Marcia Federson

University of São Paulo | USP · Institute of Mathematical and Computer Sciences (ICMC) (São Carlos)

Similarly to generalized ordinary differential equations that comprise various types of classical deterministic equations, generalized stochastic equations (GSEs) were created to contain equa- tions involving stochastic processes. The main goal of this paper is to investigate several types of stability and boundedness for non-autonomous GSEs by mea...

The study of models which deal with oscillating functions is of great importance in Physics and other sciences. In this work, we are interested in investigating the oscillatory behavior of solutions of delay differential equations subject to impulsive effects and having righthand sides which admit not only many discontinuities but also oscillations...

It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible...

The aim of this paper is to investigate the existence and uniqueness of solutions of the following boundary value problem concerning generalized ODEs $$\begin{aligned} \left\{ \begin{array}{l} \dfrac{dx}{d\tau } = D[A(t)x+F(t)], \\ \displaystyle \int _{a}^{b} d[K(s)]x(s)=r, \end{array} \right. \end{aligned}$$for operators taking values in general B...

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...

Generalized ordinary differential equations (we write generalized ODEs), introduced by J. Kurzweil in 1957, are known to encompass several other types of equations as measure functional differential equations, for instance. In this paper, we obtain converse Lyapunov theorems for generalized ODEs and, in particular, for measure functional differenti...

In this paper, we investigate the existence and uniqueness of a solution for a linear Volterra-Stieltjes integral equation of the second kind, as well as for a homogeneous and a nonhomogeneous linear dynamic equations on time scales, whose integral forms contain Perron [Formula: see text]-integrals defined in Banach spaces. We also provide a variat...

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...

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 e...

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...

A control system is a time-evolving system over which one can act through an input or control function. The purpose of control theory is to analyze properties, of such systems, with the intention of bringing a certain initial data to certain, final data. This chapter introduces new concepts of controllability and observability for abstract generali...

This chapter is dedicated to the study of semidynamical systems generated by generalized ordinary differential equations (ODEs). Besides the existence of a local semidynamical system, it shows the existence of an associated impulsive semidynamical system. The chapter concerns the existence of a local semidynamical system generated by the nonautonom...

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...

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 glob...

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...

The theory of exponential dichotomy is an important tool to the investigation of the behavior of nonautonomous differential equations. Exponential dichotomy is a kind of conditional stability which generalizes the notion of hyperbolicity of autonomous systems to nonautonomous systems. The classical notion and basic properties of dichotomies in the...

The aim of this paper is to obtain results on topological properties of flows for nonnegative time in the framework of generalized ODEs. We define the concept of generalized semiflow and we present some recursive properties as minimality and recurrence. Using correspondence theorems, we translate our results to measure differential equations which...

The aim of this paper is to obtain converse Lyapunov theorems for measure functional differential equations which are known to encompass dynamic equations on time scales, impulsive differential equations and functional differential equations (see [7], [8]). Our main tool is to consider the framework of generalized ordinary differential equations, i...

The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurz-weil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonau-tonomous ODE. By means of the correspondence between impulse differential equations...

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...

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a...

The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrö...

This paper deals with exponential dichotomy for generalized ODEs. We establish sufficient conditions to ensure that the exponential dichotomy of a linear generalized ODE is robust under certain perturbations. As a consequence, we obtain results on robustness of dichotomies for measure differential equations and, in particular, impulsive differentia...

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...

We present new criteria for the existence of oscillatory and nonoscillatory solutions of measure delay differential equations with impulses. We deal with the integral forms of the differential equations using the Perron and the Perron-Stieltjes integrals. Thus the functions involved can have many discontinuities and be of unbounded variation and ye...

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...

We present a variation-of-constants formula for functional differential equations of the form y˙=L(t)yt+f(yt,t),yt0=φ, where L is a bounded linear operator and φ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application tt↦f(yt, t) is Kurzweil integrable w...

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between gene...

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...

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...

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...

We consider a class of functional differential equations subject to perturbations, which vary in time, and we study the exponential stability of solutions of these equations using the theory of generalized ordinary differential equations and Lyapunov functionals. We introduce the concept of variational exponential stability for generalized ordinary...

Finally, the editors wish to express their gratitude to the many authors and reviewers who contributed so greatly to the success of this special issue.

This is a review paper on recent results for different types of generalized ordinary differential equations. Its scope ranges from discontinuous equations to equations on time scales. We also discuss their relation with inclusion and highlight the use of generalized integration to unify many of them under one single formulation.

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...

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 present here the linear Cauchy–Stieltjes integral on regulated func- tions with values in Banach spaces on time scales and represent a linear operator on the space of the regulated functions by means of an appropriate kernel in the integral.

We consider a class of functional differential equations with variable impulses and we establish new stability results. We discuss the variational stability and variational asymptotic stability of the zero solution of a class of generalized ordinary differential equations where our impulsive functional differential equations can be embedded and we...

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,...

In this paper, we give sufficient conditions for the uniform boundedness and uniform ultimate boundedness of solutions of a class of retarded functional differential equations with impulse effects acting on variable times. We employ the theory of generalized ordinary differential equations to obtain our results. As an example, we investigate the bo...

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 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.

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semid...

If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schrödinger-type equation. We consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional “i...

It is done a method for the definition of a closed continuous curve C in the plane, coming from an equation of the Liénard type, in such a way that C surrounds a periodic orbit of the equation itself.

In this paper, the concept of Poisson stability is investigated for impulsive semidynamical systems. Recursive properties are also investigated.

