scholarly journals On the Mann and Ishikawa iteration processes

1996 ◽  
Vol 1 (4) ◽  
pp. 341-349
Author(s):  
Zhou Haiyun ◽  
Jia Yuting
Keyword(s):  
Strong Convergence ◽  
Iteration Process ◽  
Accretive Operators ◽  
Mann Iteration ◽  
Ishikawa Iteration ◽  
Ishikawa Iteration Process ◽  
Mann Iteration Process ◽  
Iteration Processes

It is shown that a result of Chidume, involving the strong convergence of the Mann iteration process for continuous strongly accretive operators, is actually a corollary to a result by Nevanlinna and Reich. It is then shown that the Nevanlinna and Reich result can be extended to the case of an Ishikawa iteration process.

scholarly journals Approximating the zeros of accretive operators by the Ishikawa iteration process

1996 ◽  
Vol 1 (2) ◽  
pp. 153-167 ◽  
Author(s):  
Zhou Haiyun ◽  
Jia Yuting
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Iteration Process ◽  
Accretive Operators ◽  
Convergence Theorems ◽  
Ishikawa Iteration ◽  
Uniformly Smooth ◽  
Ishikawa Iteration Process ◽  
Uniformly Smooth Banach Spaces ◽  
Iteration Processes

Some strong convergence theorems are established for the Ishikawa iteration processes for accretive operators in uniformly smooth Banach spaces.

scholarly journals On New Picard-Mann Iterative Approximations with Mixed Errors for Implicit Midpoint Rule and Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Teng-fei Li ◽  
Heng-you Lan
Keyword(s):  
Optimal Control Problems ◽  
Nonexpansive Mappings ◽  
Elliptic Boundary ◽  
Iteration Process ◽  
Implicit Midpoint Rule ◽  
Mann Iteration ◽  
Midpoint Rule ◽  
Elliptic Boundary Value ◽  
Mann Iteration Process ◽  
Iteration Processes

In order to solve (partial) differential equations, implicit midpoint rules are often employed as a powerful numerical method. The purpose of this paper is to introduce and study a class of new Picard-Mann iteration processes with mixed errors for the implicit midpoint rules, which is different from existing methods in the literature, and to analyze the convergence and stability of the proposed method. Further, some numerical examples and applications to optimal control problems with elliptic boundary value constraints are considered via the new Picard-Mann iterative approximations, which shows that the new Picard-Mann iteration process with mixed errors for the implicit midpoint rule of nonexpansive mappings is brand new and more effective than other related iterative processes.

scholarly journals An iteration process for common fixed points of two nonself asymptotically nonexpansive mappings

2012 ◽  
Vol 20 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Sezgin Akbulut
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Contemporary Literature ◽  
Iteration Process ◽  
Common Fixed Points ◽  
Weak And Strong Convergence ◽  
Mann Iteration ◽  
Asymptotically Nonexpansive ◽  
Ishikawa Iteration Process ◽  
Mann Iteration Process

Abstract In this paper, we introduce an iteration process for approximating common fixed points of two nonself asymptotically nonexpansive map- pings in Banach spaces. Our process contains Mann iteration process and some other processes for nonself mappings but is independent of Ishikawa iteration process. We prove some weak and strong convergence theorems for this iteration process. Our results generalize and improve some results in contemporary literature.

scholarly journals On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Buthinah A. Bin Dehaish ◽  
Rawan K. Alharbi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Mann Iteration ◽  
Approximation Theorems ◽  
Ishikawa Iteration Process ◽  
Mann Iteration Process

The present paper seeks to illustrate approximation theorems to the fixed point for generalized α -nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and refine many of the recently reported results in the literature.

scholarly journals The modified Ishikawa iteration process with errors in CAT(0) spaces

2020 ◽  
Vol 28 (3) ◽  
pp. 217-228
Author(s):  
Sajad Ranjbar
Keyword(s):  
Strong Convergence ◽  
Iteration Process ◽  
Ishikawa Iteration ◽  
Continuous Mappings ◽  
Asymptotically Nonexpansive ◽  
Ishikawa Iteration Process ◽  
Modified Ishikawa Iteration Process

AbstractIn this article, Δ-convergence and strong convergence of the modified Ishikawa iteration process with errors are established for continuous mappings of asymptotically nonexpansive type in CAT(0) spaces. Our results extend and improve the previous results given by many authors.

scholarly journals Convergence Analysis of ParallelS-Iteration Process for System of Generalized Variational Inequalities

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
D. R. Sahu ◽  
Shin Min Kang ◽  
Ajeet Kumar
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Iteration Process ◽  
Mann Iteration ◽  
Generalized Variational Inequalities ◽  
Parallel Iteration ◽  
Mann Iteration Process ◽  
Iteration Processes ◽  
New System

We consider a new system of generalized variational inequalities (SGVI) defined on two closed convex subsets of a real Hilbert space. To find the solution of considered SGVI, a parallel Mann iteration process and a parallelS-iteration process have been proposed and the strong convergence of the sequences generated by these parallel iteration processes is discussed. Numerical example illustrates that the proposed parallelS-iteration process has an advantage over parallel Mann iteration process in computing altering points of some mappings.

scholarly journals A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Songnian He ◽  
Wenlong Zhu
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Boundary Point ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Method ◽  
Minimum Norm ◽  
Mann Iteration ◽  
Boundary Point Method

LetHbe a real Hilbert space andC⊂H a closed convex subset. LetT:C→Cbe a nonexpansive mapping with the nonempty set of fixed pointsFix(T). Kim and Xu (2005) introduced a modified Mann iterationx0=x∈C,yn=αnxn+(1−αn)Txn,xn+1=βnu+(1−βn)yn, whereu∈Cis an arbitrary (but fixed) element, and{αn}and{βn}are two sequences in(0,1). In the case where0∈C, the minimum-norm fixed point ofTcan be obtained by takingu=0. But in the case where0∉C, this iteration process becomes invalid becausexnmay not belong toC. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of Tand prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projectionPC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

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