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Claudio A. Gallegos

Publications and source records attributed to Claudio A. Gallegos.

7 recordsLinked to original sources
arXiv2026

Evolution families and variation of constants formula for abstract functional differential equations with time-dependent infinite delay

In this paper, we consider a class of first-order abstract retarded functional differential equations in Banach spaces, incorporating a time-dependent infinite delay governed by a regulated function. We establish the existence of mild solutions for the nonlinear equation and show that the family of solution maps for the linear equation forms a well-defined evolution family of bounded linear operators on an appropriate phase space. Furthermore, we leverage this evolution family to prove a variation of constants formula for the inhomogeneous linear problem.

math.CA↗
arXiv2025

The interplay of $μ$-dichotomy, bounded growth, and spectral properties via growth rate comparisons

We investigate the behavior of the dichotomy spectrum of nonautonomous linear systems under general growth rates. By introducing comparison criteria we clarify how $μ$-dichotomy and $μ$-bounded growth interact. We also study the evolution of the dichotomy spectrum under these comparisons, revealing that faster growth rates compress the spectrum, while slower growth rates expand it. Moreover, we introduce equivalence relations on the set of growth rates, which enable us to establish that, for any given system, there exists -- up to equivalence -- at most one growth rate under which both properties, bounded growth and dichotomy, hold. Finally, we show that these equivalence relations lead to a classification of dichotomy spectra. Our results are valid in both discrete and continuous time settings.

math.DS↗
arXiv2025

Discrete $μ$-dichotomy spectrum: beyond uniformity and new insights

We develop spectral theorems for nonautonomous linear difference systems, considering different types of $μ$-dichotomies, both uniform and nonuniform. In the nonuniform case, intriguing scenarios emerge -- that have been employed but whose consequences have not been thoroughly explored -- which surprisingly exhibit unconventional behavior. These particular cases motivate us to introduce two novel properties of nonautonomous systems (even in the continuous-time framework), which appear to have been overlooked in the existing literature. Additionally, we introduce a new conceptualization of a nonuniform $μ$-dichotomy spectrum, which lies between the traditional nonuniform $μ$-dichotomy spectrum and the slow nonuniform $μ$-dichotomy spectrum. Moreover, and this is particularly noteworthy, we propose a conjecture that enables the derivation of spectral theorems in this new setting. Finally, contrary to what has been believed in recent years, through the lens of optimal ratio maps, we show that the nonuniform exponential dichotomy spectrum is not preserved between systems that are weakly kinematically similar.

math.DS↗
arXiv2024

Spectrum invariance dilemma for nonuniformly kinematically similar systems

We unveil instances where nonautonomous linear systems manifest distinct nonuniform $μ$-dichotomy spectra despite admitting nonuniform $(μ, \varepsilon)$-kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform $μ$-dichotomy such as in the notion of nonuniform $(μ, \varepsilon)$-kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the $\varepsilon$-neighborhood of the nonuniform $μ$-dichotomy spectrum. These concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform $μ$-dichotomy spectrum.

nlin.CD↗
arXiv2023

On stability for generalized linear differential equations and applications to impulsive systems

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

math.CA↗
arXiv2022

Hybrid measure differential equations: existence of solutions and continuous dependence on parameters

This paper is devoted to study the qualitative properties of hybrid measure differential equations (HMDEs, for short). We establish several results on the existence of global solutions, including the existence of regulated, continuous, differentiable and S-asymptotically $ω$-periodic solutions. Furthermore, we present a result on continuous dependence of solutions in terms of parameters. Our results are based on extensions of Kasnoselskii's fixed-point theorem.

math.CA↗