Rogelio Grau Acuña - USP...Acknowledgment First of all, I want to thank God for giving me the wisdom and strength to make this a reality project. I would like to thank my advisor,

On qualitative properties of generalized ODEs

Rogelio Grau Acuña

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Assinatura:_______________________

Rogelio Grau Acuña

On qualitative properties of generalized ODEs

Doctoral thesis submitted to the Instituto de CiênciasMatemáticas e de Computação - ICMC-USP inpartial fulfillment of the requirements for the degreeof the Doctorate Program in Mathematics. FINALVERSION

Concentration Area: Mathematics

Advisor: Profe. Dra. Márcia Cristina Anderson BrazFederson

Co-advisor: Profa. Dra. Jaqueline Godoy Mesquita

USP – São Carlos September 2016

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassie Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

Acuña, Rogelio GrauA634o On qualitative properties of generalized ODEs

/ Rogelio Grau Acuña; orientadora Márcia CristinaAnderson Braz Federson; coorientador JaquelineGodoy Mesquita. – São Carlos – SP, 2016.

134 p.

Tese (Doutorado - Programa de Pós-Graduação emMatemática) – Instituto de Ciências Matemáticas e deComputação, Universidade de São Paulo, 2016.

1. Template. 2. Qualification monograph.3. Dissertation. 4. Thesis. 5. Latex. I. Federson,Márcia Cristina Anderson Braz, orient. II. Mesquita,Jaqueline Godoy, coorient. III. Título.

Rogelio Grau Acuña

Sobre propriedades qualitativas de EDOs generalizadas

Tese apresentada ao Instituto de CiênciasMatemáticas e de Computação – ICMC-USP, comoparte dos requisitos para obtenção do título deDoutor em Ciências – Matemática. VERSÃOREVISADA

Área de Concentração: Matemática

Orientadora: Profa. Dra. Márcia Cristina AndersonBraz Federson

Coorientadora: Profa.Dra. Jaqueline Godoy Mesquita

USP – São Carlos Setembro de 2016

Acknowledgment

First of all, I want to thank God for giving me the wisdom and strength to make this a

reality project.

I would like to thank my advisor, professor Marcia Cristina Anderson Braz Federson,

for all the opportunities that she provided me during my PhD course, encouraging me to

improve my career as a researcher, showing a lot of opportunities.

I am grateful to my co-advisor, professor Jaqueline Godoy Mesquita, who gave me all the

necessary support during my stay in Brasilia and Ribeirao Preto. She taught me a lot and

shared with me her knowledge. This was really important to me.

I am grateful to CNPq and CAPES for the financial support during my doctorate.

Finally, i would like to thank my loved family: Sonia Acuna, Rosa ortiz, Maria Carolina,

Maria Camila, Osvaldo Grau, Clareth Grau and the rest of my family.

i

Resumo

Neste trabalho, nosso objetivo e provar resultados sobre prolongamento de solucoes, lim-

itacao uniforme de solucoes, estabilidade uniforme e estabilidade uniforme assintotica (no

sentido classico de Lyapunov) para equacoes diferenciais em medida e para equacoes dinami-

cas em escalas temporais.

A fim de obter os nossos resultados, empregamos a teoria de EDOs generalizadas, uma

vez que estas equacoes abrangem equacoes diferenciais em medida e equacoes dinamicas

em escalas temporais. Portanto, para obter nossos resultados, vamos comecar por provar,

os resultados que queremos para EDOs generalizadas abstratas. Em seguida, usando a

correspondencia entre as solucoes de EDOs generalizadas e solucoes de equacoes diferenciais

em medida (ver [38]), estenderemos os resultados para estas ultimas equacoes. Depois disso,

usando a correspondencia entre as solucoes de equacoes diferenciais em medida e as solucoes

de equacoes dinamicas em escalas temporais (ver [21]), estenderemos todos os resultados

para estas ultimas equacoes.

Finalmente, investigamos EDOs generalizadas autonomas e mostramos que estas equacoes

nao aumentam a classe de EDOs autonomas classicas, mesmo quando consideramos uma

classe mais geral de funcoes nos lados direitos das equacoes.

Os novos resultados encontrados estao contidos em [16, 17, 18, 19].

Palavras-chaves: Equacoes diferenciais em medida, equacoes diferenciais ordinarias

generalizadas, equacoes dinamicas em escalas temporais, limitacao, estabilidade de Lya-

punov, prolongamento, integral de kurzweil-Henstock-Stieltjes, funcionais de Lyapunov.

iii

Abstract

In this work, our goal is to prove results on prolongation of solutions, uniform bounded-

ness of solutions, uniform stability as well uniform asymptotic stability (in the classical sense

of Lyapunov) for measure differential equations and for dynamic equations on time scales.

In order to get our results, we employ the theory of generalized ODEs, since these equa-

tions encompass measure differential equations and dynamic equations on time scales. There-

fore, to get our results, we start by proving the expected result for abstract generalized ODEs.

Then, using the correspondence between the solutions of these equations and the solutions of

measure differential equations (see [38]), we extend all the results to these the latter. After

that, using the correspondence between the solutions of measure differential equations and

the solutions of dynamic equations on time scales (see [21]), we extend all the results to these

last equations.

Finally, we investigate autonomous generalized ODEs and show that these equations do

not enlarge the class of classical autonomous ODEs, even when we consider a more general

class of functions as right-hand sides.

All the new results presented in this work are contained in papers [16, 17, 18, 19].

Keywords: Measure differential equations, generalized ordinary differential equations,

dynamic equations on time scales, boundedness, Lyapunov stability, prolongation, Kurzweil-

Henstock-Stieltjes integral, Lyapunov functionals.

v

Contents

Introduction 1

1 Generalized ODE 5

1.1 The Kurzweil integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Generalized ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Dynamic equations on time scales 17

2.1 Time scales calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Kurzweil-Henstock delta integrals . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Correspondences between equations 25

3.1 Measure Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Measure differential equations and Generalized ODEs . . . . . . . . . . . . . 26

3.3 Dynamic equation on time scales and measure differential equations . . . . . 36

4 Prolongation of solutions 47

4.1 Prolongation of the solutions of generalized ODEs . . . . . . . . . . . . . . . 47

4.2 Prolongation of solutions of measure differential equations . . . . . . . . . . 61

4.3 Prolongation of solutions of dynamic equation on time scales . . . . . . . . . 66

vii

viii CONTENTS

5 Boundedness of solutions 79

5.1 Boundedness of solutions of generalized ODEs . . . . . . . . . . . . . . . . . 79

5.2 Boundedness of solutions of measure differential equations . . . . . . . . . . 91

5.3 Boundedness of solutions of dynamic equations on time scales . . . . . . . . 94

6 Lyapunov stability 99

6.1 Lyapunov stability for generalized ODEs . . . . . . . . . . . . . . . . . . . . 99

6.2 Lyapunov stability for measure differential equations . . . . . . . . . . . . . 107

6.3 Lyapunov stability for dynamic equations on time scales . . . . . . . . . . . 110

7 Remarks on autonomous generalized ODEs 119

7.1 Autonomous generalized ODEs . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Correspondence between F(Ω, h, ω) and F(Ω, h, ω) . . . . . . . . . . . . . . 122

7.3 The classes F(Ω∞, h, ω) and F(Ω∞, h, ω, E) . . . . . . . . . . . . . . . . . . 127

Introduction

The theory of generalized ordinary differential equations (we write generalized ODEs, for

short) was introduced by J. Kurzweil in 1957 for Euclidean and Banach space-valued func-

tions with the purpose to generalize certain results on continuous dependence on parameters

of the solutions of ordinary differential equations (see [30]).

Since then, these equations have been attracting the attention of many researchers, be-

cause they encompass several types of differential equations, such as ordinary and functional

differential equations (see [5, 33]), measure differential equations and measure functional dif-

ferential equations (see [20, 38]), dynamic equations on time scales and functional dynamic

equations on time scales (see [20, 42]), differential equations with impulses (see [1, 23]),

among others.

We point out that by the correspondences between the solutions of a generalized ODE and

the solutions of other types of equations, we are able to translate the results from generalized

ODEs to another class of differential equations. The gain in doing so comes from the use

of a nonabsolute integral in the integral form of a differential equation. Such integral is

the Kurzweil integral whose main feature is to integrate highly oscillating functions. It also

copes well with many discontinuities.

In this thesis, our goal is to present results on prolongation of solutions of measure dif-

ferential equations and of solutions of dynamic equations on time scales, as well as results

on boundedness of solutions and Lyapunov-type stability results for these two types of equa-

tions. In order to obtain our results, we employ the theory of generalized ODEs, since these

equations encompass measure differential equations and dynamic equations on time scale.

Therefore, we start by proving our results for abstract generalized ODEs and, then, by the

correspondence between the solutions of these equations and the solutions of measure differ-

ential equations, we extend our results to this last class of equations. After that, we use the

correspondence between the solutions of measure differential equations and the solutions of

dynamic equations on time scales to extend our results to these last equations.

1

2 Introduction

Measure differential equations are present in the literature since the decade of 1960 with

a work by W. Schmaedeke (see [37]) on control theory. Since then, several works had been

developed in measure differential equations on qualitative properties of their solutions, their

asymptotic behavior and applications. See [1, 2, 6, 10, 11, 12, 13, 14, 34, 39, 40].

It worths noticing that measure differential equations represent a very important tool

to applications, since they describe impulsive systems with discontinuous solutions and,

therefore, can be used to studying evolutionary processes such as biological or physical

phenomena, optimal control models in economics, among others. See, for instance, [1, 12,

34, 37]

In this work, we consider an integral form of a measure differential equations of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (0.1)

where τ0 ≥ t0, f : O × [t0,+∞) → Rn, O ⊂ Rn is an open set, and g : [t0,+∞) → Ris a nondecreasing function. The integral on the right-hand side of (0.1) is the Kurzweil-

Henstock-Stieljes integral. In order to obtain our main results, we assume conditions on

indefinite integral instead of conditions on the integrands. This choice enables us to deal

with functions which may be highly oscillating or have many discontinuites.

On the other hand, dynamic equations on time scales also play an important role on

applications to several fields of knowledge. This is a recent theory, which was introduced

by Stefan Hilger in his doctor thesis (see [27]), in order to unify the discrete and continuous

analysis and the cases “in between”. The main idea of this theory is to prove the results for

a type of equation, called dynamic equation, where the domain of the unknown function is a

time scale, which is an arbitrary closed and nonempty subset of the real numbers. Therefore,

depending on the chosen time scale, we obtain a result for a different type of equation. For

example, when the time scale is the set of real numbers, we obtain a result for differential

equations, and when the time scale is the set of integers, we obtain a result for difference

equations. Notice that we can obtain several results for different types of equations depending

on the chosen time scale.

Since dynamic equations on time scales have several applications to real-world models,

it is very important to understand their properties on unbounded intervals, that is, when

the time is sufficiently large, because then one can investigate properties such as asymptotic

behavior, stability, boundedness, attractors, bifurcation, among others. Therefore the first

step in order to do that is to investigate existence and uniqueness of a maximal solution and

prolongation of solutions for these equations.

The present thesis is divided into seven chapters which are organized as follows. In the

first chapter, we recall the basic concepts and properties concerning the Kurzweil integral and

Introduction 3

its applications to generalized ODEs. The second chapter is devoted to recalling the theory

of dynamic equations on time scales, and to presenting the main results and basic concepts

of this theory. In the third chapter, we recall the basis of the theory of measure differential

equations and its main properties. Also, we investigate the correspondence between the

solutions of measure differential equations and the solutions of generalized ODEs. Finally,

we present a connection between the solutions of measure differential equations and the

solutions of dynamic equations on time scales.

The fourth chapter is devoted to investigating the prolongation of solutions of generalized

ODEs, measure differential equations, as well as the solutions of dynamic equations on time

scales. We start by investigating the existence and uniqueness of maximal solutions of

generalized ODEs defined in a general Banach space. New results are obtained on this

subject. Then, using the correspondence between the solutions, we obtain analogous results

for measure differential equations and for dynamic equations on time scales. Most of the

results presented in this chapter are new.

On the other hand, we are also interested in investigating the boundedness of solutions of

measure differential equations and of dynamic equations on time scales. Then we prove new

results on boundedness of solutions of generalized ODE which do not require a Lipschitz-

type condition with respect to the second variable on the Lyapunov functional, improving

the results found in the literature (see [2]). After that, we extend our results to measure

differential equations, using the correspondence between the solutions of these equations

and the solutions of generalized ODEs, improving the results found in the literature for

measure differential equations. Furthermore, we prove a result on boundedness of solutions

of dynamic equations on time scales, using the fact that a dynamic equation on time scales

is a special case of measure differential equations. Moreover, we introduce new concepts

of boundedness of solutions of dynamic equations on time scales, namely, quasi-uniformly

ultimately boundedness and uniformly ultimately boundedness. All these result are new and

are collected in Chapter 5.

In this work, we also investigate Lyapunov-type stability of the trivial solution of general-

ized ODEs in Banach spaces. The stability concepts presented here extend the concepts from

[44]. Also, we point out that our results on uniform stability and uniform asymptotic stabil-

ity of the trivial solution of a generalized ODE do not require any Lipschitz-type condition

on the Lyapunov functional, improving the results found in the literature (see [3, 22, 24]).

Using the correspondences between the solutions, we extend our results to measure differ-

ential equations and to dynamic equations on time scales. All these results are original and

are contained in Chapter 6.

Finally, we investigate autonomous generalized ODEs and prove that these equations do

not enlarge the class of classical autonomous ODEs.

All the new results presented here are contained in the papers [16, 17, 18, 19].

Chapter

1Generalized ODE

In this chapter, we recall the concept of generalized ordinary differential equations (we

write generalized ODEs, for short) and present some basic results which play a fundamental

role throughout this work.

We divide this chapter into two sections. In the first section, we present the definition

of the Kurzweil integral which is the basis to define the concept of a generalized ODE. The

second section is devoted to presenting the basic theory of generalized ODEs. The main

references for this chapter are [31, 38].

1.1 The Kurzweil integral

We start this section by recalling some basic concepts concerning the Kurzweil integral.

Let [a, b] be an interval of R, −∞ < a < b < +∞. A tagged division of [a, b] is a finite

collection of point-interval pairs D = (τi, [si−1, si]), where a = s0 ≤ s1 ≤ . . . ≤ s|D| = b

is a division of [a, b] and τi ∈ [si−1, si], i = 1, 2, . . . , |D|, where the symbol |D| denotes the

number of subintervals in which [a, b] is divided.

A gauge on a set B ⊂ [a, b] is any function δ : B → (0,∞). Given a gauge δ on [a, b], we

say that a tagged division D = (τi, [si−1, si]) is δ-fine if, for every i, we have

[si−1, si] ⊂ (τi − δ(τi), τi + δ(τi)).

Using these definitions and notations, we are able to present the concept of a Kurzweil

integrable function defined on [a, b]× [a, b] taking values in an abstract space X.

Throughout this section, let us assume that X is a Banach space with norm ‖ · ‖ .

5

6 Chapter 1 — Generalized ODE

Definition 1.1. A function U : [a, b]× [a, b]→ X is called Kurzweil integrable over [a, b], if

there is an element I ∈ X such that given ε > 0, there is a gauge δ on [a, b] such that∥∥∥∥∥∥|D|∑i=1

[U (τi, si)− U (τi, si−1)]− I

∥∥∥∥∥∥ < ε

for every δ-fine tagged division D = (τi, [si−1, si]) of [a, b]. In this case, I is called the

Kurzweil integral of U over [a, b] and it is denoted by∫ baDU (τ, t).

We use the notation S(U,D) =|D|∑i=1

[U (τi, si) − U (τi, si−1)] for the Riemann-type sum

corresponding to the function U and to the tagged division D.

Analogously to the Riemann integral, the Kurzweil integral has the usual properties of

linearity, additivity with respect to adjacent intervals and integrability on subintervals.

If the integral∫ baDU(τ, t) exists, then we define∫ a

b

DU(τ, t) = −∫ b

a

DU(τ, t), if a < b,

and ∫ b

a

DU(τ, t) = 0, if a = b.

In particular, when U(τ, t) = f(τ)t, where f : [a, b] → Rn, then we obtain the usual

Kurzweil-Henstock integral of f . More generally, when U(τ, t) = f(τ)g(t), we obtain the

Kurzweil-Henstock-Stieljes (or Perron-Stieltjes) integral of f : [a, b] → Rn with respect to a

function g : [a, b]→ R and we will denote such integral byb∫a

f(s)dg(s).

Clearly, Definition 1.1 makes sense only if, for a given gauge δ of [a, b], there exists at

least one δ-fine division D of [a, b]. This is the content of the next result. For more details,

see [26, Theorem 4.1].

Lemma 1.2 (Cousin Lemma). Given a gauge δ of [a, b], there exists a δ-fine tagged division

D of [a, b].

The next result is known as the Saks-Henstock Lemma. A proof of it can be found in

[38, Lemma 1.13].

Lemma 1.3 (Saks-Henstock Lemma). Let U : [a, b]× [a, b]→ X be Kurzweil integrable over

[a, b]. Given ε > 0, let δ be a gauge of [a, b] corresponding to ε in the definition of the

1.1 The Kurzweil integral 7

Kurzweil integral such that∥∥∥∥∥∥|D|∑i=1

[U (τi, αi)− U (τi, αi−1)]−b∫

a

DU(τ, t)

∥∥∥∥∥∥ < ε, (1.1)

for every δ-fine tagged division D = (τi, [αi−1, αi]) of [a, b]. If

a ≤ β1 ≤ ξ1 ≤ γ1 ≤ β2 ≤ ξ2 ≤ γ2 ≤ . . . ≤ βm ≤ ξm ≤ γm ≤ b

represents a δ-fine tagged partial division (ξi, [βi, γi]) : i = 1, 2, . . . ,m , that is,

ξi ∈ [βi, γi] ⊂ (ξi − δ(ξi), ξi + δ(ξi)), i = 1, 2, . . . ,m,

butm⋃i=1

[βi, γi] does not need to equal [a, b], then

∥∥∥∥∥∥m∑i=1

[U (τi, αi)− U (τi, αi−1)−

b∫a

DU(τ, t)]∥∥∥∥∥∥ < ε.

The following result is an immediate consequence of the Saks-Henstock Lemma. See [38].

Corollary 1.4. Let U : [a, b] × [a, b] → X be Kurzweil integrable over [a, b]. Given ε > 0,

let δ be a gauge on [a, b] corresponding to ε in the definition of the Kurzweil integral and

[γ, v] ⊂ [a, b]. Then the following assertions hold.

