Geometry of Banach limits and their applications

2020 ◽  
Vol 75 (4) ◽  
pp. 725-763
Author(s):  
E. M. Semenov ◽  
F. A. Sukochev ◽  
A. S. Usachev
Keyword(s):  
Banach Limits

Functional Banach Limits

2021 ◽  
pp. 143-160
Author(s):  
Gokulananda Das ◽  
Sudarsan Nanda
Keyword(s):  
Banach Limits

scholarly journals Common Attractive Points of Generalized Hybrid Multi-Valued Mappings and Applications

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1307
Author(s):  
Lili Chen ◽  
Ni Yang ◽  
Jing Zhou
Keyword(s):  
Variational Inequality ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Approximation Method ◽  
Variational Inequality Problem ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Banach Limits ◽  
Weak Convergence Theorem ◽  
The Common

In this paper, we first propose the concepts of (ζ,η,λ,π)-generalized hybrid multi-valued mappings, the set of all the common attractive points (CAf,g) and the set of all the common strongly attractive points (CsAf,g), respectively for the multi-valued mappings f and g in a CAT(0) space. Moreover, we give some elementary properties in regard to the sets Af, Ff and CAf,g for the multi-valued mappings f and g in a complete CAT(0) space. Furthermore, we present a weak convergence theorem of common attractive points for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in the above space by virtue of Banach limits technique and Ishikawa iteration respectively. Finally, we prove strong convergence of a new viscosity approximation method for two (ζ,η,λ,π)-generalized hybrid multi-valued mappings in CAT(0) spaces, which also solves a kind of variational inequality problem.

Order properties of the set of Banach limits

2015 ◽  
Vol 91 (1) ◽  
pp. 20-22
Author(s):  
E. A. Alekhno ◽  
E. M. Semenov ◽  
F. A. Sukochev ◽  
A. S. Usachev
Keyword(s):  
Banach Limits

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.

Extreme points of the set of Banach limits

Positivity ◽  
2012 ◽  
Vol 17 (1) ◽  
pp. 163-170 ◽  
Author(s):  
E. Semenov ◽  
F. Sukochev
Keyword(s):  
Extreme Points ◽  
Banach Limits

scholarly journals An application of Banach limits

1988 ◽  
Vol 103 (1) ◽  
pp. 244-244 ◽  
Author(s):  
Z. U. Ahmad ◽  
Mursaleen
Keyword(s):  
Banach Limits
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