Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal Discrete and Continuous Dynamical Systems Series B, 2021 In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.
Upper and Lower Semicontinuity of Impulsive Cocycle Attractors for Impulsive Nonautonomous Systems E. M. Bonotto, M. C. Bortolan, T. Caraballo, R. Collegari Journal of Dynamics and Differential Equations, 2021 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.
Linear generalized ordinary differential equations Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Miguel V. S. Frasson Generalized Ordinary Differential Equations in Abstract Spaces and Applications, 2021 The investigation of linear equations in the framework of generalized ordinary differential equations (ODEs) is very important as they also are in the setting of classical ODEs. This chapter deals with linear generalized ODEs presenting an appropriate environment where any initial value problem (IVP) for linear generalized ODEs admits a unique global solution. It aims to establish a variation-of-constants formula for linear perturbed generalized ODEs. The chapter presents a result on the existence and uniqueness of a global solution for IVP for the linear generalized ODE. It also presents a variation-of-constants formula for a linear perturbed generalized ODE. The chapter describes the correspondence between linear generalized ODEs and linear measure functional differential equations.
The kurzweil integral Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita Generalized Ordinary Differential Equations in Abstract Spaces and Applications, 2021 This chapter is devoted to the theory of integration introduced by Jaroslav Kurzweil in the form presented in his articles dated 1957, 1958, 1959, and 1962, and so on. It provides the heart of the theory of generalized ordinary differential equations which is precisely the Kurzweil integration theory, presented in a concise form which includes its most fundamental properties–those usually expected that a “good integral” will fulfill. The chapter aims to present a short historical background and its relation with the Kurzweil integral. A common rigid flat pendulum, whose pivot is forced to oscillate along the vertical line, has only one stable point, that is, vertically downwards. Kapitza pendulum is stabilized by minimizing the potential energy. The chapter concludes by calling the reader's attention to the fact that the main feature of the Kurzweil integral is to handle well highly oscillating functions as shown by the Kapitza equation for the inverted pendulum.
Preliminaries Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, Eduard Toon Generalized Ordinary Differential Equations in Abstract Spaces and Applications, 2021 This chapter presents two pillars of the theory of generalized ordinary differential equations (ODEs). One of these pillars concerns the spaces in which the solutions of a generalized ODE are generally placed. The other pillar concerns the theory of nonabsolute integration, due to Jaroslav Kurzweil and Ralph Henstock, for integrands taking values in Banach spaces. The solutions of a Cauchy problem for generalized ODE, with right-hand side in a class of functions introduced by J. Kurzweil in, usually belong to a certain space of functions of bounded variation. Regarding functions of bounded variation, which are known to be of bounded semivariation and, hence, of bounded S-variation, this chapter presents a coherent overview of functions of bounded S-variation over bilinear triples. The Kurzweil integral and the variational Henstock integral can be extended to Banach space-valued functions as well as to the evaluation of integrands over unbounded intervals.
Periodicity Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Maria Carolina Mesquita Generalized Ordinary Differential Equations in Abstract Spaces and Applications, 2021 The study of periodic solutions is an important and well-known branch of the theory of differential equations related, in a broad sense, to the study of periodic phenomena that arise in problems applied in technology, biology, and economics. There are many works concerning periodicity of solutions in the framework of classic ordinary differential equations (ODEs) and impulsive differential equations. This chapter investigates periodicity of solutions of linear generalized ODEs for functions taking values in ℝ n . It presents a Floquet-type theorem which provides a characterization of the fundamental matrix of periodic linear generalized ODEs. The chapter presents some results and definitions on fundamental matrices.
Linear FDEs in the frame of generalized ODEs: variation-of-constants formula Rodolfo Collegari, Márcia Federson, Miguel Frasson Czechoslovak Mathematical Journal, 2018 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.