(i) If (v − γ) < δ (γ) , then∥∥∥∥U (γ, v)− U (γ, γ)−∫ v

γ

DU (τ, t)

∥∥∥∥ < ε;

(ii) If (v − γ) < δ (v) , then∥∥∥∥U (v, v)− U (v, γ)−∫ v

γ

DU (τ, t)

∥∥∥∥ < ε.

The next result is a generalization of [38, Theorem 1.14] for general Banach space-valued

function. We omit its proof here, since it is similar to the finite dimensional case.


Theorem 1.5. If U : [a, b] × [a, b] → X is a function such that for every c ∈ [a, b), U is

Kurzweil integrable over [a, c] and the limit

limc→b−

[ c∫a

DU(τ, t)− U(b, c) + U(b, b)]

= I ∈ X

exists, then the function U is Kurzweil integrable over [a, b] and

b∫a

DU(τ, t) = I.

Similarly, if the function U is Kurzweil integrable over [c, b] for every c ∈ (a, b] and the limit

limc→a+

[ b∫c

DU(τ, t) + U(a, c)− U(a, a)]

= I ∈ X

exists, then the function U is Kurzweil integrable over [a, b] and

b∫a

DU(τ, t) = I.

The next result shows that the indefinite Kurzweil integral of U : [a, b]× [a, b]→ X, the

function s 7→∫ saDU(τ, t), is not continuous in general. A proof of this fact can be found in

[38, Theorem 1.16].

Theorem 1.6 (Hake-type Theorem). Let U : [a, b]× [a, b]→ X be Kurzweil integrable over

[a, b] and let c ∈ [a, b]. Then,

lims→c

[ s∫a

DU(τ, t)− U(c, s) + U(c, c)]

=

c∫a

DU(τ, t)

and

lims→c

[ b∫s

DU(τ, t) + U(c, s)− U(c, c)]

=

b∫c

DU(τ, t).

Remark 1.7. Notice that the indefinite Kurzweil integral is continuous at a point c ∈ [a, b],

if and only if, U(c, ·) : [a, b] → X, given by U(c, t) for t ∈ [a, b], is continuous at the point

t = c.

1.1 The Kurzweil integral 9

We recall the reader that a function f : [a, b]→ X is called regulated, if the lateral limits

f(t−) = lims→t−

f(s), t ∈ (a, b] and f(t+) = lims→t+

f(s), t ∈ [a, b)

exist. On the other hand, given a function f : [a, b] → X, its variation var[a,b] f on the

interval [a, b] is defined by

var[a,b] f := supD∈D[a,b]

|D|∑i=1

|f(ti)− f(ti−1)| ,

where D[a, b] denotes the set of all division of [a, b]. If var[a,b] f < +∞, we say that f is of

bounded variation in the interval [a, b].

The space of all regulated functions f : [a, b]→ X will be denoted by G([a, b], X) which

is a Banach space when endowed with the usual supremum norm

‖f‖∞ = sups∈[a,b]

‖f(s)‖ .

It is also known that functions of bounded variation in [a, b] are regulated. For more infor-

mation, see [25, 29].

The following result, which describes some properties of the indefinite Kurzweil-Henstock-

Stieltjes integral, is a special case of Theorem 1.6.

Theorem 1.8. Let f : [a, b] → Rn and g : [a, b] → R be functions such that g is regulated

and∫ baf(s)dg(s) exists. Then the functions

h(t) =

∫ t

a

f(s)dg(s), t ∈ [a, b] and k(t) =

∫ b

t

f(s)dg(s), t ∈ [a, b],

are regulated on [a, b] and satisfy

h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b),

h(t−) = h(t)− f(t)∆−g(t), t ∈ (a, b],

k(t+) = k(t)− f(t)∆+g(t), t ∈ [a, b),

k(t−) = k(t) + f(t)∆−g(t), t ∈ (a, b],

where

∆+g(t) = g(t+)− g(t) and ∆−g(t) = g(t)− g(t−).


Also,

g(t+) = limξ→t+

g(ξ), t ∈ [a, b) and g(t−) = limξ→t−

g(ξ), t ∈ (a, b].

Analogously, one can define h(t+), h(t−), k(t+) and k(t−).

We finish this section by presenting a result which ensures the existence of the Kurzweil-

Henstock-Stieltjes integralb∫a

f(s)dg(s). This result can be found in [20, Theorem 2.1].

Theorem 1.9. Let f : [a, b] → Rn be a regulated function on [a, b] and g : [a, b] → R be a

nondecreasing function. Then, the following conditions hold

(i) The integral

∫ b

a

f(s)dg(s) exists;

(ii)

∣∣∣∣∫ b

a

f(s)dg(s)

∣∣∣∣ ≤ ∫ b

a

|f(s)| dg(s).

1.2 Generalized ODEs

In this section, we reacall the definition of a generalized ODEs and its main properties.

The main reference is [38].

Consider a subset O ⊂ X, an interval [t0,+∞) ⊂ R and an X-valued function F : Ω→ X

defined for each (x, t) ∈ Ω, where Ω = O × [t0,+∞).

Definition 1.10. A function x : [α, β]→ X is called a solution of the generalized ODE

dx

dτ= DF (x, t) (1.2)

on the interval [α, β] ⊂ [t0,+∞), if (x(t), t) ∈ Ω, for every t ∈ [α, β], and

x(s2)− x(s1) =

s2∫s1

DF (x(τ), t), (1.3)

for every s1, s2 ∈ [α, β].

The integral on the right-hand side of (1.3) can be understood as the Kurzweil integral

of U(τ, t) = F (x(τ), t) described in Definition 1.1.

1.2 Generalized ODEs 11

Remark 1.11. It is worth mentioning that the notation in (1.2) is only symbolical. The

letter D indicates that (1.2) is a generalized ODE and this concept is defined via its solutions.

Even the expressiondx

dτdoes not mean that the solution has a derivative.

We can also define a solution of the generalized ODE (1.2) with an initial condition as

shows the next definition.

Definition 1.12. The function x : [α, β] → X is a solution of the generalized ODE (1.2)

with initial condition x(τ0) = x0 on the interval [α, β] ⊂ [t0,+∞), if τ0 ∈ [α, β], (x(t), t) ∈ Ω

for every t ∈ [α, β] and

x(s)− x0 =

s∫τ0

DF (x(τ), t)

for every s ∈ [α, β].

Although a solution x of the generalized ODE (1.2) is defined on a bounded interval [α, β]

in Definition 1.10, it is possible to extend this definition to a nondegenerated interval I, not

necessarily compact. This is the content of the next definition.

Definition 1.13. We say that x : I → X, where I is a nondegenerated subinterval of

[t0,+∞), is a solution of the generalized ODE (1.2) on I, if (x(t), t) ∈ Ω for every t ∈ I and

if the following equality

x(s2)− x(s1) =

s2∫s1

DF (x(τ), t)

is satisfied for every s1, s2 ∈ I.

In what follows, we present a class of functions F : Ω→ X which is be crucial to obtaining

important properties concerning the solutions of the generalized ODE (1.2).

Definition 1.14. Given a nondecreasing function h : [t0,+∞)→ R, we say that a function

F : Ω→ X belongs to the class F(Ω, h), if

‖F (x, t)− F (x, s)‖ ≤ |h(t)− h(s)| (1.4)

for all (x, t), (x, s) ∈ Ω, and if

‖F (x, t)− F (x, s)− F (y, t) + F (y, s)‖ ≤ ‖x− y‖ |h(t)− h(s)| (1.5)

for all (x, t), (x, s), (y, t), (y, s) ∈ Ω.

The following result is essential to our purposes. It can be found in [38, Lemma 3.9].


Lemma 1.15. Let F : Ω→ X satisfy condition (1.4). If [α, β] ⊂ [t0,+∞) and x : [α, β]→

X is such that (x(t), t) ∈ Ω for every t ∈ [α, β] and if the Kurzweil integralβ∫α

DF (x(τ), t)

exists, then for every s2, s1 ∈ [α, β], we have∥∥∥∥∥∥s2∫s1

DF (x(τ), t)

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)| .

The next result describes an important property of the solutions of the generalized ODE

(1.2). It can be found in [38, Lemma 3.10].

Lemma 1.16. Assume F : Ω → X satisfies condition (1.4). If [a, b] ⊂ [t0,+∞) and

x : [a, b]→ X is a solution of the generalized ODE (1.2), then the inequality

‖x(t)− x(s)‖ ≤ |h(t)− h(s)|

holds for all t, s ∈ [a, b]. Also, each point in [a, b] at which the function h is continuous is a

continuity point of the solution x : [a, b]→ X.

To the next result, let var[a,b] x be the variation of a function x : [a, b]→ X. If the right-

hand side of the generalized ODE (1.2) satisfies condition (1.4), then the next result ensures

that its solution is a function of bounded variation. A proof of it can be found in [38, Lemma

3.11].

Corollary 1.17. Let F : Ω → X satisfy condition (1.4). If [α, β] ⊂ [t0,+∞) and x :

[α, β] −→ X is a solution of the generalized ODE (1.2), then x is a function of bounded

variation (and, therefore, regulated) in [α, β] and

var[α,β]x ≤ h(β)− h(α) < +∞. (1.6)

Now, we present a result which ensures the existence of the integral involved in the

definition of the solution of the generalized ODE (1.2). This result is a consequence of [38,

Lemma 3.9, Corollary 3.16].

Proposition 1.18. Let F : Ω→ X be an element of the class F(Ω, h). If [α, β] ⊂ [t0,+∞)

and x : [α, β]→ X is regulated (in particular, a function of bounded variation) in [α, β] and

(x(s), s) ∈ Ω for every s ∈ [α, β], then the Kurzweil integral∫ βαDF (x(τ), t) exists and the

function s 7→∫ sαDF (x(τ), t) is of bounded variation in [α, β].

The next result describes the discontinuities of a solution of the generalized ODE (1.2),

provided F satisfies (1.4). For a proof of it, see [38, Lemma 3.12].


Lemma 1.19. Consider [α, β] ⊂ [t0,+∞), x : [α, β] → X is a solution of the generalized

ODE (1.2) and F : Ω→ X satisfies condition (1.4). Then, we have

x(t+)− x(t) = F (x(t), t+)− F (x(t), t), for every t ∈ [α, β),

and

x(t)− x(t−) = F (x(t), t)− F (x(t), t−), for every t ∈ (α, β],

where

F (x, t+) = lims→t+

F (x, s), for every t ∈ [α, β),

and

F (x, t−) = lims→t−

F (x, s), for every t ∈ (α, β].

The next result can be found in [38, Theorem 3.14].

Theorem 1.20. Let F ∈ F(Ω, h). If x : [a, b] → X, is the uniform limit of a sequence

(xk)k∈N of step functions xk : [a, b] → X such that (x(s), s)) ∈ Ω and (xk(s), s) ∈ Ω, for

every k ∈ N and for every s ∈ [a, b], then the integral∫ baDF (x(τ), t) exists and∫ b

a

DF (x(τ), t) = limk→∞

∫ β

α

DF (xk(τ), t).

The next result ensures the existence and uniqueness of a local solution for an initial

value problem of the generalized ODE (1.2). For a proof of it, see [23, Theorem 2.15].

Theorem 1.21 (Local existence and uniqueness). Let F : Ω → X be an element of the

class F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing and left-continuous.

If (x0, τ0) ∈ Ω is such that x0 + F (x0, τ+0 ) − F (x0, τ0) ∈ O, then there exist ∆ > 0 and a

function x : [τ0, τ0 + ∆]→ X which is the unique solution of the generalized ODE (1.2) for

which x(τ0) = x0.

Remark 1.22. Notice that the condition x0 +F (x0, τ+0 )−F (x0, τ0) ∈ O from Theorem 1.21

is sufficient, but not necessary. The following example illustrates such a claim.

Example 1.23. Let X = R with norm | · | (absolute value), B1 := B(0, 1) ⊂ R, [0, 1] ⊂ Rand Ω := B1 × [0, 1], where B(0, 1) = x ∈ R : |x| < 1 . Consider a function ϕ : [0, 1] → Rgiven by

ϕ(t) :=

0, t = 0

t− 1, 0 < t ≤ 1,(1.7)

and a function F : Ω → R given by F (x, t) := ϕ(t), for all (x, t) ∈ Ω. Notice that the

function F is constant with relation to the first variable, that is, F (x, t) = F (y, t) for every


x, y ∈ B1. Also, define a function h : [0, 1]→ R as follows

h(t) :=

0, t = 0

2t+ 1, 0 < t ≤ 1.

Note that, by the definition, the function h is left-continuous on (0, 1] and increasing on

[0, 1]. Consider the generalized ODE given bydx

dτ= DF (x, t) = D[ϕ(t)]

x(0) = 0.(1.8)

Assertion 1. F ∈ F(Ω, h).

i) At first, we will show that |F (x, t2)−F (x, t1)| ≤ |h(t2)−h(t1)|, for all (x, t2), (x, t1) ∈ Ω.

– Let 0 < t1 ≤ t2 ≤ 1 and x ∈ B1. Then

|F (x, t2)− F (x, t1)| = |ϕ(t2)− ϕ(t1)|= |t2 − 1− (t1 − 1)|= |t2 − t1|≤ |2(t2 − t1)|= |2t2 + 1− (2t1 + 1)|= |h(t2)− h(t1)|.

– Let t1 = 0, 0 < t2 ≤ 1 and x ∈ B1. Then

|F (x, t2)− F (x, t1)| = |ϕ(t2)− ϕ(t1)|= |t2 − 1| = 1− t2< 1 < 2t2 + 1

= h(t2)− h(0) = |h(t2)− h(t1)|.

– The case t1 = t2 = 0 follows trivially. Therefore, we have

|F (x, t2)− F (x, t1)| ≤ |h(t2)− h(t1)|, for every (x, t2), (x, t1) ∈ Ω.

ii) Also, the following inequality

|F (x, t2)− F (x, t1)− F (y, t2) + F (y, t1)| ≤ |h(t2)− h(t1)| |x− y| ,


is satisfied for all (x, t2), (x, t1), (y, t2), (y, t1) ∈ Ω. Indeed, notice that

|F (x, t2)− F (x, t1)− F (y, t2) + F (y, t1)| = |ϕ(t2)− ϕ(t1)− ϕ(t2) + ϕ(t1)|= 0 ≤ |h(t2)− h(t1)| |x− y| ,

for all (x, t2), (x, t1), (y, t2), (y, t1) ∈ Ω.

Assertion 2. For x0 = 0 and τ0 = 0, we have 0 + F (0, 0+)− F (0, 0) /∈ B1.

Indeed, notice that by the definition of F, we have

F (0, 0+)− F (0, 0) = limt→0+

F (0, t)− F (0, 0) = limt→0+

ϕ(t) = −1,

which implies that

0 + F (0, 0+)− F (0, 0) = −1 /∈ B1.

Assertion 3. The function ϕ, given by (1.7), is the unique solution of the generalized ODE

(1.8) on [0, 1].

Indeed, we will start by proving that ϕ is solution of the generalized ODE (1.8) on [0, 1]. Let

s ∈ [0, 1], then

ϕ(s)− ϕ(0) =

s∫0

D[ϕ(t)] =

s∫0

DF (ϕ(τ), t).

Therefore, ϕ is solution of (1.8) on [0, 1].

Now, suppose x : [0, 1]→ R is also a solution of (1.8) on [0, 1]. Then, given s ∈ [0, 1], we

have

x(s) = 0 +

s∫0

DF (x(τ), t)︸︷︷︸ϕ(t)

=

s∫0

D[ϕ(t)] = ϕ(s)− ϕ(0)︸︷︷︸0

= ϕ(s),

that is, x(s) = ϕ(s) for all s ∈ [0, 1]. Notice that the equality ϕ(s)−ϕ(0) =s∫

0

D[ϕ(t)] follows

directly from the definition of the Kurzweil integral.

Chapter

2Dynamic equations on time scales

In this chapter, we will present the basic concepts concerning the theory of dinamic

equations on time scales. This theory is recent and it was introduced by Stefan Hilger in his

doctor thesis (see [27]) in order to unify the discrete and continuous analysis and the cases

“in between”.

This chapter is divided into three sections. In the first section, we present the basic

definitions and notations about the theory of dynamic equations on time scales. The second

section is dedicated to presenting the basic concepts and properties of delta derivatives. The

last section is devoted to the study of the Kurzweil-Henstock ∆-integrals.

The main references for this chapter are [7, 8, 27].

2.1 Time scales calculus

In this section, our goal is to present the basic concepts and results concerning the theory

of dynamic equations on time scales.

A time scale is an arbitrary nonempty and closed subset of the real numbers (with the

standard topology of R). Thus

R, [1, 2] ∪ N, Z, the Cantor set

are examples of time scales. On the other hand, open intervals, half-open intervals, R\Q, Qare not examples of time scales, since they are not closed sets in R. Here, we denote a time

scale by the symbol T.

17

18 Chapter 2 — Dynamic equations on time scales

Let T be a time scale. We define the forward jump operator σ : T→ T by

σ(t) := inf s ∈ T : s > t ,

while the backward jump operator ρ : T→ T is given by

ρ(t) := sup s ∈ T : s < t .

In the above definitions, we consider inf ∅ = supT and sup ∅ = inf T. Then, it is not difficult

to see that σ(t) = t, if T has a maximum t, and ρ(t) = t, if T has a minimum t. Since Tis a closed set, it is clear from the definition of the forward jump operator and backward

jump operator, that the values σ(t) and ρ(t) belong to T, for all t ∈ T. Also, we define the

graininess function µ : T→ [0,+∞) by

µ(t) := σ(t)− t.

Definition 2.1. Let T be a time scale and t ∈ T. We say that

(i) t is right-scattered, if σ(t) > t;

(ii) t is left-scattered, if ρ(t) < t;

(iii) t is isolated, if it is right-scattered and left-scattered.

Definition 2.2. Let T be a time scale and t ∈ T. We say that

(i) t is right-dense, if t < supT and σ(t) = t;

(ii) t is left-dense, if inf T < t and ρ(t) = t;

(iii) t is dense, if it is right-dense and left-dense.

In what follows, we present some examples to illustrate the above definitions. They are

borrowed from [7].

Example 2.3. Let T = Z. Then for all t ∈ Z, we have

σ(t) = inf s ∈ Z : s > t = inf t+ 1, t+ 2, . . . = t+ 1 > t

ρ(t) = sup s ∈ Z : s < t = sup t− 1, t− 2, . . . = t− 1 < t;

Also, we have

µ(t) = 1.

Therefore, we can easily see that every point in Z is isolated.

2.1 Time scales calculus 19

Example 2.4. Let T = R. Then, for each t ∈ R, we have

σ(t) = inf s ∈ R : s > t = inf(t,+∞) = t

ρ(t) = sup s ∈ R : s < t = sup(−∞, t) = t.

Thus, each t ∈ R is a right-dense and left-dense point. Hence t is dense. Furthermore,

µ(t) = 0, for all t ∈ T.

