Jaqueline Godoy Mesquita

• Professor
• Professor (Assistant) at University of Brasília

This paper investigates a new class of equations called measure functional differential equations with state-dependent delays. We establish the existence and uniqueness of solutions and present a discussion concerning the appropriate phase space to define these equations. Also, we prove a version of periodic averaging principle to these equations....

A matemática desempenha um papel crucial na previsão de fenômenos e na formulação de políticas públicas diante de emergências globais. Exemplos notáveis incluem seu uso durante a pandemia da Covid-19 e na investigação de mudanças climáticas. Essa importância ressalta a necessidade de um diálogo mais amplo entre as áreas do conhecimento e de investi...

This paper is devoted to the investigation of the existence of mild solutions of abstract retarded functional differential equations with infinite state-dependent delay. We obtain the result concerning the existence of mild solutions for the equations with state-dependent delays as a fixed point of the solution operator of an associated abstract re...

We provide a criterion for instability of equilibria of equations in the form x ˙ ( t ) = g ( x t ′ , x t ) , which includes neutral delay equations with state-dependent delay. The criterion is based on a lower bound Δ > 0 for the delay in the neutral terms, on regularity assumptions of the functions in the equation, and on spectral assumptions on...

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 $\omega$-periodic solutions. Furthermore, we present a result on continuous de...

cosine and sine functions defined on a Banach space are useful tools in the study of wide classes of abstract evolution equations. In this paper, we introduce a definition of cosine and sine functions on time scales, which unify the continuous, discrete and the cases “in between.” Our definition includes several types of time scales such as real nu...

In this article, we investigate the existence and uniqueness of solutions of linear and semilinear second–order equations involving time scales. To obtain such results, we make use of exponential dichotomy and fixed point results. Also, we present some examples and applications to illustrate our main results.

In this work, we formulate the Beverton–Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton–Hol...

In this work, we introduce the measure functional differential equations (MFDEs) with infinite time-dependent delay, and we study the correspondence between the solutions of these equations and the solutions of the generalized ordinary differential equations (GODEs, for short) in Banach spaces. Using the theory of GODEs , we obtain results concerni...

It is well-known that generalized ODEs encompass several types of differential equations as, for instance, functional differential equations, measure differential equations, dynamic equations on time scales, impulsive differential equations and any combinations among them, not to mention integrals equations, among others. The aim of this paper is t...

In this work, we formulate the definition of periodicity for functions defined on isolated time scales. The introduced definition is consistent with the known formulations in the discrete and quantum calculus settings. Using the definition of periodicity, we discuss the existence and uniqueness of periodic solutions to a family of linear dynamic eq...

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized OD...

Many countries around the world are trying to fight Covid-19, and their main methods are lockdown, quarantine, isolation, and awareness programs to encourage people to adopt social distancing and maintain personal hygiene. The lockdown is aimed to restrict the movement of humans from or to certain places. Quarantine is aimed toward separating the s...

This chapter aims to investigate the results on continuous dependence on parameters for generalized ordinary differential equations (ODEs) taking values in a Banach space. It includes a new result on the convergence of solutions of a nonautonomous generalized ODE. The chapter also investigates continuous dependence results on parameters for general...

This chapter is devoted to basic properties of nonautonomous generalized ordinary differential equations (ODEs) and to applying some of the results to measure functional differential equations (FDEs) and functional dynamic equations on time scales. It deals with local existence and uniqueness of solutions of generalized ODEs, and provides analogous...

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...

This chapter introduces a new class of integral equations known as generalized ordinary differential equations (ODEs) for functions taking values in a Banach space. Kurzweil introduced the concept of generalized ODEs for vector-valued and Banach space-valued functions. The chapter describes the relation between generalized ODEs and measure function...

Averaging methods have the purpose to simplify the analysis of nonautonomous differential systems through simpler autonomous differential systems obtained as an "averaged" equation of the original equation. The averaged systems they considered were autonomous ordinary differential equations (ODEs). In the 1970s, the investigations about averaging m...

