scholarly journals Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical banach spaces

SpringerPlus ◽  
2015 ◽  
Vol 4 (1) ◽  
Author(s):  
C E Chidume ◽  
A U Bello ◽  
B Usman
Keyword(s):  
Banach Spaces ◽  
Strongly Monotone ◽  
Lipschitz Maps ◽  
Type Algorithm

Fr ´Echet Differentiability Except For Γ‎-Null Sets

2012 ◽  
Author(s):  
Joram Lindenstrauss ◽  
David Preiss ◽  
Tiˇser Jaroslav
Keyword(s):  
Banach Spaces ◽  
Common Point ◽  
Lipschitz Functions ◽  
Infinite Dimensional ◽  
Lipschitz Maps ◽  
Almost Everywhere ◽  
Fréchet Differentiability ◽  
Gateaux Derivatives ◽  
Frechet Differentiability ◽  
Porous Sets

This chapter gives an account of the known genuinely infinite dimensional results proving Fréchet differentiability almost everywhere except for Γ‎-null sets. Γ‎-null sets provide the only notion of negligible sets with which a Fréchet differentiability result is known. Porous sets appear as sets at which Gâteaux derivatives can behave irregularly, and they turn out to be the only obstacle to validity of a Fréchet differentiability result Γ‎-almost everywhere. Furthermore, geometry of the space may (or may not) guarantee that porous sets are Γ‎-null. The chapter also shows that on some infinite dimensional Banach spaces countable collections of real-valued Lipschitz functions, and even of fairly general Lipschitz maps to infinite dimensional spaces, have a common point of Fréchet differentiability.

scholarly journals A NEW COARSELY RIGID CLASS OF BANACH SPACES

2020 ◽  
pp. 1-19
Author(s):  
F. Baudier ◽  
G. Lancien ◽  
P. Motakis ◽  
Th. Schlumprecht
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Concentration Inequality ◽  
Metric Characterization ◽  
Lipschitz Maps ◽  
Hamming Graphs

We prove that the class of reflexive asymptotic- $c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic- $c_{0}$ space $Y$ , then $X$ is also reflexive and asymptotic- $c_{0}$ . In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic- $c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

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