Nonuniform Hyperbolicity and Admissibility

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Luis Barreira ◽  
Claudia Valls ◽  
Davor Dragičevič
Keyword(s):  
Exponential Dichotomy ◽  
Linear Operators ◽  
Nonuniform Hyperbolicity

AbstractFor a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓ

Lyapunov theorems for exponential dichotomies in Hilbert spaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650033 ◽  
Author(s):  
Davor Dragičević ◽  
Ciprian Preda
Keyword(s):  
Hilbert Spaces ◽  
Exponential Dichotomy ◽  
Complete Characterization ◽  
Linear Operators ◽  
Nonlinear Perturbations ◽  
Exponential Behavior ◽  
Exponential Dichotomies ◽  
Linear And Nonlinear ◽  
Unstable Direction

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

scholarly journals On Spectral Characterization of Nonuniform Hyperbolicity

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Muna Abu Alhalawa ◽  
Davor Dragičević
Keyword(s):  
Linear Operators ◽  
Sharp Contrast ◽  
Spectral Characterization ◽  
Fréchet Space ◽  
Nonuniform Hyperbolicity ◽  
Exponential Dichotomies ◽  
Infinite Dimensional ◽  
Theoretic Characterization ◽  
Principal Advantage

We give a complete functional theoretic characterization of tempered exponential dichotomies in terms of the invertibility of certain linear operators acting on a suitable Frechét space. In sharp contrast to previous results, we consider noninvertible linear cocycles acting on infinite-dimensional spaces. The principal advantage of our results is that they avoid the use of Lyapunov norms.

Exponential dichotomies with respect to a sequence of norms and admissibility

2014 ◽  
Vol 25 (03) ◽  
pp. 1450024 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomy ◽  
Linear Operators ◽  
Conditional Stability ◽  
Exponential Behavior ◽  
Exponential Dichotomies ◽  
Linear Perturbations ◽  
Nonuniform Exponential Dichotomy

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in lp spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

scholarly journals Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

2021 ◽  
pp. 1-27
Author(s):  
Tomás Caraballo ◽  
Alexandre N. Carvalho ◽  
José A. Langa ◽  
Alexandre N. Oliveira-Sousa
Keyword(s):  
Differential Equations ◽  
Continuous Dependence ◽  
Exponential Dichotomy ◽  
Nonlinear Evolution ◽  
Nonuniform Hyperbolicity ◽  
Exponential Dichotomies ◽  
Infinite Dimensional ◽  
Linear Evolution ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

On αβ-statistical convergence for sequences of fuzzy mappings and Korovkin type approximation theorem

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3749-3760 ◽  
Author(s):  
Ali Karaisa ◽  
Uğur Kadak
Keyword(s):  
Statistical Convergence ◽  
Approximation Theorem ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Bernstein Operator ◽  
Korovkin Type Approximation Theorem ◽  
Type Approximation ◽  
Inclusion Relations ◽  
Prior Investigation

Upon prior investigation on statistical convergence of fuzzy sequences, we study the notion of pointwise ??-statistical convergence of fuzzy mappings of order ?. Also, we establish the concept of strongly ??-summable sequences of fuzzy mappings and investigate some inclusion relations. Further, we get an analogue of Korovkin-type approximation theorem for fuzzy positive linear operators with respect to ??-statistical convergence. Lastly, we apply fuzzy Bernstein operator to construct an example in support of our result.

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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