This chapter investigates the following types of equations: measure functional differential equations (FDEs), impulsive FDEs, functional dynamic equations on time scales, impulsive functional dynamic equations on time scales, all of which involving Banach space-valued functions. The main advantage behind the theory of dynamic equations comes from t...

This chapter presents the concepts of uniform boundedness, quasiuniform boundedness, and uniform ultimate boundedness in the scenery of generalized ordinary differential equations (ODEs). It includes criteria of uniform boundedness and uniform ultimate boundedness for the generalized ODE. The chapter presents some results concerning the boundedness...

This chapter presents the study of the stability theory for generalized ordinary differential equations (ODEs). The results on the stability of the trivial solution in the framework of the generalized ODE are inspired in the theory, developed by Aleksandr M. Lyapunov on the stability of solutions for classic ODEs. Converse Lyapunov theorems confirm...

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 i...

In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of the delayed population model with survival rate on isolated time scales given by \begin{document}$ x^{\Delta} (t) = \gamma(t) x(t) + \dfrac{x(d(t))}{\mu(t)}e^{r(t)\mu(t)\left(1 - \frac{x(d(t))}{\mu(t)}\righ...

Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations as well, including Measure Functional Differential Equations (measure FDEs). It presents a unifo...

In this paper, we study the differentiability of mild solutions for a class of fractional abstract Cauchy problem. We consider the fractional derivative of order α∈(1,2) in the sense of Caputo. In the first part, we establish the existence of classical solutions for the homogeneous Cauchy problem in terms of the α-resolvent family corresponding to...

In this paper, we prove existence and uniqueness of solutions of Volterra–Stieltjes integral equations using the Henstock–Kurzweil integral. Also, we prove that these equations encompass impulsive Volterra–Stieltjes integral equations and prove the existence and uniqueness for these equations. Finally, we present some examples to illustrate our res...

We present new Massera-type theorems for various types of equations with periodic right-hand sides. We deal with generalized ordinary differential equations, measure differential equations, impulsive equations (all of which might have discontinuous solutions), as well as dynamic equations on time scales. For scalar nonlinear equations, we find suff...

We prove a global bifurcation result for a parameterized dynamic equation on time scales. The approach is topological and based on a notion of topological degree for compact perturbations on nonlinear Fredholm maps in Banach spaces. Also, we provide several examples considering discrete, continuous and hybrid time scales in order to illustrate our...

In this paper, we introduce the definition of a $C_0$-semigroup on a time scale, which unifies the continuous, discrete and other cases which lie between them. Also, it extends the classical theory of operator semigroups to the quantum case. We study the relationship between the semigroup and its infinitesimal generator. We apply our theory to stud...

The object of study is an autonomous impulsive system proposed as a model of drugs absorption by living organisms consisting of a linear differential delay equation and an impulsive self-support condition. We get a representation of the general solution in terms of the fundamental solution of the differential delay equation. The impulsive self-supp...

In this paper, we investigate the asymptotic behavior for measure differential equations and dynamic equations on time scales via generalized ordinary differential equations (generalized ODEs, for short). At first, we establish new results that guarantee the existence of unbounded solutions for generalized ODEs, and after that, using the known corr...

In the present work, we introduce the concept of asymptotically almost automorphic functions on time scales and study their main properties. We study nonautonomous dynamic equations on time scales given by $x^{\Delta} (t) = A(t) x(t) + f(t)$ and $x^{\Delta} (t) = A(t) x(t) + f(t, x(t))$, $t \in \mathbb T$, where $\mathbb T$ is an invariant under tr...

In this paper, we prove the stability results for measure differential equations, considering more general conditions under the Lyapunov functionals and concerning the functions f and g. Moreover, we prove these stability results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the...

In this paper, we introduce a class of measure neutral functional differential equations of type (Formula presented.)through the relation with a certain class of generalized ordinary differential equations introduced in Federson and Schwabik (Differ Integral Equ 19(11):1201–1234, 2006) (we write generalized ODEs), using similar ideas to those of Fe...

In this paper, we prove the results on existence and uniqueness of the maximal solutions for measure differential equations, considering more general conditions on functions f and g by using the correspondence between the solutions of these equations and the solutions of generalized ODEs. Moreover, we prove these results for the dynamic equations o...

