the results of Rhoades and Soltuz in some aspects.     

Banach空间中Mann迭代和Ishikawa迭代的等价性问题

用不同于已有的方法证明了任意实Banach空间中一致Lipschitz强连接伪压缩算子在具误差的修正的Mann迭代和具误差的修正的Ishikawa迭代下收敛和稳定的等价性,其中迭代参数{βn}仅需lim sup n→∞βn〈k/L（L＋1）,这推广和改进了目前需假设lim n→∞ βn=0和两迭代程序初始点的取值需相同条件下的已有结果.     

ON CONVERGENCE OF ISHIKA WAITERATION PROCEDURES WITH ERRORS

1　IntroductionLet X be a real Banach space and let Jdenotethe normalized duality mapping from X into2 X* given byJ( x) ={ f*∈ X* :〈x,f*〉 =‖ x‖ 2 ,　　‖ f*‖ =‖ x‖ } ,where X* is the dual and〈· ,·〉denotes the generalized duality pairing.Clearly,X issmooth if and only if J is a single-valued mapping.An operator T with domain D( T) =K and range R( T) in X iscalled strongly accretive ifthere exists a constant k>0 and for all x,y∈D( T) ,there exists j( x-y)∈ J( x-y) suchthat…     

A necessary and sufficient condition for the strong convergence of Lipschitzian pseudocontractive mapping in Banach spaces

The purpose of this paper is to establish some necessary and sufficient conditions for the strong convergence of the Ishikawa iterative sequence and the Mann iterative sequence to a fixed point of pseudocontractive mapping in Banach spaces. Our results, to some extent, improve and extend the well-known result of [S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1) (1974) 147–150.] to Banach spaces.     

构造非线性映射不动点的迭代法

为了构造非线性映射的不动点,许多作者对Mann迭代法进行了研究.后来Ishikawa推广了Mann迭代法,对Hilbert空间H的紧凸子集D的李卜西兹拟压缩映射T,证明了新的迭代法{x_n}强收敛于T的一不动点,{x_n}被定义为     

渐近伪压缩映象的收敛性定理

在任意的实Banach空间中,证明了一致Lipschitzian和渐近伪压缩映象的具误差的Ishikawa和Mann迭代序列的一些收敛性定理.     

Banach空间中非线性$\Phi$-伪压缩映象不动点的迭代逼近

本文研究了非线性拟$\Phi$-伪压缩映象与$\Phi$-伪压缩映象的不动点的迭代逼近问题.研究结果表明: 在任意实的Banach空间$E$中,拟$\Phi$-伪压缩映象与$\Phi$-伪压缩映象$T$~($T$不必连续)的具误差的Ishikawa与Mann迭代序列强收敛于$T$的唯一不动点.所得结果不但改进和推广了最近一些文献的相关结果,而且也改进了定理的证明方法.