Example 2.5. Let T = qZ ∪ 0 , with q > 1. Then for any t ∈ T\ 0 , we have

σ(t) = inf s ∈ T : s > t = infqt, q2t, . . .

= qt > t

ρ(t) = sup s ∈ Z : s < t = sup

t

q,t

q2, . . .

=t

q< t.

Thus, each t ∈ T\ 0 is an isolated point. On the other hand, if t = 0, then σ(0) = 0 which

implies that 0 is a right-dense point. Also, note that

µ(t) =

(q − 1)t, t ∈ T\ 0 ,0, t = 0.

Example 2.6. Let T = hZ = hk : k ∈ Z , where h > 0. Then, for every t ∈ T, we have

σ(t) = inf s ∈ T : s > t = inf t+ nh : n ∈ N = t+ h > t

ρ(t) = sup s ∈ T : s < t = sup t− nh : n ∈ N = t− h < t.

Hence, every point t ∈ T is an isolated point and

µ(t) = σ(t)− t = h, for any t ∈ T,

which implies that µ is constant.

To conclude this section, we define the set Tκ which is derived from the time scale T as

follows

Tκ =

T\(ρ(supT), supT], if supT <∞,T, if supT =∞.

(2.1)

This definition is important to define the ∆-derivative of a function f, which we will presented

in the next section.


2.2 Differentiation

In this section, we present the basic concepts and main properties about the ∆-derivative

of a function f : T→ R.

Let f : T→ R be a function. We define the so-called delta (or Hilger) derivative of f at

a point t ∈ Tκ as follows.

Definition 2.7. Suppose that f : T → R is a function and t ∈ Tκ. Then, we define f∆(t)

to be the number (provided it exists) with the property that, given any ε > 0, there is a

neighborhood U of t (that is, U = (t− δ, t+ δ) ∩ T for some δ > 0) such that∣∣(f(σ(t))− f(s))− f∆(t)(σ(t)− s)∣∣ ≤ ε |σ(t)− s| , for all s ∈ U.

We call f∆(t) the delta (ou Hilger) derivative of f at t. Moreover, we say that f is delta

differentiable on Tκ, provided f∆(t) exists for all t ∈ Tκ. The function f∆ : Tκ → R is called

the ∆-derivative of f on Tκ.

It is also possible to extend the concept of ∆-derivative presented in Definition 2.7 for an

Rn-valued functions.

Definition 2.8. Let t ∈ Tκ and f : T → Rn be an Rn-valued function, f = (f1, . . . , fn)

where each component fi is a real function defined on T. We say that f is ∆-differentiable

at t if each component fi is ∆-differentiable at t and we define

f∆(t) = (f∆1 (t), . . . , f∆

n (t)).

In the sequel, we present some examples of the ∆-derivative of a function f : T → R.

Such examples are borrowed from [7].

Example 2.9. Let f : T→ R be given by f(t) = c, for all t ∈ T, where c ∈ R is a constant.

Then f∆(t) = 0 for any t ∈ Tκ. Indeed, for every ε > 0, we have

|(f(σ(t))− f(s))− 0(σ(t)− s)| = |c− c| = 0 ≤ ε |σ(t)− s| , for all s ∈ Tκ.

Example 2.10. Let f : T → R be a function given by f(t) = t2, for any t ∈ T. Then

f∆(t) = t + σ(t), for all t ∈ Tκ. In fact, given ε > 0, let us take U = (t − δ, t + δ) with

2.2 Differentiation 21

0 < δ ≤ ε. Then, for all s ∈ U, we obtain:

|(f(σ(t))− f(s))− (t+ σ(t))(σ(t)− s)| =∣∣(σ(t))2 − s2 − tσ(t) + ts− (σ(t))2 + sσ(t)

∣∣= |σ(t)(s− t)− s(s− t)|= |s− t| |σ(t)− s|< δ |σ(t)− s| ≤ ε |σ(t)− s| .

The next result presents some important properties of the ∆-derivatives. For a proof of

it, see [7, Theorem 1.16].

Theorem 2.11. Let f : T → R be a function and t ∈ Tκ. Then the following assertions

hold

(i) If f is ∆-differentiable at t, then f is continuous at t.

(ii) If f is continuous at t and t is right-scattered, then f is ∆-differentiable at t, with

f∆(t) =f(σ(t))− f(t)

µ(t).

(iii) If t is right-dense, then f is ∆-differentiable at t if, and only if, the limit

lims→t

f(s)− f(t)

s− t

exists as a finite number. In this case, we have

f∆(t) = lims→t

f(t)− f(s)

t− s.

(iv) If f is ∆-differentiable at t, then

f(σ(t)) = f(t) + µ(t)f∆(t).

The next example illustrates the above properties. Such example is borrowed from [7].

Example 2.12. Let h > 0 and

T = hZ = hk : k ∈ Z .

Let f : T→ R be a function. As it was showed in Example 2.6, any t ∈ T is right-scattered.

Also, note that f is continuous on T = hZ (since each point t ∈ T isolated). Thus, by the


statement (ii) from Theorem 2.11, we have

f∆(t) =f(σ(t))− f(t)

µ(t)=f(t+ h)− f(t)

h, for all t ∈ T.

The next result describes algebraic properties of the ∆-derivatives. A proof of it can be

found in [7, Theorem 1.20].

Theorem 2.13. Suppose f, g : T→ R are ∆-differentiable at t ∈ Tκ. Then, we have

(i) f + g : T→ R is ∆-differentiable at t, with

(f + g)∆(t) = f∆(t) + g∆(t).

(ii) For every α ∈ R, αf : T→ R is ∆-differentiable at t, with

(αf)∆(t) = αf∆(t).

(iii) f · g : T→ R is ∆-differentiable at t, with

(f · g)∆(t) = f∆(t)g(t) + f(σ(t))g∆(t) = f(t)g∆(t) + f∆(t)g(σ(t)).

(iv) If g(t)g(σ(t)) 6= 0, thenf

gis ∆-differentiable at t and

(fg

)∆

(t) =f∆(t)g(t)− f(t)g∆(t)

g(t)g(σ(t)).

Remark 2.14. By Theorem 2.13, it is not difficult to see that if T = R, we have the usual

properties for the Frechet derivative.

2.3 Kurzweil-Henstock delta integrals

In this section, we present the basic concepts and the main properties of the Kurzweil-

Henstock ∆-integrals. This section is based on [35].

Let T be a time scale. For each pair of numbers a, b ∈ T, a ≤ b, set [a, b]T = [a, b] ∩ T.The open and half-open time scales intervals are defined in a similar way.

We say that δ = (δL, δR) is a ∆-gauge in [a, b]T, provided δL(t) > 0 in (a, b]T, δR(t) > 0

in [a, b)T, δL(a) ≥ 0, δR ≥ 0 and δR(t) ≥ µ(t) for all t ∈ [a, b)T.

2.3 Kurzweil-Henstock delta integrals 23

We say that P is a tagged division of [a, b]T if

P = a = s0 ≤ ξ1 ≤ s1 ≤ ξ2 ≤ s2 ≤ · · · ≤ sn−1 ≤ ξn ≤ sn = b

with si−1 < si for all i = 1, . . . , n and si, ξi ∈ T. We denoted such a tagged division by

P = [si−1, si]T : ξi , where [si−1, si]T denotes an interval in T and ξi is the associated “tag

point” in [si−1, si]T.

If δ is a ∆-gauge of [a, b]T, then a tagged division P is called δ-fine, whenever

ξi − δL(ξi) ≤ si−1 < si ≤ ξi + δR(ξi),

for i = 1, . . . , n.

In the sequel, we recall the concept of the Kurzweil-Henstock ∆-integral. This concept

was introduced for the first time by A. Peterson and B. Thompson in [35].

Definition 2.15. A function f : [a, b]T → R is called Kurzweil-Henstock ∆-integrable (or

KH delta integrable) over [a, b]T, if there is a number I such that given ε > 0, there exists a

∆-gauge δ on [a, b]T such that ∣∣∣∣∣I −n∑i=1

f(ξi)(si − si−1)

∣∣∣∣∣ < ε

for all δ-fine tagged division P of [a, b]T. In this case, I is called the Kurzweil-Henstock

∆-integral of f over [a, b]T and it will be denoted by KH∫ baf(t)∆t.

Remark 2.16. When it is clear, we write simply∫ baf(t)∆t instead of KH

∫ baf(t)∆t.

Notice that Definition 2.15 makes sense only if, for a given ∆-gauge δ on [a, b]T, there

exists at least one δ-fine tagged division of [a, b]T. The next result ensures this existence and

it is a generalization of the known Cousin Lemma for a ∆-gauge on a time scale interval.

This result can be found [35, Lemma 1.9].

Lemma 2.17. If δ is a ∆-gauge on [a, b]T, then there exists a δ-fine tagged division P of

[a, b]T.

Finally, we present the last result of this section. It describes a very important property

of the Kurzweil-Henstock ∆-integral.

Theorem 2.18. Let f : [a, b]T → R be a function and c ∈ [a, b]. Then f is Kurzweil-Henstock

∆-integrable over [a, b]T if, and only if, f is Kurzweil-Henstock ∆-integrable over [a, c]T and

[c, b]T. In this case, ∫ b

a

f(t)∆t =

∫ c

a

f(t)∆t+

∫ b

c

f(t)∆t.


Also, if f, g : [a, b]T → R are Kurzweil-Henstock ∆-integrable functions over [a, b]T, then

αf + βg is a Kurzweil-Henstock ∆-integrable function over [a, b]T and∫ b

a

(αf + βg)(t)∆t = α

∫ b

a

f(t)∆t+ β

∫ b

a

g(t)∆t,

for all α, β ∈ R.

Chapter

3Correspondences between equations

In this chapter, we present the correspondence between the solutions of a measure dif-

ferential equation and the solutions of a generalized ODE. Furthermore, we prove that the

dynamic equations on time scales can be viewed as a measure differential equations (we write

MDE, for short). These correspondences are fundamental to our purposes.

We divide this chapter into three sections. In the first section, we discuss the main

properties of measure differential equations (MDEs). In the second section, we present the

correspondence between MDEs and generalized ODEs. In the third section, we recall the

reader that dynamic equations on time scales are a special class of measure differential

equations. The main references for this chapter are [10, 20, 21, 38, 42].

3.1 Measure Differential Equations

Throughout this section, let Rn be the n-dimensional Euclidean space with norm | · | .Consider an open set O ⊂ Rn. Also, let f : O × [t0,+∞) → Rn and g : [t0,+∞) → R be

functions.

In this work, we consider the integral form of a measure differential equations of the type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (3.1)

where τ0 ≥ t0, and the integral on the right-hand side is in the sense of Kurzweil-Henstock-

Stieltjes taken with respect to g : [t0,+∞)→ R, which is a nondecreasing and left-continuous

function.

25

26 Chapter 3 — Correspondences between equations

Since these equations are very important to applications, their qualitative properties have

been investigated by different authors (see, for instance, [9, 10, 32, 40, 41]). It is a known fact

that these equations encompass integral form ordinary and impulsive differential equations

Remark 3.1. It is not difficult to see that if I ⊂ [t0,+∞) is a nondegenerate interval. Then,

the function x : I → Rn is a solution of the measure differential equation (3.1) if, and only

if, (x(t), t) ∈ O × I for all t ∈ I and

x(s2)− x(s1) =

s2∫s1

f(x(s), s)dg(s),

for every s1, s2 ∈ I, where the integral on the right-hand side is in the sense of Kurzweil-

Henstock-Stieltjes.

3.2 Measure differential equations and Generalized ODEs

In this section, we present a correspondence inspired in a result found in [38, Theorem

5.17]. Such correspondence describes the relation between the solutions of a measure differ-

ential equation and the solutions of a generalized ODE. This correspondence is crucial to

prove our main results.

We will show that, under certain assumptions, we can establish a correspondence between

the solutions of a integral form of a measure differential equation of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0,

and the solutions of the generalized ODE

dx

dτ= DF (x, t),

where F is given by F (x, t) =∫ tτ0f(x, s)dg(s), for x ∈ O and t ∈ [t0,+∞).

We recall the reader that G([t0,+∞),Rn) denotes the vector space of functions x :

[t0,+∞)→ Rn such that x|[α,β] belongs to the space G([α, β],Rn), for all [α, β] ⊂ [t0,+∞).

We use the symbolG0([t0,+∞),Rn) to denote the vector space of all functions x ∈ G([t0,+∞),Rn)

such that sups∈[t0,+∞)

e−(s−t0) |x(s)| < +∞. This space is endowed with the norm

‖x‖[t0,+∞) = sups∈[t0,+∞)

e−(s−t0) |x(s)| , x ∈ G0([t0,+∞),Rn),

3.2 Measure differential equations and Generalized ODEs 27

and it becomes a Banach space (see [28]).

We will use the notation x ∈ G([t0,+∞), O) for a function x ∈ G([t0,+∞),Rn) such that

x(s) ∈ O, for all s ∈ [t0,+∞). The notation x ∈ G0([t0,+∞), O) is defined in a similar way.

In what follows, we say that γ : [t0,+∞) → R+ is locally Kurzweil-Henstock-Stieltjes

integrable with respect to g if, and only if, the function [s1, s2] 3 t 7→ γ(t) is Kurzweil-

Henstock-Stieltjes integrable with respect to g on [s1, s2], for every s1, s2 ∈ [t0,+∞).

Throughout this section, let us assume the following conditions on the functions f :

O × [t0,+∞)→ Rn and g : [t0,+∞)→ R.

(A1) The function g : [t0,+∞)→ R is nondecreasing and left-continuous on (t0,+∞).

(A2) The Kurzweil-Henstock-Stieltjes integral∫ s2

s1

f(x(s), s)dg(s)

exists, for all x ∈ G([t0,+∞), O) and all s1, s2 ∈ [t0,+∞).

(A3) There exists a Kurzweil-Henstock-Stieltjes integrable function M : [t0,+∞)→ R+ with

respect to g such that ∣∣∣∣∣∣s2∫s1

f(x(s), s)dg(s)

∣∣∣∣∣∣ ≤s2∫s1

M(s)dg(s),

for all x ∈ G([t0,+∞), O) and all s1, s2 ∈ [t0,+∞), s1 ≤ s2.

(A4) There exists a Kurzweil-Henstock-Stieltjes integrable function L : [t0,+∞)→ R+ with

respect to g such that∣∣∣∣∣∣s2∫s1

[f(x(s), s)− f(z(s), s)]dg(s)

∣∣∣∣∣∣ ≤ ‖x− z‖[t0,+∞)

s2∫s1

L(s)dg(s),

for all x, z ∈ G0([t0,+∞), O) and all s1, s2 ∈ [t0,+∞), s1 ≤ s2.

The next result ensures that if the function f : O× [t0,+∞)→ Rn satisfies the conditions

(A2), (A3) and (A4), and g : [t0,+∞)→ R satisfies condition (A1), then the function F given

by F (x, t) =∫ tτ0f(x, s)dg(s), for (x, t) ∈ O×[t0,+∞) belongs to the class F(O×[t0,+∞), h),

where h(t) =∫ tτ0

(M(s) + L(s))dg(s), t ∈ [t0,+∞).

Theorem 3.2. Assume f : O × [t0,+∞) → Rn satisfies conditions (A2), (A3) and (A4),

and g : [t0,+∞)→ R satisfies condition (A1). Choose an arbitrary τ0 ∈ [t0,+∞) and define


F : O × [t0,+∞)→ Rn by

F (x, t) =

∫ t

τ0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (3.2)

Then F ∈ F(Ω, h), where Ω = O × [t0,+∞), and h : [t0,+∞)→ R given by

h(t) =

∫ t

τ0

(M(s) + L(s))dg(s), t ∈ [t0,+∞) (3.3)

is a nondecreasing function.

Proof. At first, we prove that F is well-defined. Indeed, let t ∈ [t0,+∞) and x ∈ O. Define

the following function

cx : [t0,+∞)→ O (3.4)

s 7→ cx(s) = x.

Note that cx ∈ G0([t0,+∞), O) (in particular, x ∈ G([t0,+∞), O)). Thus, condition (A2)

implies

∫ t

τ0

f(cx(s), s)dg(s) =

∫ t

τ0

f(x, s)dg(s) exists and, therefore, F is well-defined. On

the other hand, since M and L are Kurzweil-Henstock-Stieltjes integrable functions, h is

well-defined. Also, by the left-continuity of g, h is a left-continuous function.

Now, for an arbitrary x ∈ O and for t0 ≤ s1 ≤ s2 < +∞, by condition (A3), we have

|F (x, s2)− F (x, s1)| =∣∣∣∣∫ s2

τ0

f(x, s)dg(s)−∫ s1

τ0

f(x, s)dg(s)

∣∣∣∣=

∣∣∣∣∫ s2

s1

f(x, s)dg(s)

∣∣∣∣ =

∣∣∣∣∫ t

τ0

f(cx(s), s)dg(s)

∣∣∣∣≤∫ s2

s1

M(s)dg(s) ≤∫ s2

s1

(M(s) + L(s))dg(s) = (h(s2)− h(s1)).

Analogously, condition (A4) implies that if x, y ∈ O and t0 ≤ s1 ≤ s2 < +∞, then

|F (x, s2)− F (x, s1)− F (y, s2) + F (y, s1)|

=

∣∣∣∣∫ s1

s1

[f(x, s)− f(y, s)]dg(s)

∣∣∣∣ =

∣∣∣∣∫ s1

s1

[f(cx(s), s)− f(cy(s), s)]dg(s)

∣∣∣∣≤ ‖cx − cy‖[t0,+∞)

s2∫s1

L(s)dg(s) ≤ |x− y|s2∫s1

L(s)dg(s) = |x− y| (h(s2)− h(s1)).


(Note that ‖cx − cy‖[t0,+∞) = sups∈[t0,+∞)

e−(s−t0) |cx(s)− cy(s)|︸︷︷︸|x−y|

≤ |x− y|).

Remark 3.3. Since g is left-continuous, by relation (3.3), clearly h is left-continuous.

Remark 3.4. Given a compact interval [a, b] ⊂ [t0,+∞) and a function x ∈ G([a, b], O), we

identify the function x with its constant prolongation to the whole interval [t0,+∞), that is,

with the function

x(t) =

x(a), t ∈ [t0, a]

x(t), t ∈ [a, b]

x(b), t ∈ [b,+∞).

(3.5)

It is clear that x ∈ G0([t0,+∞), O) and ‖x‖[t0,+∞) ≤ ‖x‖∞ .

The next result brings an important property of the regulated function on [a, b]. It can

be found in [29].

Theorem 3.5. Any function in G([a, b], X) (in particular, any function of bounded variation

in [a, b]) can be written as the uniform limit of step functions.

