Error Estimate of Data Dependence for Discontinuous Operators by New Iteration Process with Convergence Analysis

Abstract

In this paper, we introduce a new discontinuous operator and investigate the existence and uniqueness of fixed points for the operators in complete metric spaces. We also provide rate of convergence and data dependency of S-iterative scheme for a fixed point of the discontinuous operators in Banach spaces. Moreover, we prove the estimation Collage theorems and compare error estimate between data dependency and Collage theorems. Numerical examples are provided to support our results.     

Weak Convergence Theorems for Nonself Mappings

Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {k(i)n}  [1, ∞)(i = 1, 2), and F := F(T1)∩F(T2) = . An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Fr′echet differentiable norm or its dual E*has Kadec-Klee property, then weak convergence theorems are obtained.     

关于m-增生算子零点的粘性复合迭代序列的强收敛性

在一致光滑的Banach空间的框架下,引入关于m-增生算子的一种新粘性复合迭代序列{xn},并证明了在适当的控制条件下,此迭代序列强收敛于m-增生算子的一个零点,用不同方法推广了相关文献的近代结果.     

渐近非扩张非自映射的收敛定理(英文)

设E是实的一致凸Banach空间,K是E的一个非空闭凸集,P是E到K上的非扩张的保核收缩映射.设T1,T2,T3:K→E分别是具有数列{hn},{ln},{kn}[1,∞)的渐近非扩张非自映射,使得sum (hn-1) from n=1 to ∞<∞,sum ((ln-1)) from n=1 to ∞<∞及sum (n=1(kn-1) from n=1 to ∞<∞,且F=F(T1)∩F(T2)∩F(T3)={x∈K:T1x=T2x=T3x}≠Ф.定义迭代序列{xn}:x1∈K,xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1zn),zn=P((1-γn)xn+γnT3(PT3)n-1xn),其中{αn},{βn},{γn}[ε,1-ε],ε是大于零的实数.(i)如果T1,T2,T3中有一个是全连续的或者半紧的,则{xn}强收敛于某一点q∈F;(ii)如果E具有Frechet可微范数或者满足Opial’s条件或者E的对偶空间E~*具有Kadec-Klee性质,则{xn}弱收敛于某一点q∈F.     

关于增生算子方程解的Ishikawa迭代逼近

设X是任意实Banach空间,T:X→X是Lipschitz连续的增生算子.本文证明了,带误差的Ishikawa迭代序列强收敛到方程x Tx=f的唯一解.而且,还给Ishikawa迭代序列提供了一般的收敛率估计.利用该结果,本文推得,若T:X→X是Lipschitz连续的强增生算子,则带误差的Ishikawa迭代序列强收敛到方程Tx=f的唯一解.     

Banach空间中m-d-增生算子零点迭代算法的构造

研究了Banach空间中m-d-增生算子零点的迭代算法的构造问题,获得了一个强收敛定理.     

Convergence and Data Dependency of Normal−S Iterative Method for Discontinuous Operators on Banach Space

We study convergence, rate of convergence and data dependency of normal?S iterative method for a fixed point of a discontinuous operator T on a Banach space. We also prove some Collage type theorems for T. The main aim here is to show that there is a close relationship between the concepts of data dependency of fixed points and the collage theorems and show that the latter provides better estimate. Numerical examples in support of the results obtained are also given.     

A New Hybrid Algorithm and Its Numerical Realization for a Quasi-nonexpansive Mapping

The purpose of this article is to propose a new hybrid projection method for a quasi-nonexpansive mapping.The strong convergence of the algorithm is proved in real Hilbert spaces.A numerical experiment is also included to explain the effectiveness of the proposed methods.The results of this paper axe interesting extensions of those known results.     

A Unified Convergence Theory for Newton-Type Methods for Zeros of Nonlinear Operators in Banach Spaces

The paper is concerned with the convergence problem of Newton type methods for finding zeros of nonlinear operators in Banach spaces. Some families of nonlinear operators redefined by different Lipschitz conditions and an universal constant is introduced so that unified convergence determination of these methods is established for the defined families.     

