Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767-773] in condition αn∈(0,1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder-Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73-81].     

Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type are proved in Banach spaces using an implicit iteration scheme recently introduced by Xu and Ori [Numer. Funct. Anal. Optim. 22 (2001) 767-773].     

严格伪压缩映像族隐格式迭代逼近不动点几何结果

对非线性算子迭代序列逼近不动点过程的几何结构进行研究,在提出并证明了一个H ilbert空间中收敛序列的钝角原理基础上,应用这个钝角原理研究了严格伪压缩映像族的隐格式迭代序列逼近公共不动点的几何结构.并证明了相应的钝角原理.这个钝角原理表述了严格伪压缩映像族的隐格式迭代序列逼近公共不动点时与公共不动点集形成了钝角关系.这个钝角关系是使用相应内积序列的上极限表示的.事实上这个钝角结果的表述形式也是一个几何变分不等式,迭代序列的极限点即是这个几何变分不等式的解.一方面这个钝角结果表述了严格伪压缩映像族公共不动点隐格式逼近的几何过程,另一方面,这个钝角结果自然是隐格式迭代序列逼近严格伪压缩映像族公共不动点的必要条件.     

Convergence of the new composite implicit iteration process with random errors

In this paper, strong convergence theorems for approximation of common fixed points of a finite family of asymptotically demicontractive mappings are proved in Banach spaces using the new composite implicit iteration scheme with errors. Our results of this paper improve and extend the corresponding results of Chen, Song, Zhou [R.D. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Gu [F. Gu, The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings, J. Math. Anal. Appl. 329 (2007) 766–776] and Yang and Hu [L.P. Yang, G. Hu, Convergence of implicit iteration process with random errors, Acta Math. Sinica (Chin. Ser.) 51 (1) (2008) 11–22].     

An iterative process for a finite family of pseudocontractive mappings

The purpose of this paper is to study the following implicit iteration scheme recently introduced by Xu and Ori [Numer. Funct. Anal. Optim., 22, (2001) 767–773]:

Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let be N strictly pseudocontractive self-maps of K such that , where F(T_i)={xK:T_ix=x} and let be two real sequence satisfying the conditions:

where L1 is common Lipschitz constant of . For x₀K, let be new implicit process defined by

x_n=α_nx_n−1+(1−α_n)T_ny_n,

Approximation of fixed points of a strictly pseudocontractive mapping

A fixed point of the strictly pseudocontractive mapping is obtained as the limit of an iteratively constructed sequence with error estimation in general Banach spaces.

     

Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings

To approximate a common fixed point of a countable family of continuous pseudocontractive mappings, we introduce an implicit iteration sequence. A necessary and sufficient condition for the convergence of a sequence of such iterates for countably many continuous pseudocontractive mappings is given. We also prove the convergence theorems of an implicit iteration sequence for a countable family of strictly pseudocontractive mappings.     

逼近严格伪压缩映象不动点的Ishikawa迭代程序

Let(X,‖·‖ ) be a Banach space.Let K be a nonempty closed,convex subset of Xand T∶K→K.Assume that T is Lipschitzian,i.e.there exists L>0 such that‖ T(x) -T(y)‖≤ L‖ x -y‖for all x,y∈K.Withoutloss of generality,assume that L≥ 1 .Assume also that T is strictly pseudocontractive.According to[1 ] this may be statedas:there exists k∈ (0 ,1 ) such that‖ x -y‖≤‖ x -y + r[(I -T -k I) x -(I -T -k I) y]‖for all r>0 and all x,y∈ K.Throughout,let N denote the set of positive in…     

凸度量空间中两无限族严格拟渐近伪压缩映象公共不动点的逼近

凸度量空间中,引入了（λ,{kn}）-严格拟渐近伪压缩映象.并且讨论了两无限族严格拟渐近伪压缩映象公共不动点的逼近问题.在适当的条件下,证明了一些强收敛定理.改进了和推广一些文献的结果.     

Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings

In the present paper, we prove that the modified implicit iteration sequence for a finite family of asymptotically quasi-nonexpansive mappings converges strongly to a common fixed point of the family in a uniformly convex Banach space, requiring one member T in the family to be semi-compact. Our results extend and improve some recent results.     

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.     

Browder-Petryshyn 型的严格伪压缩映射的粘滞迭代逼近方法

主要研究Browder-Petryshyn型的严格伪压缩映射的粘滞迭代逼近过程,证明了Browder-Petryshyn型的严格伪压缩映射的不动点集F(T)是闭凸集．在q-一致光滑且一致凸的Banach空间中,对于严格伪压缩映射T,利用徐洪坤在2004年引进的粘滞迭代得到的序列弱收敛于T的某个不动点．同时证明了Hilbert空间中Browder-Petryshyn型的严格伪压缩映射的相应迭代序列强收敛到T的某个不动点,其结果推广与改进了徐洪坤2004年的相应结果．     

Strong convergence theorems of an iterative scheme for strictly pseudocontractive mappings in Banach spaces

The article establishes the sufficient and necessary conditions for the strong convergence of the Ishikawa iterative scheme with random errors in the sense of Xu for a strictly pseudocontractive mapping in an arbitrary real Banach space under certain assumptions. It affirmatively answers the open question that put forth by Zeng. The results in this paper extend and improve the corresponding results of Browder, Petryshyn, Rhoades, Osilike and Zeng.     

On the implicit iterative process for strictly pseudo-contractive mappings in Banach spaces

In this paper, we consider a composite implicit iterative process for an infinite family of strictly pseudo-contractive mappings. Strong convergence theorems are established in an arbitrary real Banach space. The results presented in this paper improve and extend the recent ones announced by many others.     

Explicit averaging cyclic algorithm for common fixed points of asymptotically strictly pseudocontractive maps

Weak and strong convergence theorems for the approximation of common fixed points of a finite family of asymptotically strictly pseudocontractive mappings are proved in Hilbert spaces using an explicit averaging cyclic algorithm.     

严格伪压缩映象的Ishikawa 迭代序列的收敛率估计

It is shown that any fixed point of each l.ipschitzian, strictly pseudocontractive mapping 7“ on a closed convex subset K of a Banach space X may be norm approximated by Ishikawa iterative procedure. The argument in this paper provides a convergence rate estimate.Moreover the result in this paper improves, generalizes and summarizes some important and elegant recent results.     

A New Implicit Iteration Process with Errors for the Family of Strongly Pseudocontractive Mappings

A method is presented for generating a sequence of lower and upper bounds for the eigenvalues of the problem (i) Tu-λSu = 0, where T and S belong to a class of unbounded and nonsymmetric operators in a separable Hilbert space. Sufficient conditions are derived for the convergence of the sequence of bounds to the eigenvalues of (i), and the applicability of the method is illustrated by approximating the smallest eigenvalue of a non-selfadjoint differential eigenvalue problem.     

An Implicit Iteration Process with Errors for a Finite Family of r-strictly Asymptotically Pseudocontractive Mappings

In this paper,we will establish several strong convergence theorems for the approximation ofcommon fixed points of r-strictly asymptotically pseudocontractive mappings in uniformly convex Banachspaces using the modiied implicit iteration sequence with errors,and prove the necessary and sufficient conditionsfor the convergence of the sequence.Our results generalize,extend and improve the recent work,in thistopic.     

Strong Convergence of New Iteration Process for a Strongly Continuous Semigroup of Asymptotically Pseudocontractive Mappings

In this article, we establish strong convergence theorems for fixed points of strongly continuous semigroup of uniformly Lipschitzian asymptotically pseudocontractive mappings, in the framework of real Banach space.