In this paper, we prove the existence of positive solutions of an elliptic superlinear problem. Also, we are interested here in getting results concerning the existence of positive solutions for the discrete formulation of our problem. Therefore, in order to do it, we employ the radial solutions of the elliptic superlinear problem, obtaining a seco...

The present paper deals with existence and uniqueness of global mild solutions for a new model of Navier–Stokes equations on R² subjected to impulse effects at variable times. By using the framework of impulsive/nonautonomous dynamical systems we are able to consider impulse effects in the system as well relax conditions on the external forcing ter...

In this paper, we present versions of Massera’s theorem for linear and nonlinear q-difference equations and present some examples to illustrate our results.

In this article, we introduce the concepts of Bochner and Bohr almost periodic functions in quantum calculus and show that both concepts are equivalent. Also, we present a correspondence between almost periodic functions defined in quantum calculus and N0, proving several important properties for this class of functions. We investigate the existenc...

The existence and uniqueness of almost automorphic solutions for linear and semilinear nonconvolution Volterra equations on time scales is studied. The existence of asymptotically almost automorphic solutions is proved. Examples that illustrate our results are given.

In this paper, we are concerned with linear control systems on time scales. We show that, under appropriate hypotheses, the self-accessible trajectories have diameter greater than or equal to a certain fixed positive number.

In this paper, we investigate the boundedness results for measure differential equations. In order to obtain our results, we use the correspondence between these equations and generalized ODEs. Furthermore, we prove our results concerning boundedness of solutions for dynamic equations on time scales, using the fact that these equations represent a...

The present paper deals with existence and uniqueness of global mild solutions for the 2D Navier-Stokes equations with impulses. Using the framework of nonautonomous dynamical systems, we extend previous results considering the 2D Navier-Stokes equations with impulse effects and allowing that the nonlinear terms are explicitly time-dependent. Addit...

We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impuls...

In this paper, we prove a periodic averaging principle for quantum difference equations and present some examples to illustrate our result.

In this note, we fix some misprints in the paper of Lizama and Mesquita (2013) [2].

We consider measure functional differential equations (we write measure FDEs) of the form , where f is Perron–Stieltjes integrable, is given by , with , and and are the distributional derivatives in the sense of the distribution of L. Schwartz, with respect to functions and , , and we present new concepts of stability of the trivial solution, when...

In this paper, we prove a strong connection between almost periodic functions on time scales and almost periodic functions on $\mathbb R$. An application to difference equations on $\mathbb T = h \mathbb Z$ is given.

In the present work, we introduce the concept of almost automorphic functions on time scales and present the first results about their basic properties. Then, we study the nonautonomous dynamic equations on time scales given by and , where is a special case of time scales that we define in this article. We prove a result ensuring the existence of a...

We consider a large class of retarded functional differential equations subject to impulse effects at variable times, and we present an averaging result for this class of equations by means of the techniques and tools of the theory of generalized ordinary differential equations introduced by J. Kurzweil.

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equati...

In the present paper, we study the non-autonomous difference equations given by u(k + 1) = A(k)u(k) + f (k) and u(k + 1) = A(k)u(k)+ g(k, u(k)) for k is an element of Z, where A(k) is a given non-singular n x n matrix with elements a(ij)(k), 1 <= i, j <= n, f : Z -> E-n is a given n x 1 vector function, g : Z x E-n -> E-n and u(k) is an unknown n x...

Using a known correspondence between the solutions of impulsive measure functional differential equations and the solutions of impulsive functional dynamic equations on time scales, we prove that the limit of solutions of impulsive functional dynamic equations over a convergent sequence of time scales converges to a solution of an impulsive functio...

We study the relation between measure functional differential equations, impulsive measure functional differential equations, and impulsive functional dynamic equations on time scales. For both types of impulsive equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging. Along the way,...

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, con...

We prove a periodic averaging theorem for generalized ordinary differential equations and show that averaging theorems for ordinary differential equations with impulses and for dynamic equations on time scales follow easily from this general theorem. We also present a periodic averaging theorem for a large class of retarded equations.

We consider retarded functional differential equations in the setting of Kurzweil–Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.