Remark 3.6. Notice that by the Theorem 3.5 we can ensure that given x ∈ G([a, b], X)

and ε > 0, there exists a step function ϕ : [a, b] → X such that ‖ϕ(t)− x(t)‖ < ε for every

t ∈ [a, b]. On the other hand, it is important to point out that the result still valid if we

change X by an set O ⊂ X, that is, given x : [a, b] → O a step function, there exist a step

function φ : [a, b] → O such that ‖φ(t)− x(t)‖ < ε, for every t ∈ [a, b]. In particular, given

x : [a, b]→ O a step function, there exist a sequence of finite step functions φk : [a, b]→ O,

k = 1, 2, . . . such that

‖φk − x‖∞ = sups∈[a,b]

|φk(s)− x(s)| k→+∞→ 0.

For more details, see [42].

The next result describes the relation between the Kurzweil-Henstock-Stieltjes integral

and the Kurzweil integral. A similar version of this result was proved in [38, Proposition 5.12],

for the case where f satisfies the usual conditions of Caratheodory. Also, [38, Proposition

5.12] describes the relation between the Lebesgue-Stieltjes integral and the Kurzweil integral.


and g : [t0,+∞) → R satisfies condition (A1). Let F : O × [t0,+∞) → Rn be defined

by (3.2). If [a, b] ⊂ [t0,+∞) and x : [a, b] → O is a regulated function (in particular,

a function of bounded variation), then both the Kurzweil integral∫ baDF (x(τ), t) and the

Kurzweil-Henstock-Stieltjes integral∫ baf(x(s), s)dg(s) exist and have the same value.


Proof. Let F : O × [t0,+∞) → Rn be defined by (3.2). By Theorem 3.2, F ∈ F(Ω, h),

where Ω = O × [t0,+∞), and h is defined by (3.3). Now, suppose x : [a, b] → O is a

regulated function. Then by Proposition 1.18, the Kurzweil integral∫ baDF (x(τ), t) exists.

Also, according to condition (A2),∫ baf(x(s), s)dg(s) exists, where x is defined as in Remark

3.4. On the other hand,∫ baf(x(s), s)dg(s) =

∫ baf(x(s), s)dg(s) and, therefore, the last

integral exists.

We will show that∫ baDF (x(τ), t) =

∫ baf(x(s), s)dg(s).

Assertion 1. For any finite step function ϕ : [a, b]→ O, we have∫ a

b

f(ϕ(s), s)dg(s) =

∫ b

a

DG(ϕ(τ), t).

Indeed, let ϕ : [a, b] → O be a finite step function. Then, ϕ is a regulated function on

[a, b] and, therefore, the Kurzweil integral∫ baDF (ϕ(τ), t) exists and the Kurzweil-Henstock-

Stieltjes integral∫ baf(ϕ(s), s)dg(s) also exists.

Since ϕ is a step function, there exists a division

a = s0 < s1 < . . . < sn = b

and vectors c1, c2, . . . , cn ∈ Rn such that

x(s) = ck for every s ∈ (sk−1, sk), k = 1, 2, . . .m.

It is easy to check that if sk−1 < σ1 < σ2 < sk, then∫ σ2

σ1

DG(ϕ(τ), t) =

∫ σ2

σ1

DF (ck, t) = F (ck, σ2)− F (ck, σ1).

Since

F (ck, σ2)− F (ck, σ1) =

∫ σ2

t0

f(ck, s)dg(s)−∫ σ1

t0

f(ck, s)dg(s)

=

∫ σ2

σ1

f(ck, s)dg(s) =

∫ σ2

σ1

f(ϕ(s), s)dg(s),

we have ∫ σ2

σ1

f(ϕ(s), s)dg(s) = F (ck, σ2)− F (ck, σ1) =

∫ σ2

σ1

DF (ϕ(τ), t).


Let k ∈ 1, 2, . . . , n and suppose σ ∈ (sk−1, sk) is given. By Theorem 1.8, we obtain∫ σ

sk−1

f(ϕ(s), s)dg(s)− f(ϕ(sk−1), sk−1)∆+g(sk−1) = limξ→s+k−1

∫ σ

ξ

f(ϕ(s), s)d(s)

= limξ→s+k−1

F (ck, σ)− F (ck, ξ)

= F (ck, σ)− F (ck, s+k−1),

that is,∫ σ

sk−1

f(ϕ(s), s)dg(s) = F (ck, σ)− F (ck, s+k−1) + f(ϕ(sk−1), sk−1)∆+g(sk−1). (3.6)

On the other hand, according to Theorem 1.6, we get∫ σ

sk−1

DF (ϕ(τ), t) = limξ→s+k−1

(∫ σ

ξ

DF (ϕ(τ), t) + F (ϕ(sk−1), ξ)− F (ϕ(sk−1), sk−1))

= limξ→s+k−1

(∫ σ

ξ

DF (ck, t) +

∫ ξ

t0

f(ϕ(sk−1), s)dg(s)−∫ sk−1

t0

f(ϕ(sk−1), s)dg(s))

= limξ→s+k−1

(F (ck, σ)− F (ck, ξ) +

∫ ξ

sk−1

f(ϕ(sk−1), s)dg(s))

= F (ck, σ)− F (ck, s+k−1) + f(ϕ(sk−1), sk−1)∆+g(sk−1). (3.7)

Notice that the last equality follows directly by (i) from the Corollary 1.8. Hence, by (3.6)

and (3.7), ∫ σ

sk−1

f(ϕ(s), s)dg(s) =

∫ σ

sk−1

DF (ϕ(τ), t),

for every k ∈ 1, 2, . . . , n and for all σ ∈ (sk−1, sk).

Analogously, we can prove that∫ sk

σ

f(ϕ(s), s)dg(s) =

∫ sk

σ

DF (ϕ(τ), t),

for each k ∈ 1, 2, . . . , n and for each σ ∈ (sk−1, sk). Therefore, we get∫ b

a

f(ϕ(s), s)dg(s) =n∑k=1

∫ sk

sk−1

f(ϕ(s), s)dg(s)


=n∑k=1

∫ sk

sk−1

DF (ϕ(τ), t) =

∫ b

a

DF (ϕ(τ), t),

that is, ∫ b

a

f(ϕ(s), s)dg(s) =

∫ b

a

DF (ϕ(τ), t).

This proves Assertion 1.

Since x : [a, b] → O is a regulated function, by Theorem 3.5 (see Remark 3.6), there

exists a sequence of finite step functions ϕk : [a, b]→ O, k = 1, 2, . . . such that

‖ϕk − x‖∞ = sups∈[a,b]

|ϕk(s)− x(s)| k→+∞→ 0.

By Assertion 1, ∫ b

a

f(ϕk(s), s)dg(s) =

∫ b

a

DF (ϕk(τ), t). (3.8)

Now, using Theorem 1.20, we have

limk→∞

∫ b

a

DF (ϕk(τ), t) =

∫ b

a

DF (x(τ), t). (3.9)

Then, by (3.8) and (3.9), we obtain

limk→∞

∫ b

a

f(ϕk(s), s)dg(s) = limk→∞

∫ b

a

DF (ϕk(τ), t) =

∫ b

a

DF (x(τ), t). (3.10)

On the other hand, condition (A4) and Remark 3.4 imply∥∥∥∥∥∥b∫

a

[f(ϕk(s), s)− f(x(s), s)]dg(s)

∥∥∥∥∥∥ =

∥∥∥∥∥∥b∫

a

[f(ϕk(s), s)− f(x(s), s)]dg(s)

∥∥∥∥∥∥≤ ‖ϕk − x‖[t0,+∞)

b∫a

L(s)dg(s)

≤ ‖ϕk − x‖∞

b∫a

L(s)dg(s)k→+∞→ 0,

that is,

limk→∞

∫ b

a

f(ϕk(s), s)dg(s) =

∫ b

a

f(x(s), s)dg(s). (3.11)


Finally, by (3.10) and (3.11),∫ b

a

DF (x(τ), t) =

∫ b

a

f(x(s), s)dg(s),

obtaining the desired result.

The next result gives us a correspondence between the solutions of the integral form of

an MDE of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (3.12)

and the solutions of the generalized ODE

dx

dτ= DF (x, t),

where F is given by F (x, t) =∫ tτ0f(x, s)dg(s). This result is inspired in [38, Theorem 5.17].

Since this result is crucial to our main results, we repeat its proof here, following the ideas

of [38].


and g : [t0,+∞) → R satisfies condition (A1). Then, the function x : I → Rn is a solution

of the MDE (3.12) on I ⊂ [t0,+∞) if, and only if, x is a solution of the generalized ODE

dx

dτ= DF (x, t)

on I with the function F is given by (3.2).

Proof. Let x : I → Rn be a solution of the MDE (3.12) on I ⊂ [t0,+∞). Then, given

s1, s2 ∈ I, we have

x(t)− x(s1) =

∫ t

s1

f(x(s), s)dg(s), for all t ∈ [s1, s2]. (3.13)

Now, from Theorem 1.8, [s1, s2] 3 t 7→∫ ts1f(x(s), s)dg(s) is a regulated function and, there-

fore, x|[s1,s2] is also a regulated function for any s1, s2 ∈ I.

Let F : O × [t0,+∞) → Rn be defined by (3.2). By Theorem 3.7, given s1, s2 ∈ I, the

integrals∫ s2s1DF (x(τ), t) and

∫ s2s1f(x(s), s)dg(s) exist and∫ s2

s1

DF (x(τ), t) =

∫ s2

s1

f(x(s), s)dg(s).


Thus

x(s2)− x(s1) =

s2∫s1

f(x(s), s)dg(s) =

∫ s2

s1

DF (x(τ), t),

that is,

x(s2)− x(s1) =

∫ s2

s1

DF (x(τ), t).

This shows that x is a solution of the generalized ODE

dx

dτ= DF (x, t).

Conversely, let x : I → Rn be a solution of the generalized ODEdx

dτ= DF (x, t) on I, with

the function F given by (3.2).

Let s1, s2 ∈ I be such that s1 ≤ s2. According to the Theorem 3.2, F ∈ F(Ω, h), where

h is a nondecreasing function. Thus, by Corollary 1.17, x|[s1,s2] is a function of bounded

variation in [s1, s2]. Again, using Theorem 3.7, we have

x(s2)− x(s1) =

∫ s2

s1

DF (x(τ), t) =

∫ s2

s1

f(x(s), s)dg(s),

that is,

x(s2)− x(s1) =

∫ s2

s1

f(x(s), s)dg(s).

This shows that x is a solution of the MDE (3.12), which completes the proof.

We finish this section by presenting an example which illustrates the previous theorem.

Example 3.9. Let R be the 1-dimensional Euclidean space with norm | · | (absolute value).

Consider the MDE in its integral form

x(t) = x(0) +

∫ t

0

f(x(s), s)dg(s), t ≥ 0 (3.14)

where f : R× [0,+∞)→ R and g : [0,+∞)→ R are given by respectively

f(x, t) =t sinx

4t + [|t|],


for all (x, t) ∈ R× [0,+∞) and

g(t) =

t, t ∈ [0, 1]

t+ 1, t ∈ (1,+∞),

where the symbol [|t|] denotes the integer part of t, and the symbol t := t − [|t|] denotes

the fractional part of t.

We will show that f satisfies conditions (A2), (A3), (A4), and g satisfies condition (A1).

Indeed, clearly g satisfies condition (A1). Consider an arbitrary x ∈ G([0,+∞),R) and let

s1, s2 ∈ [0,+∞) be such that s1 ≤ s2. Notice that [s1, s2] 3 t 7→ f(x(t), t) is a regulated

function on [s1, s2]. Thus by Theorem 1.9 (item (i)),∫ s2s1f(x(s), s)dg(s) exists and, therefore,

f satisfies (A2).

Now, define M : [0,+∞)→ R by M(t) = t , for t ∈ [0,+∞). Evidently, M is a locally

Kurzweil-Henstock-Stieltjes integrable with respect to g and

∣∣∣∣∫ s2

s1

f(x(s), s)dg(s)

∣∣∣∣ Theorem 1.9↓≤

∫ s2

s1

|f(x(s), s)| dg(s)

=

∫ s2

s1

∣∣∣∣s sin(x(s))

4s + [|s|]

∣∣∣∣ dg(s)

≤∫ s2

s1

s dg(s) =

∫ s2

s1

M(s)dg(s),

for every x ∈ G([0,+∞),R) and s1, s2 ∈ [0,+∞), s1 ≤ s2. Thus, f satisfies condition (A3).

On the other hand, define L : [0,+∞)→ R by L(t) := M(t) for t ∈ [0,+∞). Then

∣∣∣∣∫ s2

s1

[f(x(s), s)− f(y(s), s)] dg(s)

∣∣∣∣ Theorem 1.9↓≤

∫ s2

s1

|f(x(s), s)− f(y(s), s)| dg(s)

≤∫ s2

s1

s e−s∣∣∣ sin(x(s))− sin(y(s))

∣∣∣dg(s)

≤∫ s2

s1

L(s)e−s |x(s)− y(s)| dg(s)

≤∫ s2

s1

L(s) ‖x− y‖[0,+∞) dg(s)

= ‖x− y‖[0,+∞)

∫ s2

s1

L(s)dg(s),

for all x, y ∈ G0([0,+∞),R) and s1, s2 ∈ [0,+∞), s1 ≤ s2. Hence f and g fulfill all the

hypotheses of Theorem 3.8.


To obtain the corresponding generalized ODE, we choose an arbitrary τ0 ∈ [0,+∞) and

define

F (x, t) :=

∫ t

τ0

f(x, s)dg(s), (3.15)

for (x, t) ∈ R× [0,+∞). Notice that

F (x, t) = ϕ(t) sinx,

where

ϕ(t) =

∫ t

τ0

s4s + [|s|]

dg(s),

for t ∈ [0,+∞).

On the other hand, if x : I → R, where 0 ∈ I ⊂ [0,+∞), is a solution of the MDE

(3.14), by Theorem 3.8, x is also a solution of the generalized ODEdx

dτ= DF (x, t) with the

function F given by (3.15) and, therefore,

x(ξ) = x(0) +

∫ ξ

0

DF (x(τ), t) = x(0) +

∫ ξ

0

D[ϕ(t) sinx(τ)] = x(0) +

∫ ξ

0

sinx(s)dϕ(s),

for all ξ ∈ I.

3.3 Dynamic equation on time scales and measure dif-

ferential equations

In this section, we recall some definitions and concepts concerning dynamic equations on

time scales and prove that these equations are a special case of MDEs.

We recall that, given a time scale T, for each pair of numbers a, b ∈ T, a ≤ b, we define a

closed time scale interval by [a, b]T = [a, b]∩T. The open and half-open intervals are defined

in a similar way.

Let T be a time scale. The next notation is borrowed from [42]. Given a real number

t ≤ supT, define

t∗ = inf s ∈ T : s ≥ t . (3.16)

By this definition, t ≤ t∗, for every real number t ≤ supT and t∗ = t for any t ∈ T.Note that t∗ may differ from σ(t). The following example illustrates this fact.

Example 3.10. Let T = N and t ∈ T. Then

t∗ = inf s ∈ T : s ≥ t = inf t, t+ 1, t+ 2, . . . = t

3.3 Dynamic equation on time scales and measure differential equations 37

aand

σ(t) = inf s ∈ T : s > t = inf t+ 1, t+ 2, t+ 3, . . . = t+ 1.

Moreover, t∗ 6= σ(t).

Coming back to time scales T, since T is a closed set, we have t∗ ∈ T. Now, define the

extension of T by

T∗ =

(−∞, supT], if supT <∞,(−∞,∞), otherwise.

(3.17)

Given a function f : T→ Rn, we define its extension f ∗ : T∗ → Rn by

f ∗(t) = f(t∗), t ∈ T∗.

Analogously, for each function f : B × T→ Rn, B ⊂ Rn, we define f ∗ : B × T∗ → Rn by

f ∗(x, t) = f(x, t∗), x ∈ B, t ∈ T∗.

Note that, while the function f is defined on T, its extension f ∗ is defined on the whole

interval (−∞, supT], whenever supT < +∞, and (−∞,+∞), whenever supT = +∞.

Let a, b ∈ T, a ≤ b, and f : [a, b]T → Rn be a function. Notice that if t ∈ [a, b], then t∗

is well-defined and t∗ ∈ [a, b]T, since a∗ = a ≤ t ≤ b = b∗ ≤ supT. Thus, we conclude that

f ∗ : [a, b]→ Rn given by f ∗(t) = f(t∗), for every t ∈ [a, b], is well-defined.

On the other hand, given a real number t0 ∈ T and let f : [t0,+∞)T → Rn be a function,

it is clear that we may have points t ∈ [t0,+∞) such that t > supT. Therefore, t∗ is not well-

defined. Thus, in order to ensure that the function t 7→ t∗ is well-defined, for t ∈ [t0,+∞),

it is necessary to require that supT = +∞.

In this work, we consider a dynamic equation on time scales of type

x∆(t) = f(x∗, t) (3.18)

where f : O × [t0,+∞)T → Rn, where O ⊂ Rn is an open subset. Also, x∆ denotes the

∆-derivative of x, as defined in Chapter 2. The integral form of the dynamic equation on

time scales (3.18) is given by

x(t)− x(t0) =

∫ t

t0

f(x∗(s), s)∆s, t ≥ t0, (3.19)

where the integral on the right-hand side is the Kurzweil-Henstock ∆-integral.


The next result describes the properties of the extension f ∗, based on the properties of

f . This result can be found in [42, Lemma 4].

Lemma 3.11. If f : T → Rn is a regulated function, then f ∗ : T∗ → Rn is also regulated.

If f is left-continuous on T, then f ∗ is left-continuous on T∗. If f is right-continuous on T,then f ∗ is right-continuous at right-dense points of T∗.

Lemma 3.12. Let T be a time scale such that supT = +∞, and t0 ∈ T. Let g : [t0,+∞)→ Rbe given by g(t) = t∗, for all t ∈ [t0,+∞). Then g satisfies the following conditions

(i) g is a nondecreasing function;

(ii) g is left-continuous on (t0,+∞).

Proof. Item (i) follows immediately from the definition of the function g.

On the other hand, item (ii) is an immediate consequence of Lemma 3.11 by considering

the identity function i : [t0,+∞)T → R given by i(t) = t.

The next result describes a correspondence between Kurzweil-Henstock-Stieltjes integrals

and Kurzweil-Henstock ∆-integrals. This result is very important to establish a correspon-

dence between dynamic equations on time scales and measure differential equations. It can

be found in [21, Theorem 4.2].

Theorem 3.13. Let T be a time scale and f : [a, b]T → Rn be a function. Define g(t) = t∗,

for every t ∈ [a, b]. Then, the Kurzweil-Henstock ∆-integral∫ baf(t)∆t exists if, and only if,

the Kurzweil-Henstock-Stieltjes integral∫ baf ∗(t)dg(t) exists; In this case, both integrals have

the same value.

