scholarly journals Relaxed iterative methods for an infinite family of d-accretive mappings in a Banach space and their applications

2018 ◽  
Vol 2018 (1) ◽  
pp. 1-16 ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Infinite Family ◽  
Accretive Mappings

scholarly journals Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuanheng Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

Convergence of derivative free iterative methods

2019 ◽  
Vol 28 (1) ◽  
pp. 19-26
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
SANTHOSH GEORGE ◽  
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Local Convergence ◽  
Practical Interest ◽  
Systems Of Equations ◽  
Hölder Continuous ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Derivative Free ◽  
Special Cases

We present the local as well as the semi-local convergence of some iterative methods free of derivatives for Banach space valued operators. These methods contain the secant and the Kurchatov method as special cases. The convergence is based on weak hypotheses specializing to Lipschitz continuous or Holder continuous hypotheses. The results are of theoretical and practical interest. In particular the method is compared favorably ¨ to other methods using concrete numerical examples to solve systems of equations containing a nondifferentiable term.

scholarly journals Local solvability of a constrainedgradient system of total variation

2003 ◽  
Vol 2003 (6) ◽  
pp. 353-365 ◽  
Author(s):  
C. E. Chidume ◽  
H. Zegeye
Keyword(s):  
Banach Space ◽  
Iterative Methods ◽  
Total Variation ◽  
Local Solvability ◽  
Independent Interest ◽  
Smooth Banach Space ◽  
Uniformly Smooth Banach Space ◽  
Uniformly Smooth ◽  
Compact Subsets

SupposeXis a realq-uniformly smooth Banach space andF,K:X→XwithD(K)=F(X)=Xare accretive maps. Under various continuity assumptions onFandKsuch that0=u+KFuhas a solution, iterative methods which converge strongly to such a solution are constructed. No invertibility assumption is imposed onKand the operatorsKandFneed not be defined on compact subsets ofX. Our method of proof is of independent interest.

scholarly journals Two-Step Viscosity Approximation Scheme for Variational Inequality in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Liping Yang ◽  
Weiming Kong
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Approximation Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Viscosity Approximation ◽  
Convergence Conditions ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings{Ti}i=1∞in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of{Ti}i=1∞, which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.

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 Unified Semi-Local Convergence for k—Step Iterative Methods with Flexible and Frozen Linear Operator

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 233 ◽  
Author(s):  
Ioannis Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Iterative Method ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Local Convergence ◽  
Small Region ◽  
Convergence Region ◽  
Local Convergence Analysis ◽  
Lipschitz Conditions

The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton’s, or Stirling’s, or Steffensen’s, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis.

scholarly journals A further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

1995 ◽  
Vol 38 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Zong-Ben Xu ◽  
Yao-Lin Jiang ◽  
G. F. Roach
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Accretive Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

Iterative Methods for Strictly Pseudo-Contractive Mappings

2013 ◽  
Vol 333-335 ◽  
pp. 1402-1405
Author(s):  
Yang Liu ◽  
Yan Hao
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Smooth Banach Space ◽  
Convergence Theorems ◽  
Uniformly Smooth Banach Space ◽  
Contractive Mappings ◽  
Uniformly Smooth

The aim of this work is to consider an iterative method for a-strict pseudo-contractions. Strong convergence theorems are established in a real 2-uniformly smooth Banach space.

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