The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings

Convergence theorems for approximation of common fixed points of strictly pseudocontractive mappings of Browder-Petryshyn type are proved in Banach spaces using a new composite implicit iteration scheme with errors. The results presented in this paper generalize 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].     

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].     

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.     

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.

     

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.     

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.     

Strong convergence theorems for fixed points of asymptotically pseudocontractive semi-groups

Let K be a nonempty closed convex and bounded subset of a real Banach space E. Let be a strongly continuous uniformly asymptotically regular and uniformly L-Lipschitzian semi-group of asymptotically pseudocontractive mappings from K into K. Then for a given u∈K there exists a sequence {y_n}∈K satisfying the equation y_n=(1−α_n)(T(t_n))ⁿy_n+α_nu for each , where α_n∈(0,1) and t_n>0 satisfy appropriate conditions. Suppose further that E is uniformly convex and has uniformly Gâteaux differentiable norm, under suitable conditions on the mappings T, the sequence {y_n} converges strongly to a fixed point of . Furthermore, an explicit sequence {x_n} generated from x₁∈K by x_n+1:=(1−λ_n)x_n+λ_n(T(t_n))ⁿx_n−λ_nθ_n(x_n−x₁) for all integers n?1, where {λ_n}, {θ_n} are positive real sequences satisfying appropriate conditions, converges strongly to a fixed point of .     

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 common fixed points of weakly compatible pairs using the Jungck iteration

We show that the Jungck iteration scheme can be used to approximate the common fixed points of some weakly compatible pairs of generalized quasicontractive operators defined on metric spaces. The existence of coincidence points are also discussed for those pair of maps. The results are generalizations of well known results of the convergence of Picard iterations for single self maps of Banach spaces. In particular, the results improve, generalize and extend the recent results of Berinde [V. Berinde, A common fixed point theorem for compatible quasi contractive self mappings in metric spaces, Applied Mathematics and Computation 213 (2009) 348-354] and answers the open question posed in the paper.     

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．     

伪压缩族公共不动点的复合隐格式迭代逼近问题

设H是一实Hillber空间,K是H之一非空间凸子集,设{Ti}Ni=1是N个Lipschitz伪压缩映象使得F=∩Ni=1F(Ti)≠0,其中F(Ti)={x∈K:Tix=x}并且{αn}n∞=1,{βn}∞n=1[0,1]是满足如下条件的实序列(i)∑∞n=1(1-αn)2= ∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)< ∞;(iv)(1-αn)L2<1,n1;(v)αn(1-βn)2 αn[βn L(1-βn)]2<1,其中L1是{Ti}iN=1的公共Lipschitz常数,对于x0∈K,设{xn}n∞=1是由下列定义的复合隐格式迭代xn=αnxn-1 (1-αn)Tnyn,yn=βnxn (1-βn)Tnxn,其中Tn=TnmodN,则(i)limn→∞‖xn-p‖存在,对于所有的p∈F;(ii)limn→∞d(xn,f)存在,其中d(xn,F)=infp∈F‖xn-p‖;(iii)liminfn→∞‖xn-Tnxn‖=0.本文的结果推广并且改进H-K.Xu和R.G.Ori在2001年的结果和Osilike在2004年的结果,并且在这篇文章中,主要的证明方法也不同与H-K.Xu和Osilike的方法.     

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping on a closed, convex subset of a Banach space is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on , generalizing theorems of Liu.

     

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.     

非扩张映象有限族的公共不动点的隐式迭代方法

本文在实一致凸和q一致光滑Banach空间中研究了一类新的有限族非扩张映象的公共不动点的具误差和具扰动映射的隐式迭代程序并且得到了一些收敛性定理.特别地,获得了该隐式迭代程序强收敛性的充要条件.本文所得结论推广了文[1,2]中的相应结果.     

Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or ?p spaces, 1<p<∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I−T) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.     

On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings

The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of [H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150-159; B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961; P.L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris, Ser. A 284 (1977), 1357-1359; S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287-292; Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) 351-358; R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491; H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optimiz. 22 (2001) 767-773; Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optimiz. 23 (2002) 911-921].