The following result is a consequence of [21, Lemma 4.4]. We repeat its proof here,

following the ideas of [21].

Theorem 3.14. Let T be a time scale such that supT = +∞, t0 ∈ T and g(t) = t∗ for every

t ∈ [t0,+∞). If f : [t0,+∞) → Rn is such that the Kurzweil-Henstock-Stieltjes integral∫ dcf(t)dg(t) exists for every c, d ∈ [t0,+∞), then∫ d

c

f(t)dg(t) =

∫ d∗

c∗f(t)dg(t),

for every t0 ≤ c < d < ∞, where the integral on the right-hand side is in the sense of

Kurzweil-Henstock-Stieltjes and c∗, d∗ are defined as in (3.16).


Proof. Let c, d ∈ [t0,+∞) be such that c < d. We will show that g is constant on [c, c∗] and

[d, d∗]. Indeed, for any s ∈ [c, c∗] and t ∈ [d, d∗], statement (i) from Lemma 3.12 implies that

g(c) ≤ g(s) ≤ g(c∗) and g(d) ≤ g(t) ≤ g(d∗).

Since g(c) = g(c∗) = c∗ and g(d) = g(d∗) = d∗, we obtain g(s) = c∗, for all s ∈ [c, c∗], and

g(t) = d∗, for all t ∈ [d, d∗].

Now, using the definition of the Kurzweil-Henstock-Stieltjes integral and the fact that g

is constant on [c, c∗] and [d, d∗], we obtain∫ c∗cf(t)dg(t) = 0 and

∫ d∗df(t)dg(t) = 0. Hence∫ d

c

f(t)dg(t) =

∫ c∗

c

f(t)dg(t) +

∫ d

c∗f(t)dg(t) +

∫ d∗

d

f(t)dg(t) =

∫ d∗

c∗f(t)dg(t)

and this completes the proof.

The next result describes the relation between the Kurzweil-Henstock ∆-integral and

the Kurzweil-Henstock-Stieltjes integral, where the function f is defined on unbounded time

scales interval. Such result can be found in [21, Theorem 4.5].

Theorem 3.15. Let T be a time scale such that supT = +∞, t0 ∈ T and f : [t0,+∞)T → Rn

be a function such that the Kurzweil-Henstock ∆-integral∫ b

a

f(s)∆s

exists for every a, b ∈ [t0,+∞)T and a < b. Choose an arbitrary a ∈ [t0,+∞)T and define

F1(t) =

∫ t

a

f(s)∆s, t ∈ [t0,+∞)T,

F2(t) =

∫ t

a

f ∗(s)dg(s), t ∈ [t0,+∞),

where g(s) = s∗, for every s ∈ [t0,+∞). Then F2 = F ∗1 . In particular, F2(t) = F1(t) for all

t ∈ [t0,+∞)T.


Proof. Let t ∈ [t0,+∞). Then, by Theorems 3.13 and 3.14 and the fact a = a∗, we have

F2(t) =

∫ t

a

f ∗(s)dg(s) =

∫ t∗

a∗f ∗(s)dg(s)

=

∫ t∗

a

f ∗(s)dg(s)

=

∫ t∗

a

f(s)∆s

= F1(t∗) = F ∗1 (t)

and the proof is finished.

The next result is essential to prove that dynamic equations on time scales are a special

case of MDEs. A proof of it can be found in [20, Theorem 4.2.].

Theorem 3.16. Let T be a time scale, g : [a, b] → R be defined by g(s) = s∗, for every

s ∈ [a, b], and [a, b] ⊂ T∗. Consider a pair of functions f1, f2 : [a, b]→ Rn such that f1(t) =

f2(t) for every t ∈ [a, b] ∩ T. If the Kurzweil-Henstock-Stieltjes integral∫ baf1(s)dg(s) exists,

then the Kurzweil-Henstock-Stieltjes integral∫ baf2(s)dg(s) also exists and both integrals have

the same value.

Proof. Suppose the Kurzweil-Henstock-Stieltjes integral

∫ b

a

f1(s)dg(s) exists. Given ε > 0,

there is a gauge δ : [a, b]→ R+ such that∥∥∥∥∥∥|D|∑i=1

f1(τi)(g(ti)− g(ti−1))−∫ b

a

f1(s)dg(s)

∥∥∥∥∥∥ < ε,

for every δ-fine tagged division D = ([ti−1, ti], τi) of [a, b]. Let us define a gauge δ : [a, b]→R+ by

δ(t) =

δ(t), t ∈ [a, b] ∩ T,min

δ(t), 1

2inf |t− s| : s ∈ T

, t ∈ [a, b]\T.

Note that δ(t) ≤ δ(t) for any t ∈ [a, b]. Then, evidently, every δ-fine tagged division D of

[a, b] is also δ-fine. Consider an arbitrary δ-fine tagged division D = ([ti−1, ti], τi) of [a, b].

For every i ∈ 1, 2, . . . , |D| , there are two possibilities

either [ti−1, ti] ∩ T = ∅ or τi ∈ T.


In the first case, g is constant on [ti−1, ti]. Consequently, g(ti−1) = g(ti) and, therefore,

f2(τi)(g(ti)− g(ti−1)) = 0 = f2(τi)(g(ti)− g(ti−1)).

In the second case, f1(τi) = f2(τi) and

f1(τi)(g(ti)− g(ti−1)) = f2(τi)(g(ti)− g(ti−1)).

In any case,

f1(τi)(g(ti)− g(ti−1)) = f2(τi)(g(ti)− g(ti−1)), for i = 1, 2, . . . , |D| .

Thus, we have∥∥∥∥∥∥|D|∑i=1

f2(τi)(g(ti)− g(ti−1))−∫ b

a

f1(s)dg(s)

∥∥∥∥∥∥ =

∥∥∥∥∥∥|D|∑i=1

f1(τi)(g(ti)− g(ti−1))−∫ b

a

f1(s)dg(s)

∥∥∥∥∥∥< ε.

Since ε > 0 can be arbitrarily small, we conclude that∫ baf2(s)dg(s) exists and

∫ baf2(s)dg(s) =∫ b

af1(s)dg(s).

The next result establishes a correspondence between the solutions of the dynamic equa-

tion on time scales (3.18) and the solutions of the integral form of an MDE of type

x(t) = x(t0) +

∫ t

t0

f ∗(x, s)dg(s), t ≥ t0, (3.20)

where f : O × [t0,+∞)T → Rn and g : [t0,+∞) → R are given by g(s) = s∗. This result is

a slightly modified version of [21, Theorem 5.2] (which is the special case when the domain

of the solution is a compact interval). The proof from [21] can be carried out without any

changes and we reproduce it here.

Theorem 3.17. Let T be a time scale such that supT = +∞ and [t0,+∞)T be a time scale

interval. Let O ⊂ Rn be an open subset, f : O× [t0,+∞)T → Rn be a function. Assume that

for every x ∈ G([t0,+∞)T,Rn), the function t 7→ f(x(t), t) is Kurzweil-Henstock ∆-integrable

on [s1, s2]T, for every s1, s2 ∈ [t0,+∞)T. Define g : [t0,+∞) → R by g(s) = s∗, for every

s ∈ [t0,+∞). Also, let J ⊂ [t0,+∞) be a nondegenerate interval such that J ∩T is nonempty

and for each t ∈ J, we have t∗ ∈ J ∩ T. If x : J ∩ T → Rn is a solution of the initial value

problem given by x∆(t) = f(x∗, t), t ∈ J ∩ T,x(s0) = x0,

(3.21)


where x0 ∈ O and s0 ∈ J ∩ T, then x∗ : J → Rn is a solution of the initial value problem

y(t) = x0 +

∫ t

s0

f ∗(y(s), s)dg(s) = x0 +

∫ t

s0

f(y(s), s∗)dg(s) (3.22)

Conversely, if y : J → Rn satisfies the initial value problem (3.22), then it must have the

form y = x∗, where x : J ∩ T→ Rn is a solution of the initial value problem (3.21).

Proof. Assume that x : J ∩ T → Rn satisfies the dynamic equation on time scales (3.21).

Then

x(t)− x0 =

∫ t

s0

f(x∗(s), s)∆s, t ∈ J ∩ T.

By Theorem 3.15,

x∗(t)− x0 =

∫ t

s0

f(x∗(s∗), s∗)dg(s), t ∈ J.

Since f(x∗(s∗), s∗) = f(x∗(s), s∗) for all s ∈ [s0, t] ∩ T, by Theorem 3.16, we get

x∗(t)− x0 =

∫ t

s0

f(x∗(s), s∗)dg(s), t ∈ J,

that is, x∗ : J → Rn satisfies the initial value problem (3.22), which proves the first part of

the theorem.

Conversely, suppose y : J → Rn satisfies the initial value problem (3.22). Then

y(t) = x0 +

∫ t

s0

f(y(s), s∗)dg(s), t ∈ J.

Note that g is constant on every interval (α, β], where β ∈ T and α = sup s ∈ T : s < β .Thus, y has the same property and it follows that y = x∗ for some x : J ∩ T → Rn. By

reversing our previous reasoning of proof of the first part of the theorem, we conclude that

x satisfies the dynamic equation on time scale (3.21).

Corollary 3.18. Let T be a time scale such that supT = +∞ and [t0,+∞)T be a time scale

interval. Let O ⊂ Rn be an open subset, f : O× [t0,+∞)T → Rn be a function. Assume that

for every x ∈ G([t0,+∞)T,Rn), the function t 7→ f(x(t), t) is Kurzweil-Henstock ∆-integrable

on [s1, s2]T, for every s1, s2 ∈ [t0,+∞)T. Define g : [t0,+∞) → R by g(s) = s∗, for every

s ∈ [t0,+∞). Also, let J ⊂ [t0,+∞) be a nondegenerate interval such that J ∩T is nonempty

and for each t ∈ J, we have t∗ ∈ J ∩ T. If x : J ∩ T → Rn is a solution of the dynamic

equation on time scales given by

x∆(t) = f(x∗, t), t ∈ J ∩ T. (3.23)


then x∗ : J → Rn is a solution of the MDE

y(t2)− y(t1) =

∫ t2

t1

f ∗(y, s)dg(s), t1, t2 ∈ J. (3.24)

Conversely, if y : J → Rn satisfies the MDE (3.24), then it must have the form y = x∗,

where x : J ∩ T→ Rn is a solution of (3.23).

Proof. Suppose J ∩T is nonempty and x : J ∩T→ Rn is a solution of the dynamic equation

on time scales x∆(t) = f(x∗, t). Since J ∩ T is nonempty, there exists τ0 ∈ R such that

τ0 ∈ J ∩ T. Hence, x : J ∩ T→ Rn is a solution of the initial value problemx∆(t) = f(x∗, t), t ∈ J ∩ T,x(τ0) = z0,

(3.25)

where z0 := x(τ0). Thus, by Theorem 3.17, x∗ : J → Rn is a solution of

y(t) = z0 +

∫ t

τ0

f ∗(y(s), s)dg(s), t ∈ J. (3.26)

Thus, by Remark 3.1, x∗ : J → Rn satisfies

y(t2)− y(t1) =

∫ t2

t1

f ∗(y(s), s)dg(s),

for all t1, t2 ∈ J.

The converse assertion can be proved similarly.

We finish this section by presenting an example which illustrates Theorem 3.16.

Example 3.19. Let R be the 1-dimensional Euclidean space with norm | · | (absolute value)

and let T be a time scale such that supT = +∞ and 0 ∈ T. Consider the dynamic equation

on time scales given by x∆(t) = f(x∗, t), t ∈ [0,+∞)T

x(0) = x0,(3.27)

where x0 ∈ R and f : R × [0,+∞)T → R is defined by f(x, t) =5 cos(2x)

et + t+ 1, for all (x, t) ∈

R× [0,+∞)T.


We will show that f satisfies conditions (B1), (B2) and (B3). Indeed, consider an arbi-

trary y ∈ G([0,+∞)T,R) and let s1, s2 ∈ [0,+∞)T with s1 ≤ s2. Notice that the function

[s1, s2] 3 t 7→ f(y(t∗), t∗) =5 cos(2y(t∗))

et∗ + t∗ + 1=

5 cos(2y∗(t))

eg(t) + g(t) + 1

is regulated in [s1, s2], and g : [0,+∞)→ R is given by g(t) = t∗. Thus, by Theorem 1.9 (item

(i)),∫ s2s1f(y(t∗), t∗)dg(t) exists and, therefore, according to Theorem 3.13,

∫ s2s1f(y(t), t)∆t

exists and∫ s2s1f(y(t∗), t∗)dg(t) =

∫ s2s1f(y(t), t)∆t.

Now, define M : [0,+∞) → R by M(t) = t + 5, for t ∈ [0,+∞). Evidently, M is a

Kurzweil-Henstock-Stieltjes integrable function with respect to g and∣∣∣∣∫ s2

s1

f(y(s), s)∆s

∣∣∣∣ =

∫ s2

s1

f(y(s∗), s∗)dg(s) =

∫ s2

s1

∣∣∣∣ 5 cos(2y∗(s))

eg(s) + g(s) + 1

∣∣∣∣ dg(s)

≤∫ s2

s1

5dg(s) ≤∫ s2

s1

M(s)dg(s),

for every y ∈ G([0,+∞),R) and s1, s2 ∈ [0,+∞)T, s1 ≤ s2. Thus f satisfies condition (B2).

On the other hand, define L : [0,+∞)→ R by L(t) := 10, for t ∈ [0,+∞), then∣∣∣∣∫ s2

s1

[f(y(s), s)− f(w(s), s)] ∆s

∣∣∣∣ =

∫ s2

s1

[f(y(s∗), s∗)− f(w(s∗), s∗)] dg(s)

≤∫ s2

s1

|f(y(s∗), s∗)− f(w(s∗), s∗)| dg(s) ≤∫ s2

s1

5e−g(s)∣∣∣ cos(2y∗(s))− cos(2w∗(s))

∣∣∣dg(s)

≤∫ s2

s1

10e−g(s) |y∗(s)− w∗(s)| dg(s) =

∫ s2

s1

L(s)e−s∗ |y(s∗)− w(s∗)| dg(s),

≤∫ s2

s1

L(s) ‖y − w‖[0,+∞)Tdg(s) = ‖y − w‖[0,+∞)T

∫ s2

s1

L(s)dg(s),

for all y, w ∈ G0([0,+∞)T,R) and s1, s2 ∈ [0,+∞), s1 ≤ s2. Hence, f and g fulfill all the

hypotheses Theorem 3.17.

Then the corresponding measure differential equation in the integral form is given by

x(t) = x0 +

∫ t

0

f ∗(x(s), s)dg(s), t ≥ 0, (3.28)

where f ∗(x, t) = f(x, t∗) =5 cos(2x)

et∗ + t∗ + 1, for all (x, t) ∈ R× [0,+∞).

Now, let J ⊂ [0,+∞) be a nondegenerate interval such that J ∩ T is nonempty and for

each t ∈ J, we have t∗ ∈ J ∩ T. Theorem 3.17 says that if x : J ∩ T→ R is a solution of the


dynamic equation on time scales (3.27). Then the function x∗ : J → R is a solution of the

measure differential equation (3.28).

Conversely, every solution y : J → R of the measure differential equation (3.28) has the

form y = x∗, where x : J ∩T→ R is a solution of the dynamic equation on time sales (3.27).

Chapter

4Prolongation of solutions

In this chapter, we present results on the prolongation of solutions of generalized ODEs,

measure differential equations (MDEs) and dynamic equations on time scales.

It is important to mention that, in Section 4.1, we present some new results in the

theory of generalized ODEs. Furthermore, all the results presented in Sections 4.2 and 4.3

of this chapter are new and either they follow by using the correspondence between MDEs

and generalized ODEs or they follow by the correspondence between MDEs and dynamic

equations on time scales (see Chapter 3). These results are contained in [16].

4.1 Prolongation of the solutions of generalized ODEs

In this section, our goal is to prove a result on the prolongation of solutions of an abstract

generalized ODE.

Let X be a Banach space. Consider O ⊂ X an open set, an interval [t0,+∞) ⊂ R and

Ω = O × [t0,+∞). Consider the generalized ODE

dx

dτ= DF (x, t), (4.1)

where F ∈ F(Ω, h) and the function h : [t0,+∞)→ R is nondecreasing and left-continuous.

The next result ensures the prolongation of a solution of the generalized ODE (4.1). The

proof of the following result is inspired in the proof of [38, Proposition 4.15]. Our result

generalizes the result found in [38, Lemma 4.4]. In fact, [38, Lemma 4.4] can be obtained as

a consequence of our result (see Corollary 4.2 in the sequel).

47

48 Chapter 4 — Prolongation of solutions

Throughout this section, for t0 < β < ϑ < +∞, define Γ∞β,ϑ := [β, ϑ], [β, ϑ), [β,+∞).

Theorem 4.1. Let F ∈ F(Ω, h), where the function h : [t0,+∞)→ R is nondecreasing and

left-continuous. If x : [γ, β) → X and y : I → X, I ∈ Γ∞β,ϑ, are solutions of the generalized

ODE (4.1) on [γ, β) and I, respectively, where t0 ≤ γ < β < ϑ < +∞, and if the limit

limt→β−

x(t) exists and limt→β−

x(t) = y(β), then the function z : [γ, β) ∪ I → X defined by

z(t) =

x(t), t ∈ [γ, β)

y(t), t ∈ I,

is a solution of the generalized ODE (4.1) on [γ, β) ∪ I=

[γ, ϑ], if I = [β, ϑ],

[γ, ϑ), if I = [β, ϑ),

[γ,+∞), if I = [β,+∞).

Proof. Suppose the limit limt→β−

x(t) exists and

limt→β−

x(t) = y(β). (4.2)

Define z : [γ, β) ∪ I → X by

z(t) =

x(t), t ∈ [γ, β),

y(t), t ∈ I.

Notice that the function z is well-defined and it is a regulated function, since by Lemma

1.17, x and y are regulated functions. We will show that z is a solution of the generalized

ODE (4.1) on [γ, β) ∪ I.

Since β is an accumulation point of the set [γ, β), there exists a sequence tnn∈N ⊂ [γ, β)

such that tnn→∞→ β. Therefore, by (4.2), we have

limn→∞

x(tn) = y(β). (4.3)

4.1 Prolongation of the solutions of generalized ODEs 49

Let s1, s2 ∈ [γ, β)∪ I be such that s1 ∈ [γ, β) and s2 ∈ I. Then, for n ∈ N sufficiently large,

we have tn ∈ (s1, β) and

s2∫s1

DF (z(τ), t) =

β∫s1

DF (z(τ), t) +

s2∫β

DF (z(τ), t)

=

∫ tn

s1

DF (x(τ), t) +

∫ β

tn

DF (z(τ), t) +

∫ s2

β

DF (y(τ), t)

= x(tn)− x(s1) +

β∫tn

DF (z(τ), t) + y(s2)− y(β),

that is,s2∫s1

DF (z(τ), t) = x(tn)− y(β) + y(s2)− x(s1) +

β∫tn

DF (z(τ), t).

