scholarly journals New Conditions for the Exponential Stability of Pseudolinear Difference Equations in Banach Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Rigoberto Medina
Keyword(s):  
Exponential Stability ◽  
Banach Spaces ◽  
Nonlinear Perturbations ◽  
Small Perturbations ◽  
Norm Estimates ◽  
Infinite Dimensional ◽  
Varying Coefficients ◽  
Freezing Method ◽  
Difference Systems ◽  
Combined Use

We study the local exponential stability of evolution difference systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in infinite-dimensional Banach spaces, in the sense that the exponential stability for a given pseudolinear equation persists under sufficiently small perturbations. The main methodology is based on a combined use of new norm estimates for operator-valued functions with the “freezing” method.

scholarly journals Exponential Stability Criteria for Nonautonomous Difference Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Rigoberto Medina
Keyword(s):  
Exponential Stability ◽  
Linear Systems ◽  
Difference Equations ◽  
Linear Operators ◽  
Stability Criteria ◽  
Bounded Linear ◽  
Varying Coefficients ◽  
Nonautonomous Systems ◽  
Freezing Method ◽  
Difference Systems

The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived.

scholarly journals Operators constructed by means of basic sequences and nuclear matrices

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ahmed Morsy ◽  
Nashat Faried ◽  
Samy A. Harisa ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Schauder Basis ◽  
Countable Number ◽  
Nuclear Operator ◽  
Approximation Numbers ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Nuclear Operators

AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

2019 ◽  
Vol 28 (2) ◽  
pp. 191-198
Author(s):  
T. M. M. SOW
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Mann Iteration ◽  
Infinite Dimensional ◽  
Mann Algorithm

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

An analog of Wiener’s theorem for infinite-dimensional Banach spaces

2015 ◽  
Vol 97 (1-2) ◽  
pp. 179-189
Author(s):  
A. V. Zagorodnyuk ◽  
M. A. Mitrofanov
Keyword(s):  
Banach Spaces ◽  
Infinite Dimensional ◽  
Wiener’S Theorem

scholarly journals Asymptotic Infinite-Dimensional Theory of Banach Spaces

1995 ◽  
pp. 149-175 ◽  
Author(s):  
Bernard Maurey ◽  
Vitali Milman ◽  
Nicole Tomczak-Jaegermann
Keyword(s):  
Banach Spaces ◽  
Dimensional Theory ◽  
Infinite Dimensional

Conjugacies between linear and nonlinear non-uniform contractions

2008 ◽  
Vol 28 (1) ◽  
pp. 1-19 ◽  
Author(s):  
LUIS BARREIRA ◽  
CLAUDIA VALLS
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Linear Operators ◽  
Nonlinear Perturbations ◽  
Exponential Dichotomies ◽  
Locally Lipschitz ◽  
Linear And Nonlinear

AbstractWe construct conjugacies between linear and nonlinear non-uniform exponential contractions with discrete time. We also consider the general case of a non-autonomous dynamics defined by a sequence of maps. The results are obtained by considering both linear and nonlinear perturbations of the dynamics xm+1=Amxm defined by a sequence of linear operators Am. In the case of conjugacies between linear contractions we describe them explicitly. All the conjugacies are locally Hölder, and in fact are locally Lipschitz outside the origin. We also construct conjugacies between linear and nonlinear non-uniform exponential dichotomies, building on the arguments for contractions. All the results are obtained in Banach spaces.

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.

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