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Manuel Pinto

Publications and source records attributed to Manuel Pinto.

At least 19 recordsLinked to original sources
arXiv2024

Poincar\'e-Perron problem for high order differential equations in the class of almost periodic type functions

We address the Poincar\'e-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order $n\ge 3$, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.

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arXiv2024

Compact almost automorphic dynamics of non-autonomous differential equations with exponential dichotomy and applications to biological models with delay

In the present work, we prove that, if $A(\cdot)$ is a compact almost automorphic matrix and the system $$x'(t) = A(t)x(t)\, ,$$ possesses an exponential dichotomy with Green function $G(\cdot, \cdot)$, then its associated system $$y'(t) = B(t)y(t)\, ,$$ where $B(\cdot) \in H(A)$ (the hull of $A(\cdot)$) also possesses an exponential dichotomy. Moreover, the Green function $G(\cdot, \cdot)$ is compact Bi-almost automorphic in $\mathbb{R}^2$, this implies that $G(\cdot, \cdot)$ is $\Delta_2$ - like uniformly continuous, where $\Delta_2$ is the principal diagonal of $\mathbb{R}^2$, an important ingredient in the proof of invariance of the compact almost automorphic function space under convolution product with kernel $G(\cdot, \cdot)$. Finally, we study the existence of a positive compact almost automorphic solution of non-autonomous differential equations of biological interest having non-linear harvesting terms and mixed delays.

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arXiv2024

A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. We also give some examples showing the applicability of our results.

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arXiv2023

Exponential trichotomy and global linearization of non-autonomous differential equations

Hartman-Grobman theorem was initially extended to the non-autonomous cases by Palmer. Usually, dichotomy is an essential condition of Palmer's linearization theorem. Is Palmer's linearization theorem valid for the systems with trichotomy? In this paper, we obtain new versions of the linearization theorem if linear system admits exponential trichotomy on $\mathbb{R}$. { Furthermore, the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems are H\"{o}lder continuous}. In addition, if a system is periodic, we find the equivalent function $\mathscr H(t,x)$ and its inverse $\mathscr L(t,y)$ of our linearization theorems do not have periodicity or asymptotical periodicity. To the best of our knowledge, this is the first paper studying the linearization with exponential trichotomy.

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arXiv2022

Sharpness of $C^0$ conjugacy for the non-autonomous differential equations with Lipschitzian perturbation

The classical $C^0$ linearization theorem for the non-autonomous differential equations states the existence of a $C^0$ topological conjugacy between the nonlinear system and its linear part. That is, there exists a homeomorphism (equivalent function) $H$ sending the solutions of the nonlinear system onto those of its linear part. It is proved in the previous literature that the equivalent function $H$ and its inverse $G=H^{-1}$ are both H\"{o}lder continuous if the nonlinear perturbation is Lipschitzian. Questions: is it possible to improve the regularity? Is the regularity sharp? To answer this question, we construct a counterexample to show that the equivalent function $H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely H\"{o}lder continuous. Furthermore, we propose a conjecture that such regularity of the homeomorphisms is sharp (it could not be improved anymore). We prove that the conjecture is true for the systems with linear contraction. Furthermore, we present the special cases of linear perturbation, which are closely related to the spectrum.

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arXiv2022

Higher regularity of homeomorphisms in the Hartman-Grobman theorem and a conjecture on its sharpness

Hartman-Grobman theorem states that there is a homeomorphism H sending the solutions of the nonlinear system onto those of its linearization under suitable assumptions. Many mathematicians have made contributions to prove H\"older continuity of the homeomorphisms. However, is it possible to improve the H\"older continuity to Lipschitzian continuity? This paper gives a positive answer. We formulate the first result that the homeomorphism is Lipschitzian, but not $C^1$, while its inverse is merely H\"{o}lder continuous, but not Lipschitzian. It is interesting that the regularity of the homeomorphism is different from its inverse. Moreover, some illustrative examples are presented to show the effectiveness of our results. Further, motivated by our example, we also propose a conjecture, saying, the regularity of the homeomorphisms is sharp and it could not be improved any more.

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arXiv2022

Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations

Hein and Pr\"{u}ss [J. Differential Equations, 261(2016)4709-4727] presented a version of Hartman-Grobman type $C^{0}$ linearization result for semilinear hyperbolic evolution equations. They showed that the linearising map (homomorphism) and its inverse are H\"{o}lder continuous. An important question: is it possible to improve the regularity of the homomorphisms? In the present paper, we prove that if the mild solutions of semilinear system are bounded, then the regularity of the homomorphisms is Lipchitzian, but the inverse is merely H\"{o}lder continuous. We also give a generalized local linearization result in this paper. Finally, some applications end the paper. As pointed out by Backes [J. Differential Equations, 297 (2021) 536-574], even if the diffeomorphism $F$ is $C^{\infty}$, the homomorphism can fail to be locally Lipschitz. The homomorphisms are in general only locally H\"older continuous. However, by establishing two effective dichotomy integral inequalities, we prove that the conjugacy is Lipchitzian, but the inverse is H\"{o}lder continuous. Our result is the first one to observe the higher regularity of homomorphisms in the Hartman-Grobman theorem.