By the definition of z, we have

s2∫s1

DF (z(τ), t) = x(tn)− y(β) + z(s2)− z(s1) +

β∫tn

DF (z(τ), t). (4.4)

Thus, by Lemma 1.15, we get∥∥∥∥∥∥β∫

tn

DF (z(τ), t)

∥∥∥∥∥∥ ≤ |h(β)− h(tn)| , for sufficiently large n. (4.5)

On the other hand, since h is left-continuous and tnn→∞→ β, tn < β for all n ∈ N,

limn→∞

|h(β)− h(tn)| = 0. (4.6)

Thus, by (4.5) and (4.6), we obtain

limn→∞

β∫tn

DF (z(τ), t) = 0. (4.7)


Finally, applying the limit when n→∞ in (4.4) and using (4.3) and (4.7), we have

s2∫s1

DF (z(τ), t) = z(s2)− z(s1),

for every s1, s2 ∈ [γ, β) ∪ I such that s1 ∈ [γ, β) and s2 ∈ I.

The cases where s1, s2 ∈ [γ, β) or s1, s2 ∈ I follow immediately. Therefore z is a solution

of the generalized ODE (4.1) on [γ, β) ∪ I.

The next result is inspired in [38, Lemma 4.4] for the case I = [β, ϑ]. Here, we present a

different proof of it.

Corollary 4.2. Let F ∈ F(Ω, h), where the function h : [t0,+∞)→ R is nondecreasing and

left-continuous. If x : [γ, β] → X and y : I → X, I ∈ Γ∞β,ϑ, are solutions of the generalized

ODE (4.1) on [γ, β] and I respectively, where t0 ≤ γ < β < ϑ < +∞ and x(β) = y(β), then

the function z : [γ, β] ∪ I → X given by

z(t) =

x(t), t ∈ [γ, β]

y(t), t ∈ I(4.8)

is a solution of the generalized ODE (4.1) on [γ, β] ∪ I.

Proof. Suppose

x(β) = y(β). (4.9)

Since F ∈ F(Ω, h) and h is a left-continuous function on [t0,+∞), by Lemma 1.16, x is

left-continuous on (γ, β]. In particular, we have

limt→β−

x(t) = x(β).

Thus, by (4.9), it follows that

limt→β−

x(t) = y(β).

On the other hand, since x : [γ, β] → X is a solution of the generalized ODE (4.1) on

[γ, β], x|[γ,β) is a solution of the equation (4.1) on [γ, β). Note that all the hypotheses of

Theorem 4.1 are satisfied. Then z defined by (4.8) is a solution of the generalized ODE (4.1)

on [γ, β] ∪ I and the proof is complete.

Now, our goal is to prove that, under certain conditions, the unique solution x : [τ0, τ0 +

∆]→ X of (4.1) satisfying x(τ0) = x0, which is ensured by Theorem 1.21, can be extended


to intervals greater than [τ0, τ0 + ∆], up to a maximal interval, at least while the graph of

the solution does not reach the boundary of Ω.

In the sequel, we recall the concept of maximal solution of a generalized ODE. See [38].

Let F ∈ F(Ω, h) and (x0, τ0) ∈ Ω be such that

x0 + F (x0, τ+0 )− F (x0, τ0) ∈ O,

where we recall the reader that F (x0, τ+0 ) = lim

t→τ+0F (x0, t). Fix (x0, τ0) ∈ Ω and define the

following set

Sτ0,x0 := x : Ix ⊂ [t0,+∞)→ X : Ix is an interval with the left endpoint τ0,

x is a solution of the generalized ODE (4.1), x(τ0) = x0 .

Definition 4.3. Given x : Ix → X and z : Iz → X in Sτ0,x0 , we say that x is smaller than

or equal to z (x z), if and only if, Ix ⊂ Iz and z|Ix = x.

The next result shows that the relation defines a partial order in Sτ0,x0 .

Proposition 4.4. The relation defines a partial order relation in Sτ0,x0 .

Proof. We must show that is reflexive, antisymmetric and transitive.

Reflexivity. Let x : Ix → X in Sτ0,x0 . Then Ix ⊂ Ix and x|Ix ≡ x. Therefore, x x.

Antisymmetry. Let x : Ix → X and z : Iz → X belong to Sτ0,x0 , with x z and z x.

Since x z, we have

Ix ⊂ Iz and z|Ix = x. (4.10)

On the other hand, since z x, we have

Iz ⊂ Ix and x|Iz = z. (4.11)

Thus, by (4.10) and (4.11), we obtain Ix = Iz and x = z.

Transitivity. Let x : Ix → X, z : Iz → X and y : Iy → X belong to Sτ0,x0 such that x z

and z y. Since x z, we have

Ix ⊂ Iz and z|Ix = x. (4.12)

By z y, we obtain

Iz ⊂ Iy and y|Iz = z. (4.13)

Thus, by (4.12) and (4.13), Ix ⊂ Iy and y|Ix = x, that is, x y.


Notice that to be able to investiigate the forward maximal solution, it is necessary to

require

x0 + F (x0, τ+0 )− F (x0, τ0) ∈ O.

Otherwise, it could happen that x(t) /∈ O for t > τ0, contradicting the definition of a solution

of a generalized ODE. Therefore, throughout this chapter, we assume

x+ F (x, t+)− F (x, t) ∈ O, for every x ∈ O and t ∈ [t0,+∞),

that is, we consider

Ω = ΩF :=

(x, t) ∈ Ω : x+ F (x, t+)− F (x, t) ∈ O. (4.14)

This means that there are no points in Ω to which the solution of the generalized ODE (4.1)

can escape from the set O.

Definition 4.5. (Prolongation of solution) Let τ0 ≥ t0 and let x : I → X, I ⊂ [t0,+∞),

be a solution of (4.1) on the interval I, I with left endpoint s0. The solution y : J → X,

J ⊂ [t0,+∞), with left endpoint τ0, of the generalized ODE (4.1) is called a prolongation

to the right of x, if I ⊂ J and x(t) = y(t) for all t ∈ I. If I ( J, then y is called a proper

prolongation of x to the right.

In the next definition, the concept of a maximal solution of a generalized ODE is pre-

sented.

Definition 4.6. (Maximal solution) Let (x0, τ0) ∈ Ω. We say that x : J → X is a maximal

solution of the generalized ODE dx

dτ= DF (x, t),

x(τ0) = x0,(4.15)

if x ∈ Sτ0,x0 and, for every z : I → X in Sτ0,x0 such that x z, we have x = z. In other words,

x is a maximal solution of (4.15), whenever x ∈ Sτ0,x0 , and there is no proper prolongation

of x to the right.

The following result is crucial to prove the existence and uniqueness of a maximal solution

of the generalized ODE (4.15), for the case where X is a Banach space and Ω = O×[t0,+∞),

where O being an open subset of X.

Lemma 4.7. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing and

left-continuous and Ω = ΩF , where ΩF is given by (4.14). Let (x0, τ0) ∈ Ω and consider the

generalized ODE (4.15). If y : Jy → X and z : Jz → X are solutions of the generalized


ODE (4.15), where Jy and Jz are intervals, both with left endpoint τ0, then y(t) = z(t) for

all t ∈ Jy ∩ Jz.

Proof. Let (x0, τ0) ∈ Ω be fixed. Suppose y : Jy → X and z : Jz → X are solutions of the

generalized ODE (4.15), where Jy and Jz are intervals with left endpoint τ0. Then,

y(τ0) = z(τ0) = x0. (4.16)

We will show that y(t) = z(t) for all t ∈ Jy ∩ Jz. Note that Jy ∩ Jz is an interval of the form

[τ0, d], or [τ0, d), d ≤ +∞.

Therefore, we will consider the next two cases.

Case 1. Jy ∩ Jz = [τ0, d]. We define

Λ := t ∈ [τ0, d] : y(s) = z(s), for all s ∈ [τ0, t] .

Note that Λ 6= ∅, since τ0 ∈ Λ. We denote λ = sup Λ. It is clear that λ ≤ d and [τ0, λ) ⊂ Λ.

Therefore,

y(t) = z(t), for all t ∈ [τ0, λ). (4.17)

On the other hand, since F ∈ F(Ω, h), h is a left-continuous function. Also, since by the

hypotheses, the functions y and z are solutions of the generalized ODE (4.15) on Jy ∩ Jz, it

follows by Lemma 1.16 that y and z are left-continuous on (τ0, λ]. Then we obtain

y(λ) = limt→λ−

y(t) = limt→λ−

z(t) = z(λ),

that is,

y(λ) = z(λ). (4.18)

Thus, by (4.17) and (4.18), we get

λ ∈ Λ and [τ0, λ] ⊂ Λ. (4.19)

Finally, we will show that λ = d. If λ < d. Then since (y(λ), λ) ∈ ΩF = Ω, by Theorem 1.21,

there are δ > 0 (we can take, for instance, λ+δ < d) and a unique solution x : [λ, λ+δ]→ X

of the generalized ODE dx

dτ= DF (x, t)

x(λ) = y(λ) = z(λ).(4.20)


On the other hand, y∣∣[λ,λ+δ] and z

∣∣[λ,λ+δ] are solutions of (4.20). Then, by the uniqueness,

x(t) = z(t) = y(t), for all t ∈ [λ, λ+ δ]. (4.21)

Thus, by (4.19) and (4.21), λ+ δ ∈ Λ which contradicts the definition of λ. Then d = λ and

y(t) = z(t), for all t ∈ Jy ∩ Jz.

Case 2. Jy ∩ Jz = [τ0, d), with d ≤ +∞.

In order to prove Case 2, we will prove two assertions.

Assertion 1. For every τ ∈ (τ0, d), y(s) = z(s), for all s ∈ [τ0, τ).

Define

Λ := t ∈ (τ0, d) : y(s) = z(s), for all s ∈ [τ0, t) .

Let us prove that Λ 6= ∅. Since (x0, τ0) ∈ Ω = ΩF , Theorem 1.21 ensures the existence of

η > 0 (we can take, for instance, τ0 + η < d) and a unique solution x : [τ0, τ0 + η] → X of

the generalized ODE dx

dτ= DF (x, t)

x(τ0) = x0 = y(τ0) = z(τ0).(4.22)

On the other hand, since y∣∣[τ0,τ0+η] and z

∣∣[τ0,τ0+η] are solutions of (4.22). Then,

x(t) = y(t) = z(t), for all t ∈ [τ0, τ0 + η]. (4.23)

In particular, y(t) = z(t) for all t ∈ [τ0, τ0 + η), that is, τ0 + η ∈ Λ. Thus Λ is a nonempty

set. Denote λ = sup Λ. Note that λ ≤ d and (τ0, λ) ⊂ Λ. Let us prove that λ = d. Suppose

the contrary, that is, λ < d. We will show that y(t) = z(t), for t ∈ [τ0, λ) and y(λ) = z(λ).

Indeed, let t ∈ [τ0, λ). If t = τ0, follows immediately by (4.16). On the other hand, if

t ∈ (τ0, λ), then t ∈ Λ (since (τ0, λ) ⊂ Λ). Thus, y(s) = z(s), for all s ∈ [τ0, t). Since y∣∣(τ0,t]

and z∣∣(τ0,t] are left-continuous functions, we get

y(t) = lims→t−

y(s) = lims→t−

z(s) = z(t).

Therefore

y(t) = z(t), for all t ∈ [τ0, λ). (4.24)

Thus, by (4.24) and by the left continuity of the functions y|(τ0,λ] and z|(τ0,λ], we have

y(λ) = z(λ). (4.25)


Then, by (4.24) and (4.25), we obtain

y(t) = z(t), for all t ∈ [τ0, λ]. (4.26)

Since (y(λ), λ) ∈ ΩF = Ω, Theorem 1.21 implies there are δ > 0 (we can take, for instance,

λ+ δ < d) and a unique solution x : [λ, λ+ δ]→ X of the generalized ODEdx

dτ= DF (x, t)

x(λ) = y(λ) = z(λ).(4.27)

On the other hand, since z∣∣[λ,λ+δ] and y

∣∣[λ,λ+δ] are solutions of (4.27), the uniqueness of a

solution yields

x(t) = z(t) = y(t), for all t ∈ [λ, λ+ δ]. (4.28)

Thus, by (4.26) and (4.28), we have λ + δ ∈ Λ, which contradicts the definition of λ. Then

d = λ and Λ = (τ0, d).

Assertion 2. For all t ∈ (τ0, d), y(t) = z(t).

Indeed, let t ∈ (τ0, d). Then, from Assertion 1, y(s) = z(s), for all s ∈ [τ0, t). Thus, by the

left-continuity of the functions y|(τ0,λ] and z|(τ0,λ], we have

y(t) = lims→t−

y(s) = lims→t−

z(s) = z(t),

and Assertion 2 follows.

Finally, by (4.16) and Assertion 2, we get

y(t) = z(t), for all t ∈ [τ0, d) = Jy ∩ Jz,

proving the result.

The next result gives sufficient conditions to ensure the existence and uniqueness of a

maximal solution of the generalized ODE (4.1) with initial condition x(τ0) = x0, that is,dx

dτ= DF (x, t)

x(τ0) = x0.

A version of this result was proved in [38, Proposition 4.13], considering the case X = Rn

and Ω = O × (a, b), where O = B(0, c) ⊂ Rn and (a, b) ⊂ R. Here, we present a proof for

the case where X is a Banach space and Ω = O× [t0,+∞), where O is an open subset of X.

We recall the reader that ΩF = (x, t) ∈ Ω : x+ F (x, t+)− F (x, t) ∈ O .


Theorem 4.8. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing

and left-continuous. If Ω = ΩF , then, for every (x0, τ0) ∈ Ω, there exists a unique maximal

solution x : J → X of (4.1), where x(τ0) = x0 and J is an interval with left endpoint τ0.

Proof. Suppose Ω = ΩF . Let (x0, τ0) ∈ Ω be fixed. At first, we will show the existence of a

maximal solution.

Existence. Consider the set

S := x : Jx ⊂ [t0,+∞)→ X : Jx is an interval with left endpoint τ0 and

x is a solution of the generalized ODE (4.1), x(τ0) = x0

The set S is nonempty by the local uniqueness and existence of a solution given by Theorem

1.21.

Define J :=⋃y∈S

Jy and x : J → X by the relation x(t) = y(t), where y ∈ S and t ∈ Jy.

Note that if y and z belongs to S, then y(s) = z(s), for all s ∈ Jy ∩Jz, by Lemma 4.7. Thus,

we conclude that x is well-defined. Note that J is an interval with the left endpoint τ0 (since

J is union connected with a common point) and x is a maximal solution of the generalized

ODE (4.1), proving the existence of a maximal solution.

It remains to ensure the uniqueness of a maximal solution.

Uniqueness. Assume that x1 : J1 → X and x2 : J2 → X are two maximal solutions of the

generalized ODE (4.1) with x1(τ0) = x2(τ0) = x and J1, J2 are intervals with left endpoint

τ0. Thus, by Lemma 4.7, we have to show that x1(t) = x2(t), for all t ∈ J1 ∩ J2.

Define x3 : J3 → X, J3 := J1 ∪ J2 by

x3(t) =

x1(t), t ∈ J1

x2(t), t ∈ J2.

It is clear that x3 is a solution of the generalized ODE (4.1) with initial condition x3(τ0) = x0,

J1, J2 ⊂ J3 and x3|J1 = x1, x3|J2 = x2. Since the solutions x1 and x2 are assumed to be

maximal, J3 = J2 = J1 and x3(t) = x2(t) = x1(t), for all t ∈ J3, that is, x1 = x2.

The following theorem shows that the maximal interval J of definition of a maximal

solution is half-open. This result was proved in [38, Proposition 4.14], for the case X = Rn

and Ω = O × (a, b), where O = B(0, c) ⊂ Rn and (a, b) ⊂ R. Here, we generalize the result

for the case where X is a Banach space and Ω = O × [t0,+∞), exhibiting a different proof

from the one presented in [38].


Theorem 4.9. Let F ∈ F(Ω, h), where the function h : [t0,+∞) is nondecreasing and left-

continuous, and Ω = ΩF . Suppose (x0, τ0) ∈ Ω and x : J → X is the maximal solution of the

generalized ODE (4.1) with x(τ0) = x0 and J is an interval with the left endpoint τ0. Then

J = [τ0, ω) with ω ≤ +∞.

Proof. It is clear that the maximal interval J satisfies J ⊂ [t0,+∞). Define ω := sup J. Then

clearly ω ≤ +∞. If ω = +∞, the result follows immediately. Assume that ω < +∞. Let us

prove that ω /∈ J. Suppose the contrary, that is, ω ∈ J. Then J = [τ0, ω] and by the definition

of a solution, that (x(ω), ω) ∈ Ω = ΩF . Therefore, by Theorem 1.21, there exists η > 0 such

that, on [ω, ω+η], there is a unique solution z : [ω, ω+η]→ X of the generalized ODE (4.1)

such that z(ω) = x(ω). Thus, by Corollary 4.2,

y(t):=

x(t), t ∈ J = [τ0, ω]

z(t), t ∈ [ω, ω + η]

is a solution of the generalized ODE (4.1) with y(τ0) = x0. It is easy to see that z is a proper

prolongation of x, which is assumed to be maximal. Therefore we have a contradiction.

Hence ω /∈ J and J = [τ0, ω), obtaining the desired result.

The following result is a generalization of [38, Proposition 4.15], for functions F taking

values in a Banach space X. The proof of this result follows the same ideas of [38].

Theorem 4.10. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing

and left-continuous, and Ω = ΩF . Suppose (x0, τ0) ∈ Ω and x : [τ0,+∞) → X is the

maximal solution of (4.1) with x(τ0) = x0. Then, for every compact set K ⊂ Ω, there exists

tK ∈ [τ0, ω) such that (x(t), t) /∈ K, for all t ∈ (tK , ω).

Proof. Suppose the contrary. Then, we will obtain the existence of a compact set K ⊂ Ω

and a sequence tnn∈N ⊂ [τ0, ω) such that

tnn→+∞→ ω and (x(tn), tn) ∈ K, for all n ∈ N.

Let us consider two cases: ω = +∞ and ω < +∞.

Case 1. Suppose ω = +∞. Since K is compact, the sequence (x(tn), tn)n∈N contains a

convergent subsequence which, without loss of generality, we denote again by (x(tn), tn)n∈N.

Then there exists (y, τ) ∈ K such that

limn→∞

(x(tn), tn) = (y, τ).