We present a version of the Poincare-Bendixson Theorem on the Klein bottle K2 for continuous vector fields. As a consequence, we obtain the fact that K2 does not admit continuous vector fields having a !-recurrent

We consider a certain type of second-order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations (called impulses). As a matter of fact, some particular non-impulsive cases of the system are oscillatory already. Thus, we are interested in fi...

It is known that impulsive retarded functional dierential equations can be regarded as Banach-space valued generalized ordinary dierential equations (general- ized ODEs). In the present paper, we discuss the variational stability and variational asymptotic stability of the zero solution of generalized ODEs and we apply these results to obtain the L...

We consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy, compactness and connectedness. As a consequence we obtain a version of the Poincaré–Bendixson Theorem for impulsive semidynamical systems.

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the diffe...

We consider the linear integral equations of Fredholm and Volterra Z b a α(t, s) x(s) dg(s) = f (t), t 2 (a, b), and x(t) Z t a α(t, s) x(s) dg(s) = f (t), t 2 (a, b), in the frame of the Henstock-Kurzweil integral and we prove results on the existence and uniqueness of solutions. More precisely, we consider the above equations in the sense of Hens...

This paper is concerned with systems of impulsive second order delay differential equations. We prove that unstable systems can be stabilized by imposition of impulsive controls. The main tools used are Lyapunov functionals, stability theory and control by impulses.

We prove several results concerning topological conjugation of two impulsive semidynamical systems. In particular, we prove that the homeomorphism which defines the topological conjugation takes impulsive points to impulsive points; it also preserves limit sets, prolongational limit sets and properties as the minimality of positive impulsive orbits...

It is known that retarded functional differential equations (RFDEs) can be regarded as generalized ordinary differential equations (GODEs) [see M. Federson, P. Táboas, J. Differ. Equations 195, No. 2, 313–331 (2003; Zbl 1054.34102); C. Imaz, Z. Vorel, Bol. Soc. Mat. Mex., II. Ser. 11, 47–59 (1966; Zbl 0178.44203) and F. Oliva, Z. Vorel, Bol. Soc. M...

We consider a certain second-order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed. An example is given.

We consider certain impulsive second order delay differential equations and give conditions for the existence of solutions. Moreover we prove that the non-impulsive equations can be stabilized by the imposition of proper impulse controls generalizing recent results by Li and Weng. We also comment on some possible applications and give examples.

It is known that retarded functional dierential equations (RFDEs) can be re- garded as generalized ordinary dierential

We consider the multidimensional abstract linear integral equation of Volterra type $$ x{\left( t \right)} + {\left( * \right)}{\int_{R_{t} } {\alpha {\left( s \right)}x{\left( s \right)}ds = f{\left( t \right)},{\kern 1pt} t \in R} } $$ (1) , as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous soluti...

We prove that a local flow can be constructed for a general class of nonautonomous retarded functional differential equations (RFDE). This is an extension to a result of Artstein (J. Differential Equations 23 (1977) 216) and fits in the classical theory of R. Miller and G. Sell. The main tool in this paper are generalized ordinary differential equa...

Some examples, due to G. Birkhoff, are used to explore the differences and peculiarities of the Henstock and Kurzweil integrals in abstract spaces. We also include a proof, due to C. S. Hönig, of the fact that the Bochner-Lebesgue integral is equivalent to the variational Henstock-McShane integral.

The Kurzweil-Henstock integral formalism is applied to establish the existence of solutions to the linear integral equations of Volterra-type (1) where the functions are Banach-space valued. Special theorems on existence of solutions concerning the Lebesgue integral setting are obtained. These sharpen earlier results.

Impulsive retarded differential equations were discussed. Integral of Henstock was used to establish conditions for existence and uniqueness of solutions as well as continuous dependence on the initial conditions of the system. Application of retarded functional differential equations were also studied.

We prove two versions of the Monotone Convergence Theorem for the vector integral of Kurzweil, \smallint R \textda\text(t\text)f(t)\smallint _R {\text{d}}\alpha {\text{(}}t{\text{)}}\;f(t) , where R is a compact interval of \mathbbRn \mathbb{R}^n , \smallint R a(t)\textd f(t)\smallint _R \alpha (t)\;{\text{d}}{\kern 1pt} f(t) , as well as to...

We apply the Kurzweil-Henstock integral setting to prove a Fredholm Alternative-type result for the integral equation x (t) – K ∫ [a, b] α (t, s) x (s) ds = ƒ (t), t ∈ [a, b],where x and ƒ are Kurzweil integrable functions (possibly highly oscillating) defined on a compact interval [a, b] of the real line with values on Banach spaces. An applicat...

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.

Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the generalized Riemann integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach space-valued co...

In 1990, C. S. Hönig proved that the linear Volterra integral equation x(t)-(K)∫ [a,t] α(t,s)x(s)ds=f(t),t∈[a,b] where the functions are Banach space-valued and f is a Kurzweil integrable function defined on a compact interval [a,b] of the real line ℝ, admits one and only one solution in the space of the Kurzweil integrable functions with resolvent...

In the present paper we give the Fundamental Theorem of Calculus for the variational or Henstock vector integrals $^{K}\!\!\!\int_{\mathbb R} \alpha \, df$ and $^{K}\!\!\!\int_{\mathbb R} d\alpha \, f$ of multidimensional Banach space-valued functions.

We establish conditions for the existence of solutions of the linear integral equation of Volterra \begin{equation} x\left( t\right) +^{\ast }\int\nolimits_{[ a,t] }\alpha ( s) x ( s )\, ds=f ( t ) ,\quad t\in [ a,b ] ,\tag{$V_{\ast}$} \end{equation} where the functions are Banach space-valued and $^{\ast }\int $ denotes either the Bochner-Lebesgue...