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arXiv2022

A Hartman-Grobman theorem for algebraic dichotomies

Algebraic dichotomy is a generalization of an exponential dichotomy (Lin, JDE2009). This paper gives a version of Hartman-Grobman linearization theorem assuming that linear system admits an algebraic dichotomy, which generalizes the Palmer's linearization theorem. Besides, we prove that the homeomorphism in the linearization theorem (and has a H\"{o}lder continuous inverse). Comparing with exponential dichotomy, algebraic dichotomy is more complicate. The exponential dichotomy leads to the estimates $\int_{-\infty}^{t}e^{-\alpha(t-s)}ds$ and $\int_{t}^{+\infty}e^{-\alpha(s-t)}ds$ which are convergent. However, the algebraic dichotomy will leads us to $\int_{-\infty}^{t}\left(\frac{\mu(t)}{\mu(s)}\right)^{-\alpha}ds$ or $\int_{t}^{+\infty}\left(\frac{\mu(s)}{\mu(t)}\right)^{-\alpha}ds$, whose the convergence is unknown in the sense of Riemann.

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arXiv2021

On a Poincar\'e-Perron problem for high order differential equation

We address asymptotic formulae for the classical Poincar\'e-Perron problem of linear differential equations with almost constant coefficients in a half line $[t_0,+\infty)$ for high order equation $n\ge 5$ and some $t_0\in\mathbb{R}$. By using a scalar nonlinear differential equation of Riccati type of order $n-1$, we recover Poincar\'e's and Perron's results and provide asymptotic formulae with the aid of Bell's polynomials. Furthermore, we obtain some weaker versions of Levinson, Hartman-Wintner and Harris-Lutz type Theorems without the usual diagonalization process. For an arbitrary $n\ge 5$, these are corresponding versions to known results for cases $n=2,3$ and $4$.

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arXiv2020

On almost automorphic type solutions of abstract Integral equations, a Bohr-Neugebauer type property and some applications

In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions, both situations are posed on Banach spaces. Also, we develop a Bohr-Neugebauer type result for the abstract integral equations. Before that, we introduce the notion of $λ$-bounded functions, develop the appropriate abstract theory and discuss the almost periodic situation. As applications, we study the existence of an asymptotically almost automorphic solution to integro-differential equations modeling heat conduction in materials with memory and also the existence of the almost automorphic solution to semilinear parabolic evolution equations with finite delay.

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arXiv2019

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar $et\ al$, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.

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arXiv2018

Boundness and Linearisation of a class of differential equations with piecewise constant argument

The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded, we study the bounded solution and global topological linearisation of a class of DEPCAs of general type. One of the purpose of this paper is to obtain a new criterion for the existence of a unique bounded solution, which improved the previous results. The other aim of this paper is to establish a generalized Grobman-Hartman-type theorem for the topological conjugacy between a nonlinear perturbation system and its linear system. The method is based on the new obtained criterion for bounded solution. The obtained results generalized and improved some previous papers. Some novel techniques are employed.

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arXiv2016

L$^p$-Solutions of a Nonlinear Third Order Differential Equation and Asymptotic Behavior of Linear Fourth Order Differential Equations

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory $L^p$-solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (i) the characteristic polynomial associated with the linear part has simple and real roots, (ii) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (iii) the polynomial coefficients are in $L^p([t_0,\infty[)$. These results are applied to study a fourth order linear differential equation of Poincaré type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems can not be applied.

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arXiv2016

Sufficient conditions for existence of positive periodic solution of a generalized nonresident computer virus model

In this paper, we introduce a nonresident computer virus model and prove the existence of at least one positive periodic solution. The proposed model is based on a biological approach and is obtained by considering that all rates (rates that the computers are disconnected from the Internet, the rate that the computers are cured, etc) are time dependent real functions. Assuming that the initial condition is a positive vector and the coefficients are positive $ω-$periodic and applying the topological degree arguments we deduce that generalized nonresident computer virus model has at least one positive $ω-$periodic solution. The proof consists of two big parts. First, an appropriate change of variable which conserves the periodicity property and implies the positive behavior. Second, a reformulation of transformed system as an operator equation which is analyzed by applying the continuation theorem of the coincidence degree theory.

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arXiv2016

Almost periodic evolution systems with impulse action at state-dependent moments

We study the existence of almost periodic solutions for semi-linear abstract parabolic evolution equations with impulse action at state-dependent moments. In particular, we present conditions excluding the beating phenomenon in these systems. The main result is illustrated with an example of impulsive diffusive logistic equation.

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arXiv2015

Monotone waves for non-monotone and non-local monostable reaction-diffusion equations

We propose a criterion for the existence of monotone wavefronts in non-monotone and non-local monostable diffusive equations of the Mackey-Glass type. This extends recent results by Gomez et al proved for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (up to a translation) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then this uniqueness actually holds within the class of all wavefronts and therefore the minimal fronts under consideration (either pulled or pushed) should be monotone. Similarly to the case of the KPP-Fisher equations, our approach is based on the construction of an appropriate fundamental solution for associated boundary value problem for linear integral-differential equation.

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