In particular,

tnn→+∞→ τ,


which contradicts tnn→+∞→ ω = +∞.

Case 2. Suppose ω < +∞. Since K is compact, the sequence (x(tn), tn)n∈N contains a

convergent subsequence which, without loss of generality, we denote again by (x(tn), tn)n∈N.

Then there exists (y, τ) ∈ K ⊂ Ω such that

limn→∞

(x(tn), tn) = (y, τ).

In particular,

tnn→+∞→ τ.

By the uniqueness of the limit, τ = ω. Thus, since (y, ω) = (y, τ) ∈ Ω = ΩF , we have

y + F (y, ω+)− F (y, ω) ∈ O.

Therefore, by Theorem 1.21, there exists a η > 0 such that on [ω, ω + η], there is a unique

solution z : [ω, ω + η]→ X of the generalized ODE (4.1) satisfying z(ω) = y.

Define u : [τ0, ω + η]→ X by the relation

u(t) =

x(t), t ∈ [τ0, ω),

z(t), t ∈ [ω, ω + η].

Thus, by Theorem 4.1, u : [τ0, ω + η] → X is a solution of the generalized ODE (4.1) with

initial condition u(τ0) = x0. But notice that u is clearly a proper prolongation of the solution

x, which is assumed to be maximal. This contradiction proves the result.

The following results are completely new in the theory of generalized ODEs, even for the

case X = Rn.

Corollary 4.11. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing

and left-continuous, and Ω = ΩF . Let (x0, τ0) ∈ Ω and x : [τ0, ω) → X be the maximal

solution of (4.1) with x(τ0) = x0. If x(t) ∈ N for all t ∈ [τ0, ω), where N is a compact subset

of O, then ω = +∞.

Proof. Suppose the contrary, that is, ω < +∞. Then K := N × [τ0, ω] is a compact subset

of Ω and by the hypotheses, we get

(x(t), t) ∈ K, for all t ∈ [τ0, ω). (4.29)

Since all the hypotheses from Theorem 4.10 are satisfied, there exists tK ∈ [τ0, ω) such that

(x(t), t) /∈ K, for all t ∈ (tK , ω),


which contradicts (4.29).

Corollary 4.12. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing

and left-continuous, and Ω = ΩF . Let (x0, τ0) ∈ Ω and x : [τ0, ω) → X be the maximal

solution of (4.1) with x(τ0) = x0. If ω < +∞, then the following conditions hold

(i) The limit limt→ω−

x(t) exists;

(ii) (y, ω) ∈ ∂Ω and (x(t), t)t→ω−→ (y, ω), where y := lim

t→ω−x(t).

Proof. Suppose ω < +∞. Let ε > 0. Since ω ∈ (τ0,+∞) and h is left-continuous on

(τ0,+∞), the limit limt→ω−

h(t) exists. Thus, by the Cauchy condition, there exists δ > 0 (we

can take τ0 < ω − δ) such that

|h(t)− h(s)| < ε, for all t, s ∈ (ω − δ, ω). (4.30)

Then, by (4.30) and Lemma 1.16, the following inequality holds

‖x(t)− x(s)‖ ≤ |h(t)− h(s)| < ε,

for every t, s ∈ (ω − δ, ω). Then again by the Cauchy condition, the limit limt→ω−

x(t) exists,

that is, there exists y ∈ X such that

y = limt→ω−

x(t) (4.31)

and it proves item (i).

Now, we will prove (ii). Note that

(x(t), t)t→ω−→ (y, ω).

Since ω is an accumulation point of the set [τ0, ω), there exists a sequence tnn∈N ⊂ [τ0, ω)

such that tnn→∞→ ω. Thus, by (4.31), we have

x(tn)n→∞→ y.

Since (x(tn), tn)k∈N ⊂ Ω and (x(tn), tn)n→+∞→ (y, ω), we obtain

(y, ω) ∈ Ω. (4.32)


Let us prove that (y, ω) /∈ Ω. Suppose the contrary, that is, (y, ω) ∈ Ω = ΩF . Theorem

1.21 yields that there exists ∆ > 0 such that, on [ω, ω + ∆], there is a unique solution

z : [ω, ω + ∆]→ X of the generalized ODE (4.1) with z(ω) = y.

Define u : [t0, ω + ∆]→ X by

u(t) :=

x(t), t ∈ [τ0, ω)

z(t), t ∈ [ω, ω + ∆].

Thus, by Theorem 4.1, u is a solution of the generalized ODE (4.1) with initial condition

u(τ0) = x(τ0) = x0, which is clearly a proper prolongation of the solution x. But x is assumed

to be maximal, hence we have a contradiction. Therefore we conclude that (y, ω) /∈ Ω, that

is, (y, ω) ∈ Ωc, which implies that

(y, ω) ∈ Ωc. (4.33)

By (4.32) and (4.33), we conclude that

(y, ω) ∈ ∂Ω, (4.34)

obtaining the desired result.

Even considering O = X, it is possible to ensure that the maximal solution is defined on

[τ0,+∞), when x(τ0) = x0. This is the content of the next result.

Corollary 4.13. If Ω = X × [t0,+∞) and F ∈ F(Ω, h), where h : [t0,+∞) → R is

nondecreasing and left-continuous, then for every (x0, τ0) ∈ Ω, there exists a unique maximal

solution defined on [τ0,+∞) of the generalized ODE (4.1), with x(τ0) = x0.

Proof. At first, we need to ensure that Ω = ΩF . Let (z0, s0) ∈ Ω. Since h is a nondecreasing

function, the limit lims→s+0

h(s) exists. Therefore, by the Cauchy condition, given ε > 0, there

exists δ > 0 such that if s, t ∈ (s0, s0 + δ), then |h(t)− h(s)| < ε. Then, since F ∈ F(Ω, h),

we have

‖F (z0, t)− F (z0, s)‖ ≤ |h(t)− h(s)| < ε,

for all t, s ∈ (s0, s0 + δ), that is, the limit lims→s+0

F (z0, s) exists and we denote it by F (z0, s+0 ).

Therefore

x0 + F (z0, s+0 )− F (z0, s0) ∈ X,

that is, (z0, s0) ∈ ΩF .

Now, let (x0, τ0) ∈ Ω and x : [τ0, ω)→ X be the maximal solution of the generalized ODE

(4.1) with x(τ0) = x0. Note that the existence of such a solution is guaranteed by Theorems

4.8 and 4.9.

4.2 Prolongation of solutions of measure differential equations 61

Suppose, by contradiction, ω < +∞. Thus, by Corollary 4.12, the limit limt→ω−

x(t) exists.

Let

y := limt→ω−

x(t) ∈ X. (4.35)

Since ω is an accumulation point of the set [τ0, ω), there exists a sequence tnn∈N ⊂ [τ0, ω)

such that tnn→∞→ ω. Therefore, by (4.35), we have

x(tn)n→∞→ y.

It is easy to check that N := x(tn)n∈N∪y is a compact subset of X. Thus N× [τ0, ω] is a

compact subset of Ω. By Theorem 4.10, there exists t ∈ [τ0, ω) such that (x(t), t) /∈ N×[τ0, ω],

for all t ∈ (t, ω), which contradicts the fact that x(tn) ∈ N for all n ∈ N and tnn→∞→ ω.

Therefore ω = +∞ and we obatain the desired result.

4.2 Prolongation of solutions of measure differential equa-

tions

In this section, our goal is to investigate the prolongation of solutions of measure differ-

ential equations.

Let Rn be the n-dimensional Euclidean space with norm ‖ · ‖ , and O ⊂ Rn be an open

set. Consider the integral form of a measure differential equation (MDE, for short) of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (4.36)

where τ0 ≥ t0, f : O × [t0,+∞)→ Rn and g : [t0,+∞)→ R are functions.

We start by presenting a definition of prolongation to the right of a solution x : J → Rn

of the MDE (4.36)

Definition 4.14. (Prolongation to the right) Let τ0 ≥ t0 and x : J → Rn, J ⊂ [t0,+∞), be

a solution of the MDE (4.36) on the interval J with left endpoint τ0. The solution y : J → X,

J ⊂ [t0,+∞) with left endpoint τ0, of the MDE (4.36) is called a prolongation to the right

of x, if J ⊂ J and x(t) = y(t) for all t ∈ J. If J ( J , then y is called a proper prolongation

of x to the right.

In what follows, we present a definition of maximal solution of the MDE (4.36).


Definition 4.15. (Maximal solution) Let (x0, τ0) ∈ O× [t0,+∞). The solution y : I → Rn,

I ⊂ [t0,+∞) and I with the left endpoint τ0, of the MDE

y(t) = x0 +

∫ t

τ0

f(y(s), s)dg(s), t ≥ τ0,

is called maximal, if there is no proper prolongation of y to the right.

In what follows, we present a result which ensures the existence and uniqueness of a

solution of the MDE (4.36).

Theorem 4.16. Suppose f : O × [t0,+∞)→ Rn satisfies conditions (A2), (A3) and (A4),

and g : [t0,+∞) → R satisfies condition (A1). Also, assume that for all (z0, s0) ∈ O ×[t0,+∞), we have z0 + f(z0, s0)∆+g(s0) ∈ O. Then, for every (x0, τ0) ∈ O × [t0,+∞), there

exists a unique maximal solution x : J → Rn of the MDE (4.36) with x(τ0) = x0 and J (an

interval) with left endpoint τ0.

Proof. Let (x0, τ0) ∈ O×[t0,+∞) (arbitrary, but fixed). Define a function F : O×[t0,+∞)→Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (4.37)

Then, by Theorem 3.2 and Remark 3.3 there is a nondecreasing and left-continuous function

h : [t0,+∞)→ R such that F ∈ F(Ω, h), where Ω = O × [t0,+∞) and

h(t) =

∫ t

t0

(L(s) +M(s))dg(s).

On the other hand, for each (z0, s0) ∈ Ω, we have

z0 + F (z0, s+0 )− F (z0, s0) = z0 + lim

s→s+0

s∫t0

f(z0, τ)dg(τ)−s0∫t0

f(z0, τ)dg(τ) (4.38)

= z0 + lims→s+0

s∫s0

f(z0, τ)dg(τ)

= z0 + f(z0, s0)∆+g(s0) ∈ O,

by Theorem 1.8. Thus

z0 + F (z0, s+0 )− F (z0, s0) ∈ O,


that is, Ω = ΩF . Thus all the hypotheses of Theorem 4.8 are satisfied. Therefore, there exists

a unique maximal solution x : J → X of the generalized ODE

dx

dτ= DF (x, t), (4.39)

with x(τ0) = x0, where F is given by (4.37) and J an interval with left endpoint τ0. Thus, by

Theorem 3.8, x : J → Rn is also a solution of the MDE (4.36) with x(s0) = x0. Let us prove

that x (as a solution of (4.36)) does not admit proper extension to the right. Suppose the

contrary, that is, there exists a solution x : J → Rn of the MDE (4.36) with x(τ0) = x0, where

J is an interval with left endpoint τ0, such that x extends x properly. Thus, by Theorem

3.8, x is a solution of the generalized ODE

dx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.37), which contradicts the maximality of x (as a

solution of the generalized ODE (4.39)). Therefore x (as a solution of the MDE (4.36)) does

not admit a proper extension to the right, that is, x is a maximal solution of the MDE (4.36)

with x(τ0) = x0. Again by Theorems 4.8 and 3.8, we have the uniqueness of the maximal

solution x of the MDE (4.36), obtaining the desired result.

Remark 4.17. Notice that by the proof of Theorem 4.16, x : J → Rn is the maximal

solution of the MDE (4.36) with x(τ0) = x0 if, and only if, x : J → Rn is the maximal

solution of the generalized ODEdx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.37).


and g : [t0,+∞) → R satisfies condition (A1). Also, assume that for all (z0, s0) ∈ O ×[t0,+∞), we have z0 +f(z0, s0)∆+g(s0) ∈ O. Suppose (x0, τ0) ∈ O×[t0,+∞) and x : J → Rn

is the maximal solution of (4.36) with x(τ0) = x0 and J an interval with left endpoint τ0.

Then J = [τ0, ω), with ω ≤ +∞.

Proof. Let (x0, τ0) ∈ O × [t0,+∞) and x : J → Rn be the maximal solution of the MDE

(4.36) with x(τ0) = x0 and J be an interval with left endpoint τ0. Note that the existence of

such a solution is guaranteed by Theorem 4.16.

On the other hand, define a function F : O × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (4.40)


Then using the same arguments as in the proof of Theorem 4.16, we can prove that

Ω = ΩF and F ∈ F(Ω, h),

where Ω = O × [t0,+∞) and h : [t0,+∞) → R is a nondecreasing and left-continuous

function. Also, by Remark 4.17, x : J → Rn is the maximal solution of the generalized ODE

dx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.40). Thus, all the hypotheses of Theorem 4.9 are

satisfied. Then J = [τ0, ω), where ω ≤ +∞. This completes the proof.


and g : [t0,+∞) → R satisfies condition (A1). Also, assume that for all (z0, s0) ∈ O ×[t0,+∞), we have z0 + f(z0, s0)∆+g(s0) ∈ O. Suppose (x0, τ0) ∈ O × [t0,+∞) and x :

[τ0, ω) → Rn is the maximal solution of the MDE (4.36) with x(τ0) = x0. Then, for every

compact set K ⊂ O × [t0,+∞), there exists tK ∈ [τ0, ω) such that (x(t), t) /∈ K, for all

t ∈ (tK , ω).

Proof. Let (x0, τ0) ∈ O× [t0,+∞) and x : [τ0, ω)→ Rn be the maximal solution of the MDE

(4.36) with x(τ0) = x0. Also, assume that K is a compact subset of Bc × [t0,+∞).

On the other hand, define a function F : O × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (4.41)

Similarly as in the proof of Theorem 4.16, we can prove that



function. Also, by Remark 4.17, x : [τ0, ω) → Rn is the maximal solution of generalized

ODEdx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.41). Hence all the hypotheses of Theorem 4.10 are

satisfied. Then there exist tK ∈ [τ0, ω) such that (x(t), t) /∈ K, for all t ∈ (tK , ω), obtaining

the desired result.

Corollary 4.20. Suppose f : O × [t0,+∞)→ Rn satisfies conditions (A2), (A3) and (A4),



[τ0, ω)→ Rn is the maximal solution of the MDE (4.36) with x(τ0) = x0. If x(t) ∈ N for all

t ∈ [τ0, ω), where N is a closed subset of O, then ω = +∞.

Proof. Let (x0, τ0) ∈ O× [t0,+∞) and x : [τ0, ω)→ Rn be the maximal solution of the MDE

(4.36) with x(τ0) = x0. Moreover, suppose x(t) ∈ N for all t ∈ [τ0, ω), where N is a closed

subset of O.

Define a function F : O × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (4.42)

Then, using the same arguments as in the proof of Theorem 4.16, we can prove that



function. Also, by Remark 4.17, x : [τ0, ω) → Rn is the maximal solution of generalized

ODEdx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.42).

On the other hand, note that N is compact (since N is closed and bounded in Rn).

Thus, all the hypotheses of Corollary 4.11 are satisfied. Then ω = +∞ and the proof is

complete.

Corollary 4.21. Suppose f : O × [t0,+∞)→ Rn satisfies conditions (A2), (A3) and (A4),


[τ0, ω)→ Rn is the maximal solution of the MDE (4.36) with x(τ0) = x0. If ω < +∞, then

the following conditions hold


x(t) exists;

(ii) (y, ω) ∈ ∂Ω and (x(t), t)t→ω−→ (y, ω), where y := lim

t→ω−x(t).

Proof. Suppose ω < +∞. Let (x0, τ0) ∈ O × [t0,+∞) and x : [τ0, ω) → Rn be the maximal

solution of (4.36) with x(τ0) = x0.

Define a function F : O × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ O × [t0,+∞). (4.43)


Proceding as in the proof of Theorem 4.16, we can prove that



function.

Also, by Remark 4.17 x : [τ0, ω)→ Rn is the maximal solution of generalized ODE

dx

dτ= DF (x, t),

with x(τ0) = x0, where F given by (4.43). Therefore, all the hypotheses of Corollary 4.12

are satisfied. Then the limit y := limt→ω−

x(t) exists, (y, ω) ∈ ∂Ω and (x(t), t)t→ω−→ (y, ω).

Corollary 4.22. Suppose f : Rn × [t0,+∞) → Rn satisfies conditions (A2), (A3) and

(A4) (with O = Rn), and g : [t0,+∞) → R satisfies condition (A1). Then for every

(x0, τ0) ∈ Rn× [t0,+∞), the maximal solution of the MDE (4.36) with x(τ0) = x0, is defined

in [τ0,+∞).

Proof. Let (x0, τ0) ∈ Rn × [t0,+∞). Define a function F : Rn × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (x, t) ∈ Ω. (4.44)

By Theorem 3.2 and Remark 3.3, there exists a nondecreasing and left-continuous function

h : [t0,+∞)→ R such that F ∈ F(Ω, h), where Ω = Rn× [t0,+∞). Thus all the hypotheses

of Corollary 4.13 are satisfied. Then there exist a unique maximal solution x : [τ0,+∞)→ Rn

of the generalized ODEdx

dτ= DF (x, t),

with x(τ0) = x0, where F is given by (4.44). Now, the result follows from Remark 4.17.

4.3 Prolongation of solutions of dynamic equation on

time scales

In this section, our goal is to prove results on prolongation of solutions of dynamic

equations on time scales.

Let T be a time scale and consider the dynamic equation on time scales given by

x∆(t) = f(x∗, t), (4.45)

4.3 Prolongation of solutions of dynamic equation on time scales 67

where f : O × [t0,+∞)T → Rn is a function and O ⊂ Rn is a open set.

Definition 4.23. Let x : IT → Rn, IT ⊂ [t0,+∞)T, be a solution of (4.45) on the interval

IT, with τ0 = min IT. The solution y : JT → Rn, with JT ⊂ [t0,+∞)T and τ0 = min JT, of the

dynamic equation on time scales (4.45) is called a prolongation of x to the right, if IT ⊂ JTand x(t) = y(t) for all t ∈ IT. If IT ( JT, then y is called a proper prolongation of x to the

right.

Definition 4.24. (Maximal solution) Let (x0, τ0) ∈ O×[t0,+∞)T. The solution y : IT → Rn,

with IT ⊂ [t0,+∞)T and τ0 = min IT, of the dynamic equation on time scalesy∆(t) = f(y∗, t)

y(τ0) = x0

is called maximal, if there is no proper prolongation of y to the right.

