scholarly journals The Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings in Banach Space

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 512
Author(s):  
Huancheng Zhang ◽  
Yunhua Qu ◽  
Yongfu Su
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Implicit Midpoint Rule ◽  
Numerical Examples ◽  
Midpoint Rule

This paper constructs the generalized viscosity implicit midpoint rule for nonexpansive mappings in Banach space. It obtains strong convergence conclusions for the proposed algorithm and promotes the related results in this field. Moreover, this paper gives some applications. Finally, the paper gives six numerical examples to support the main results.

scholarly journals Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

1991 ◽  
Vol 43 (1) ◽  
pp. 153-159 ◽  
Author(s):  
J. Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive ◽  
Fourth Kind

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

2013 ◽  
Vol 756-759 ◽  
pp. 3628-3633
Author(s):  
Yuan Heng Wang ◽  
Wei Wei Sun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

scholarly journals Strong Convergence on Iterative Methods of Cesàro Means for Nonexpansive Mapping in Banach Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Zhichuan Zhu ◽  
Rudong Chen
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Nonexpansive Mappings ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means

Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained asn→∞. Our main results extend and improve the corresponding results of Xu (2004), Song and Chen (2007), and Yao et al. (2009).

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.

New hybrid algorithms for global minimization of common best proximity points of some generalized nonexpansive mappings

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2381-2391
Author(s):  
Jenwit Puangpee ◽  
Suthep Suantai
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Best Proximity Point ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Hybrid Algorithms ◽  
Global Minimization ◽  
Numerical Examples ◽  
Common Best Proximity Point ◽  
Two Hybrid

In this paper, we introduce two hybrid algorithms for finding a common best proximity point of two best proximally nonexpansive mappings. We establish strong convergence theorems of the proposed algorithms under some control conditions in a real Hilbert space. Moreover, some numerical examples are given for supporting our main theorems.

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