Impulsive surfaces on dynamical systems

This work is devoted to the construction of impulsive sets in Rn. In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in Rn which satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

Attractors for impulsive non-autonomous dynamical systems and their relations

In this work, we deal with several different notions of attractors that may appear in the impulsive non-autonomous case and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics, such as the cocycle attractor, the uniform attractor and the global attractor for the impulsive skew-product semiflow. Lastly, we illustrate our theory by exhibiting an example of a non-classical non-autonomous parabolic equation with subcritical nonlinearity and impulses.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía)Fundação de Amparo à Pesquisa do Estado de São Paul

A survey on impulsive dynamical systems

In this survey we provide an introduction to the theory of impulsive dynamical systems in both the autonomous and nonautonomous cases. In the former, we will show two different approaches which have been proposed to analyze such kind of dynamical systems which can experience some abrupt changes (impulses) in their evolution. But, unlike the autonomous framework, the nonautonomous one is being developed right now and some progress is being obtained over the recent years. We will provide some results on how the theory of autonomous impulsive dynamical systems can be extended to cover such nonautonomous situations, which are more often to occur in the real world.Fundação de Amparo à Pesquisa do Estado de São PauloConselho Nacional de Desenvolvimento Científico e TecnológicoFondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

[Book of abstracts]

USPFAPESPCAPESICMC Summer Meeting on Differential Equations (2015 São Carlos

Linear generalized ordinary differential equations and application to linear functional differential equations

Neste trabalho, apresentamos uma fórmula da variação das constantes para EDOs generalizadas lineares em espaços de Banach. Mais especificamente, estamos interessados em estabelecer uma relação entre as soluções do problema de Cauchy para uma EDO generalizada linear \'dx SUP. d \'tau\' =D[A(t )x], x(\'t IND. 0\') = \'x SOB. ~\' e as soluções do problema de Cauchy perturbado \'dx SUP. d \'tau\' =D[A(t )x +F(x, t )], x(\'t IND. 0\') = x(\'t IND. 0\') = \'x SOB. ~\' , em que as funções envolvidas são Perron integráveis e, portanto, admitem muitas descontinuidades e oscilações. Também provamos a existência de uma correspondência biunívoca entre o problema de Cauchy para uma EDF linear da forma { \' y PONTO\' =L(t )\'y IND. t\' , \'y IND. t IND. 0 = \\varphi\', , em que L é um operador linear e limitado e \'varphi\' é uma função regrada, e uma certa classe de EDOs generalizadas lineares. Como consequência, obtemos uma fórmula da variação das constantes relacionando as soluções da EDF linear e as soluções do problema perturbado { \'y PONTO\' = L(t )\'y IND.t\' + f (\'yIND. t\' , \'y IND. t IND. 0\' = \'\\varphi \', em que a aplicação \'t SETA \' f (\'y IND. t\' , t) é Perron integrável, com t em um intervalo de R, para cada função regrada yIn this work, we present a variation-of-constants formula for linear generalized ordinary differential equations in Banach spaces. More specifically, we are interested in establishing a relation between the solutions of the Cauchy problem for a linear generalized ordinary differential equation \'dx SUP. d \\tau\' =D[A(t )x], x(\'t IND. 0\') = x (\'t IND. 0\') = \'x SOB. ~\' and the solutions of the perturbed Cauchy problem \'dx SUP. \'d \\tau\' =D[A(t )x +F(x, t )], x(\'t IND. \'0) = \'x SOB.~\', where the functions involved are generalized Perron integrable and, hence, admit many discontinuities and oscillations. We also prove that there exists a one-to-one correspondence between the Cauchy problem for a linear functional differential equations of the form { \'y PONTO\' = L(t) \'y IND. t, \'y IND> 0 = \\varphi, where L is a bounded linear operator and \" is a regulated function, and a certain class of linear generalized ordinary differential equations. As a consequence, we are able to obtain a variation-of-constants formula relating the solutions of the linear functional differential equation and the solutions of the perturbed problem { \'y PONTO\' = L(T)\'y IND.t´+ f (\'y IND. t\', t), \'y IND.t IND. 0\' = \\varphi, where the application t \'ARROW\' f(\'y IND. t\', t) is Perron integrable, with t in an interval of R, for each regulated function

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

summary:We present a variation-of-constants formula for functional differential equations of the form $$\dot {y}={\mathcal L}(t)y_t+f(y_t,t), \quad y_{t_0}=\varphi ,$$ where ${\mathcal L}$ is a bounded linear operator and $\varphi$ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $\mathbb R$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type $$\frac {{\rm d}x}{{\rm d}\tau } = D[A(t)x],\quad x(t_0)=\widetilde {x}$$ and the solutions of the perturbed Cauchy problem $$\frac {{\rm d}x}{{\rm d}\tau } = D[A(t)x+F(x,t)], \quad x(t_0)=\widetilde {x}.$$ Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form $$\dot {y}={\mathcal L}(t)y_t, \quad y_{t_0}=\varphi ,$$ where $\mathcal L$ is a bounded linear operator and $\varphi$ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs

Upper and Lower Semicontinuity of Impulsive Cocycle Attractors for Impulsive Nonautonomous Systems

In this work we present results to ensure a weak upper semicontinuity for a family of impulsive cocycle attractors of nonautonomous impulsive dynamical systems, as well as an example of nonautonomous dynamical system generated by an ODE in the real line to illustrate our results. Moreover, we present theoretical results regarding lower semicontinuity of impulsive cocycle attractors