On nonuniform exponential dichotomy of evolution operators in Banach spaces

2002 ◽  
Vol 44 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Mihail Megan ◽  
Bogdan Sasu ◽  
Adina Lumini?a Sasu
Keyword(s):  
Banach Spaces ◽  
Exponential Dichotomy ◽  
Evolution Operators ◽  
Nonuniform Exponential Dichotomy

scholarly journals Roughness of(ℤ+,ℤ-)-Nonuniform Exponential Dichotomy for Difference Equations in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nicolae Lupa
Keyword(s):  
Banach Spaces ◽  
Explicit Form ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
General Context ◽  
Boundedness Condition ◽  
Infinite Dimensional ◽  
Perturbed Equation ◽  
Nonuniform Exponential Dichotomy

In this paper we study the roughness of(ℤ+,ℤ-)-nonuniform exponential dichotomy for nonautonomous difference equations in the general context of infinite-dimensional spaces. An explicit form is given for each of the dichotomy constants of the perturbed equation in terms of the original ones. We emphasize that we do not assume any boundedness condition on the coefficients.

scholarly journals On uniform exponential splitting for noninvertible evolution operators in Banach Spaces

2015 ◽  
Vol 53 (2) ◽  
pp. 121-131 ◽  
Author(s):  
Claudia Luminiţa Mihiţ ◽  
Codruţa Simona Stoica ◽  
Mihail Megan
Keyword(s):  
Banach Spaces ◽  
Lyapunov Functions ◽  
Exponential Dichotomy ◽  
Evolution Operator ◽  
General Concept ◽  
Integral Inequalities ◽  
Evolution Operators ◽  
Exponential Splitting

Abstract The paper considers the general concept of uniform exponential splitting as a generalization of uniform exponential dichotomy property for evolution operators in Banach spaces. Two characterizations in terms of integral inequalities of Datko-type respectively Lyapunov functions for uniform exponential splitting of a noninvertible evolution operator with respect to invariant projections families are obtained.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals On some dichotomy concepts with different growth rates for evolution operators in Banach spaces

2015 ◽  
Vol 53 (2) ◽  
pp. 133-150
Author(s):  
Nicolae Marian Seimeanu
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Growth Rates ◽  
Real Numbers ◽  
Evolution Operators

Abstract This paper treats three concepts of (h, k)-dichotomy and their correspondents in the uniform cases. The connections between them are established through examples and counterexamples presented on the Banach space of square-summable sequences of real numbers.

scholarly journals Connections between Some Concepts of Polynomial Trichotomy for Noninvertible Evolution Operators in Banach Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mihai-Gabriel Babuţia ◽  
Nicolae Marian Seimeanu
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Evolution Operators

The present paper treats three concepts of nonuniform polynomial trichotomies for noninvertible evolution operators acting on Banach spaces. The connections between these concepts are established through numerous examples and counterexamples for systems defined on the Banach space of square-summable sequences.

NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

2015 ◽  
Vol 58 (2) ◽  
pp. 279-291
Author(s):  
LUIS BARREIRA ◽  
LIVIU HORIA POPESCU ◽  
CLAUDIA VALLS
Keyword(s):  
Asymptotic Behaviour ◽  
Ergodic Theory ◽  
Canonical Form ◽  
Exponential Dichotomy ◽  
Topological Equivalence ◽  
Exponential Dichotomies ◽  
Exponential Behaviour ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy ◽  
Topological Conjugacies

AbstractWe show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

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