一类Sylvester矩阵方程的一个迭代算法

考虑这样一类Sylvester矩阵方程:AX XB=C,A,B分别为n阶正半定、正定矩阵,C为n阶矩阵.给出了一个收敛的迭代算法.     

Banach空间中关于(H,η)-增生算子的一类新变分包含组问题

In this paper,we first introduce a new class of generalized accretive operators named(H,η)-accretive in Banach space.By studying the properties of(H,η)-accretive,we extend the concept of resolvent operators associated with m-accretive operators to the new(H,η)-accretive operators.In terms of the new resolvent operator technique,we prove the existence and uniqueness of solutions for this new system of variational inclusions.We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.     

Iterative Algorithm for Finding Approximate Solutions of a Class of Mixed Variational-like Inequalities

The purpose of this paper is to investigate the iterative algorithm for finding approximate solutionsof a class of mixed variational-like inequalities in a real Hilbert space,where the iterative algorithm is presentedby virtue of the auxiliary principle technique.On one hand,the existence of approximate solutions of this classof mixed variational-like inequalities is proven.On the other hand,it is shown that the approximate solutionsconverge strongly to the exact solution of this class of mixed variational-like inequalities.     

有限个非扩张非自映像隐迭代格式的逼近

假设E为一致凸Banach空间,K为E的非空闭凸子集且为E的非扩张收缩,P为非扩张收缩映像.{Ti:i=1,2,…,N}:K→E为非扩张映像且F(T)=∩ from i=1 to N F(Ti)≠■.定义{xn}如下:x0∈K,xn=P(αnxn-1+(1-αn)TnP[βnxn-1+(1-βn)Tnxn]),n≥1,这里{αn},{βn}为[δ,1-δ]中的实序列,其中δ∈(0,1).若{Ti:i=1,2,…,N}满足条件(B),则{xn}强收敛于x*∈F(T).     

Lipschitz强增生算子方程逼近解的带误差的Ishikawa迭代程序

设E是任意实Banach空间，T：E→E是Ligpschitz的强增生算子。证明了带误差的Ishikawa迭代序列强收敛到方程Tx=f的唯一解。特别地，还给出了Ishikawa迭代序列的收敛率估计。另一方面，一个相关结果，讨论了E中Lipschitz强伪压缩映象的不动点的带误差的Ishikawa迭代序列的收敛性。     

Strong convergence of a general viscosity explicit rule for the sum of two monotone operators in Hilbert spaces

In this paper, we study a general viscosity explicit rule for approximating the solutions of the variational inclusion problem for the sum of two monotone operators. We then prove its strong convergence under some new conditions on the parameters in the framework of Hilbert spaces. As applications, we apply our main result to the split feasibility problem and the LASSO problem. We also give some numerical examples to support our main result. The results presented in this paper extend and improve the corresponding results in the literature.     

非扩张映射具有误差修改Ishikawa迭代算法的强收敛

在自反的Banach空间中,通过引进一个新的具有误差修改的Ishikawa迭代算法,在适当条件下,得到了关于一族非扩张映射公共不动点的强收敛定理,所获结果推广和改进了一些已知结论,最后给出了一个例子说明结果的应用.     

广义单调集值混合变分不等式的一类新算法

首先证明了广义单调集值混合变分不等式等价于一个新的不动点问题，在此基础上提出了解广义集值混合变分不等式及其相关优化问题的迭代算法，并给出了这类新算法的收敛性分析，我们的结果推广和综合了该领域的一些最新结论．     

B值随机元序列加权和的强收敛性

本文研究了B值随机元序列加权和的强收敛性.利用Banach空间的几何性质(p型或p阶光滑),在较弱的条件下得到了B值随机元序列加权和的强大数律,这些结果推广和改进了已知的一些文献中相应的结论.