scholarly journals On the computational content of convergence proofs via Banach limits

2012 ◽  
Vol 370 (1971) ◽  
pp. 3449-3463 ◽  
Author(s):  
U. Kohlenbach ◽  
L. Leuştean
Keyword(s):  
Proof Theory ◽  
Quantitative Information ◽  
Prima Facie ◽  
New Developments ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
Banach Limits ◽  
Convergence Proofs ◽  
The Axiom Of Choice ◽  
Highly Uniform

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X . Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

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.

scholarly journals The smooth variational principle and generic differentiability

1991 ◽  
Vol 43 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Pando Grigorov Georgiev
Keyword(s):  
Banach Space ◽  
Variational Principle ◽  
Continuous Function ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
Generic Differentiability

A modified version of the smooth variational principle of Borwein and Preiss is proved. By its help it is shown that in a Banach space with uniformly Gâteaux differentiable norm every continuous function, which is directionally differentiable on a dense Gδ subset of the space, is Gâteaux differentiable on a dense Gδ subset of the space.

scholarly journals Viscosity Approximation Methods for Two Accretive Operators in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Jun-Min Chen ◽  
Tie-Gang Fan
Keyword(s):  
Iterative Scheme ◽  
Convex Banach Space ◽  
Approximation Methods ◽  
Accretive Operators ◽  
Viscosity Approximation ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
The Common ◽  
Viscosity Approximation Methods

We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of Ceng et al. (2009) and some others.

scholarly journals Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-17
Author(s):  
Jong Soo Jung
Keyword(s):  
Strong Convergence ◽  
Reflexive Banach Space ◽  
Iterative Scheme ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Weakly Compact ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
Iteration Methods

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

scholarly journals Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
N. Djitte ◽  
M. Sene
Keyword(s):  
Banach Space ◽  
Integral Equations ◽  
Iteration Method ◽  
Real Banach Space ◽  
Nonlinear Integral Equations ◽  
Nonlinear Operators ◽  
Hammerstein Equation ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

Amorphous Semiconductor Multilayer Structures: Interface and Layer Thickness Effects in Photoluminescence

1985 ◽  
Vol 49 ◽  
Author(s):  
T. Tiedje
Keyword(s):  
Amorphous Semiconductor ◽  
Amorphous Semiconductors ◽  
Multilayer Structures ◽  
Length Scale ◽  
Structural Perfection ◽  
New Developments ◽  
Transmission Electron ◽  
High Degree ◽  
Highly Uniform ◽  
Superlattice Period

A number of new developments have occurred recently in research on the synthesis and properties of amorphous semiconductor multilayer structures (“amorphous superlattices”) since the discovery of this class of materials in 1983.1 This and more recentwork have shown that tetrahedrally bonded amorphous semiconductors can be fabricated in the form of multilayer structures, with highly uniform layers and atomically abrupt interfaces. The remarkably high degree of structural perfection in these materials on the length scale of the superlattice period (> 5A) has been demonstrated by transmission electron microscopy.

Properties and Predicates, Objects and Names

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Prima Facie ◽  
Axiom Of Choice ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient

Hale has articulated and defended a thesis that properties are tied to predicates in possible languages. The same goes for functions. A necessary and sufficient condition for the existence of a property or function is that it is expressed by a predicate in a possible language that can be understood by finite beings, like us humans. The purpose of this chapter is to assess this view and determine how it fares against the output of standard mathematics. In order to interpret mathematics, Hale must defend some prima facie implausible claims about what sorts of languages are possible, for us finite beings. The only alternative is for Hale to reject, on purely philosophical grounds, large chunks of contemporary mathematics. The relevant mathematics is whatever relies on impredicative definitions and the axiom of choice. This includes his, and Crispin Wright’s, abstraction-based neologicism, which has been shown to rely indispensably on impredicative definitions.

scholarly journals Strong Convergence of Viscosity Methods for Continuous Pseudocontractions in Banach Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-11
Author(s):  
Filomena Cianciaruso ◽  
Giuseppe Marino ◽  
Luigi Muglia ◽  
Haiyun Zhou
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Viscosity Method ◽  
Reflexive Banach Spaces ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm ◽  
Convex Subsets

We define a viscosity method for continuous pseudocontractive mappings defined on closed and convex subsets of reflexive Banach spaces with a uniformly Gâteaux differentiable norm. We prove the convergence of these schemes improving the main theorems in the work by Y. Yao et al. (2007) and H. Zhou (2008).

scholarly journals An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Youli Yu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Reflexive Banach Space ◽  
Contractive Mapping ◽  
Bounded Subset ◽  
Point Property ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm ◽  
Uniformly Continuous

LetEbe a real reflexive Banach space with a uniformly Gâteaux differentiable norm. LetKbe a nonempty bounded closed convex subset ofE,and every nonempty closed convex bounded subset ofKhas the fixed point property for non-expansive self-mappings. Letf:K→Ka contractive mapping andT:K→Kbe a uniformly continuous pseudocontractive mapping withF(T)≠∅. Let{λn}⊂(0,1/2)be a sequence satisfying the following conditions: (i)limn→∞λn=0; (ii)∑n=0∞λn=∞. Define the sequence{xn}inKbyx0∈K,xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for alln≥0. Under some appropriate assumptions, we prove that the sequence{xn}converges strongly to a fixed pointp∈F(T)which is the unique solution of the following variational inequality:〈f(p)−p,j(z−p)〉≤0, for allz∈F(T).

scholarly journals Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Thanomsak Laokul
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergence Theorem ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Multivalued Mappings ◽  
Uniformly Convex ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

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