The symbol G([t0,+∞)T,Rn) denotes the set of all regulated functions x : [t0,+∞)T →Rn. Also, the symbol G0([t0,+∞)T,Rn) denotes the set of all functions x ∈ G([t0,+∞)T,Rn)

such that sups∈[t0,+∞)T

e−(s−t0) |x(s)| is finite. This space is endowed with the norm

‖x‖[t0,+∞)T= sup

s∈[t0,+∞)T

e−(s−t0) |x(s)| , x ∈ G0([t0,+∞)T,Rn),

and it becomes a Banach space.

We use the notation x ∈ G([t0,+∞)T, O) for a function x ∈ G([t0,+∞)T,Rn) such that

x(s) ∈ O, for all s ∈ [t0,+∞)T. The notation x ∈ G0([t0,+∞)T, Bc) is defined in an similar

way.

Remark 4.25. Note that if x ∈ G0([t0,+∞), O), then y := x|[t0,+∞)T ∈ G0([t0,+∞)T, O)

and ‖y‖[t0,+∞)T≤ ‖x‖[t0,+∞) .

From now on, we assume the following conditions on the function f : O×[t0,+∞)T → Rn:

(B1) The Kurzweil-Henstock ∆-integral

∫ t2

t1

f(y(t), t)∆t exists, for all y ∈ G([t0,+∞)T, O)

and all t1, t2 ∈ [t0,+∞)T.

(B2) There exists a Kurzweil-Henstock ∆-integrable function M : [t0,+∞)T → R such that∣∣∣∣∫ t2

t1

f(y(t), t)∆t

∣∣∣∣ ≤ ∫ t2

t1

M(t)∆t,

for all y ∈ G([t0,+∞)T, O) and all t1, t2 ∈ [t0,+∞)T, t1 ≤ t2.


(B3) There exists a Kurzweil-Henstock ∆-integrable function L : [t0,+∞)T → R such that∣∣∣∣∫ t2

t1

[f(y(t), t)− f(w(t), t)]∆t

∣∣∣∣ ≤ ‖y − w‖[t0,+∞)T

∫ t2

t1

L(t)∆t,

for all y, w ∈ G0([t0,+∞)T, O) and all t1, t2 ∈ [t0,+∞)T, t1 ≤ t2.

The next result describes a relation between the conditions on the Kurzweil-Henstock

∆-integrals and those on Kurzweil-Henstock-Stieltjes integrals. Such result is essential to

our purposes. Our proof is inspired in [21], Lemma 5.3.

Theorem 4.26. Let T be a time scale such that supT = +∞ and t0 ∈ T, and let f :

O × [t0,+∞)T → Rn be a function. Define g(t) = t∗ and f ∗(y, t) = f(y, t∗) for every y ∈ Oand t ∈ [t0,+∞).

1. If f : O×[t0,+∞)T → Rn satisfies the condition (B1), then the integral

∫ t2

t1

f ∗(x(t), t)dg(t)

exists, for all x ∈ G([t0,+∞), O) and for all t1, t2 ∈ [t0,+∞).

2. If f : O × [t0,+∞)T → Rn satisfies the condition (B2), then f ∗ : O × [t0,+∞) → Rn

satisfies the following condition∣∣∣∣∫ t2

t1

f ∗(x(t), t)dg(t)

∣∣∣∣ ≤ ∫ t2

t1

M∗(t)dg(t),

for all t1, t2 ∈ [t0,+∞), t1 ≤ t2, and for all x ∈ G([t0,+∞), O), where g(t) = t∗.

3. If f : O × [t0,+∞)T → Rn satisfies the condition (B3), then f ∗ : O × [t0,+∞) → Rn

satisfies the condition∣∣∣∣∫ t2

t1

[f ∗(x(t), t)− f ∗(z(t), t)]dg(t)

∣∣∣∣ ≤ ‖x− z‖[t0,+∞)

∫ t2

t1

L∗(t)dg(t),

for all t1, t2 ∈ [t0,+∞), t1 ≤ t2, and for all x, z ∈ G([t0,+∞), O), where g(t) = t∗.

Proof. Consider an arbitrary x ∈ G([t0,+∞), O). Let t1, t2 ∈ [t0,+∞). Then t∗1, t∗2 ∈

[t0,+∞)T. According to Remark 4.25, y := x|[t0,+∞)T ∈ G([t0,+∞)T, O) and, therefore, by

hypothesis (B1), the Kurzweil-Henstock ∆-integrals

∫ s2

s1

f(y(t), t)∆t =

∫ s2

s1

f(x(t), t)∆t ex-

ists, for every s1, s2 ∈ [t0,+∞)T. In particular,

∫ t∗2

t∗1

f(x(t), t)∆t exists. Then, by Theorems

3.13 and 3.14, we have∫ t∗2

t∗1

f(x(t), t)∆tTheorem 3.13

↓=

∫ t∗2

t∗1

f(x(t∗), t∗)dg(t)Theorem 3.16

↓=

∫ t∗2

t∗1

f(x(t), t∗)dg(t)


Theorem 3.14↓=

∫ t2

t1

f(x(t), t∗)dg(t) =

∫ t2

t1

f ∗(x(t), t)dg(t),

that is, the last integral exists for every t1, t2 ∈ [t0,+∞) as well. This proves 1.

The remaining two statements follow from Theorems 3.13 and 3.14 For 2., we have∣∣∣∣∫ t2

t1

f ∗(x(t), t)dg(t)

∣∣∣∣ Theorem 3.16↓=

∣∣∣∣∫ t2

t1

f(x(t∗), t∗)dg(t)

∣∣∣∣ Theorem 3.14↓=

∣∣∣∣∣∫ t∗2

t∗1

f(x(t∗), t∗)g(t)

∣∣∣∣∣Theorem 3.13

↓=

∣∣∣∣∣∫ t∗2

t∗1

f(x(t), t)∆t

∣∣∣∣∣ =

∣∣∣∣∣∫ t∗2

t∗1

f(y(t), t)∆t

∣∣∣∣∣ ≤∫ t∗2

t∗1

M(t)∆t

=

∫ t∗2

t∗1

M∗(t)dg(t) =

∫ t2

t1

M∗(t)dg(t),

obtaining the desired on concerning the function f ∗.

For 3., we have∣∣∣∣∫ t2

t1

[f ∗(x(t), t)− f ∗(z(t), t)]dg(t)

∣∣∣∣ Theorem 3.16↓=

∣∣∣∣∫ t2

t1

[f(x(t∗), t∗)− f(z(t∗), t∗)]dg(t)

∣∣∣∣Theorem 3.14

↓=

∣∣∣∣∣∫ t∗2

t∗1

[f(x(t∗), t∗)− f(z(t∗), t∗)]dg(t)

∣∣∣∣∣ Theorem 3.13↓=

∣∣∣∣∣∫ t∗2

t∗1

[f(x(t), t)− f(z(t), t)]∆t

∣∣∣∣∣=

∣∣∣∣∣∫ t∗2

t∗1

[f(y(t), t)− f(w(t), t)]∆t

∣∣∣∣∣ ≤ ‖y − w‖[t0,+∞)T

∫ t∗2

t∗1

L(t)∆t

= ‖y − w‖[t0,+∞)T

∫ t∗2

t∗1

L∗(t)dg(t) = ‖y − w‖[t0,+∞)T

∫ t2

t1

L∗(t)dg(t)

Remark 4.25↓≤ ‖x− z‖[t0,+∞)

∫ t2

t1

L∗(t)dg(t),

for every x, z ∈ G0([t0,+∞), O), where y := x|[t0,+∞)T and w := z|[t0,+∞)T .

The next result ensures the existence and uniqueness of a maximal solution of dynamic

equations on time scales.

Theorem 4.27. Let T be a time scale such that supT = +∞ and t0 ∈ T. Suppose f : O ×[t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3). Also, assume that for all (z0, s0) ∈O× [t0,+∞)T, we have z0 + f(z0, s0)µ(s0) ∈ O. Then, for all (x0, τ0) ∈ O× [t0,+∞)T, there

exists a unique maximal solution x : [τ0, ω)T → Rn, ω ≤ +∞ of the dynamic equation on


time scales given by x∆(t) = f(x∗, t).

x(τ0) = x0

(4.46)

Also, if ω < +∞, we get ω ∈ T and ω is left-dense.

Proof. Let (x0, τ0) ∈ O × [t0,+∞)T (arbitrary, but fixed).

Define f ∗ : O × [t0,+∞) → Rn by f ∗(x, t) = f(x, t∗), for all (x, t) ∈ O × [t0,+∞)

and g(t) = t∗, for every t ∈ [t0,+∞). Since f satisfies conditions (B1), (B2) and (B3), by

Theorem 4.26, f ∗ satisfies conditions (A2), (A3) and (A4). Also, if (z0, s0) ∈ O× [t0,+∞)T,

we have

z0 + f ∗(z0, s0)∆+g(s0) = z0 + f(z0, s∗0)(g(s+

0 )− g(s0)) (4.47)

= z0 + f(z0, s∗0)(σ(s∗0)− s∗0)

= z0 + f(z0, s∗0)µ(s∗0)

= z0 + f(z0, s0)µ(s0) ∈ O

that is,

z0 + f ∗(z0, s0)∆+g(s0) ∈ O,

for all (z0, s0) ∈ O × [t0,+∞)T. On the other hand, if (z0, s0) ∈ O × [t0,+∞), but s0 /∈ T,then

z0 + f ∗(z0, s0)∆+g(s0) = z0 + f(z0, s∗0)(g(s+

0 )− g(s0)) (4.48)

= z0 + f(z0, s∗0)(s∗0 − s∗0)

= z0 ∈ O.

It implies that for each (z0, s0) ∈ O × [t0,+∞), we obtain z0 + f ∗(z0, s0)∆+g(s0) ∈ O. Note

that, by Lemma 3.12, g is a nondecreasing and left-continuous function on (t0,+∞). Then

f ∗ and g fulfill all the hypotheses of Theorems 4.16 and 4.18. Thus, there exists a unique

maximal solution y : [τ0, ω)→ Rn, ω ≤ +∞, of the MDE

y(t) = x0 +

∫ t

τ0

f ∗(y(s), s)dg(s), (4.49)

where g(s) = s∗.

Let us consider two cases.

Case 1. Suppose ω = +∞.


By Theorem 3.17, y : [τ0,+∞)→ Rn must have the form

y = x∗, (4.50)

where x : [τ0,+∞)T → Rn is a solution of the dynamic equation on time scales (4.46).

Evidently, x is a maximal solution of the dynamic equation on time scales (4.46).

Case 2. Now, assume ω < +∞.

In order to prove that ω ∈ T and ω is left-dense, we will prove two assertions.

Assertion 1. ω ∈ T.

Suppose the contrary, that is, ω /∈ T. We define

B := s ∈ T : s < ω .

Note that B is nonempty, because τ0 ∈ B. Since ω /∈ T, we have B = T ∩ (−∞, ω] and,

therefore, B is closed subset of R. We denote β := supB. Since B is a closed, we have

β ∈ B. Also, notice that β ≤ ω. But ω /∈ T, hence β < ω.

Note that g is constant on (β, ω] and, therefore,∫ tsf(y(s), s)dg(s) = 0, for all s, t ∈ (β, ω].

Now, let σ ∈ (β, ω) be fixed, and define a function u : [τ0, ω]→ Rn by

u(t) =

y(t), t ∈ [τ0, ω)

y(σ), t = ω.(4.51)

Note that u is well-defined and u|[τ0,ω) = y. We will show that u is a solution of the MDE

(4.49) on [τ0, ω]. Indeed, let s1, s2 ∈ [τ0, ω] be such that s1 ∈ [τ0, ω) and s2 = ω. Then

u(s2)− u(s1) = y(σ)− y(s1)

(4.49)

↓=

∫ σ

s1

f ∗(y(s), s)dg(s)

=

∫ σ

s1

f ∗(y(s), s)dg(s) +

∫ ω

σ

f ∗(y(s), s)dg(s)︸︷︷︸=0

=

∫ ω

s1

f ∗(y(s), s)dg(s)

s2=ω

↓=

∫ s2

s1

f ∗(y(s), s)dg(s)

=

∫ s2

s1

f ∗(u(s), s)dg(s),

that is,

u(s2)− u(s1) =

∫ s2

s1

f ∗(u(s), s)dg(s),


for all s1 s2 ∈ [τ0, ω] such that s1 ∈ [τ0, ω) and s2 = ω.

The case s1, s2 ∈ [τ0, ω) follows immediately by the definition of u. Then u is a solution

of the MDE (4.49) on [τ0, ω]. Note that u is clearly a proper prolongation of y : [τ0, ω)→ Rn

to the right, which contradicts the fact that y is the maximal solution of the MDE (4.49).

Therefore, we conclude that ω ∈ T.

Assertion 2. ω is left-dense, that is, ρ(ω) = ω.

Suppose the contrary, that is, ρ(ω) = sup s ∈ T : s < ω < ω. Thus, g is constant on

(ρ(ω), ω] and, therefore, using the same arguments as in the proof of Assertion 1 (with

β = ρ(ω)), we can prove that there exists a proper prolongation of y : [τ0, ω) → Rn to the

right, which contradicts the fact that y is the maximal solution of the MDE (4.49). Then, ω

is left-dense.

On the other hand, by Theorem 3.17, y : [τ0, ω) → Rn must have the form y = x∗,

where x : [τ0, ω)T → Rn is a solution of the dynamic equation on time scales (4.46). We

assert that x : [τ0, ω)T → Rn is a maximal solution of the dynamic equation on time scales

(4.46). Suppose the contrary, that is, let φ : JT → Rn be a proper prolongation of x :

[τ0, ω)T → Rn to the right. Then, without loss of generality, we can consider JT = [τ0, ω]T.

Since φ : [τ0, ω]T → Rn is a solution of the dynamic equation on time scales (4.46) (because

φ is a proper prolongation of x), by Theorem 3.17, φ∗ : [τ0, ω] → Rn is solution of MDE

(4.49). Also, note that φ∗|[τ0,ω) = y. Thus, φ∗ : [τ0, ω] → Rn is a proper prolongation of

y : [τ0, ω) → Rn, which contradicts the fact that y is the maximal solution of the MDE

(4.49). Thus, x : [τ0, ω)T → Rn is a maximal solution of the dynamic equation on time scales

(4.46).

Finally, we will show the uniqueness of the maximal solution x. Suppose ψ : LT → Rn is

also a maximal solution of the dynamic equation on time scales (4.46).

Statement 1. x(t) = ψ(t), for all t ∈ [τ0, ω)T ∩ LT.

Indeed, by Theorem 3.17, ψ∗ : L→ Rn is a solution of the MDE (4.49). But y : [τ0, ω)→ Rn

is the unique maximal solution of (4.49). Then y(t) = ψ∗(t), for every t ∈ [τ0, ω) ∩ L. In

particular, since

[τ0, ω)T ∩ LT = [τ0, ω) ∩ L ∩ T ⊂ [τ0, ω) ∩ L,

we have

y(t) = ψ∗(t), for all t ∈ [τ0, ω)T ∩ LT,

and, therefore, for t ∈ [τ0, ω)T ∩ LT, we get

x(t) = x(t∗) = x∗(t)

(4.50)

↓= y(t) = ψ∗(t) = ψ(t∗) = ψ(t),

that is, x(t) = ψ(t), for all t ∈ [τ0, ω)T ∩ LT. This ends the proof of Statement 1.


Now, define λ : ET → Rn, E = [τ0, ω) ∪ L, by

λ(t) =

x(t), t ∈ [τ0, ω)T

ψ(t), t ∈ LT.

By Statement 1, λ is well-defined. Notice that λ is a solution of (4.46) and the time scales

intervals [τ0, ω)T and LT are contained in ET and λ|[τ0,ω)T = x, λ|LT = ψ. Since x and ψ are

maximal solutions of (4.46), we have ET = [τ0, ω)T = LT and λ(t) = x(t) = ψ(t), for all

t ∈ ET, that is, x(t) = ψ(t), for all t ∈ ET = [τ0, ω)T = LT, proving the uniqueness of x.

Corollary 4.28. Let T be a time scale such that supT = +∞ and t0 ∈ T. Suppose f :

O × [t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3). Also, assume that for all

(z0, s0) ∈ O × [t0,+∞)T, we have z0 + f(z0, s0)µ(s0) ∈ O. Let (x0, τ0) ∈ O × [t0,+∞)T and

x : [τ0, ω)T → Rn be the maximal solution of the dynamic equation on time scales (4.46)

(which is ensured by Theorem 4.27). If each point of T is left-scattered, then ω = +∞.

Proof. Suppose the contrary, that is, ω < +∞. Thus, by Theorem 4.27, ω ∈ T and ω is left-

dense, which contradicts the fact that each point of T is left-scattered. Then ω = +∞.

Remark 4.29. Notice that in the proof of Theorem 4.27, y = x∗ : [τ0, ω) → Rn is the

maximal solution of the measure differential equation (4.49) if, and only if, x : [τ0, ω)T → Rn

is the maximal solution of the dynamic equation on time scales (4.46), where (x0, τ0) ∈O × [t0,+∞)T.

Theorem 4.30. Let T be a time scale such that supT = +∞ and t0 ∈ T. Suppose f :


(z0, s0) ∈ O × [t0,+∞)T, we have z0 + f(z0, s0)µ(s0) ∈ O. Let (x0, τ0) ∈ O × [t0,+∞)Tand x : [τ0, ω)T → Rn be the maximal solution of the dynamic equation on time scales

(4.46). Then, for every compact set K ⊂ O × [t0,+∞)T, there exists tK ∈ [τ0, ω) such that

(x(t), t) /∈ K, for t ∈ (tK , ω) ∩ T.

Proof. Let (x0, τ0) ∈ O × [t0,+∞)T and x : [τ0, ω)T → Rn be the maximal solution of the

dynamic equation on time scales (4.46). Also, let K ⊂ O × [t0,+∞)T be a compact set. In

particular, K ⊂ O × [t0,+∞).

On the other hand, by Remark 4.29, x∗ : [τ0, ω) → Rn is the maximal solution of the

following integral form of the MDE

y(t) = x0 +

∫ t

τ0

f ∗(y(s), s)dg(s), t ≥ τ0,

where f ∗ : O × [t0,+∞) → Rn is given by f ∗(x, t) = f(x, t∗), for all (x, t) ∈ O × [t0,+∞)

and g(t) = t∗, for all t ∈ [t0,+∞).


Since f satisfies conditions (B1), (B2) and (B3), by Theorem 4.26, f ∗ satisfies conditions

(A2), (A3) and (A4). Also, by Lemma 3.12, g is a nondecreasing and left-continuous function

on (t0,+∞). Note that, using the same arguments as in the proof of Theorem 4.27, we can

prove that

z0 + f ∗(z0, s0)∆+g(s0) ∈ O,

for all (z0, s0) ∈ O × [t0,+∞). Thus, f ∗, g and x∗ fulfill all the hypotheses of Theorem

4.19 and, therefore, there exists tK ∈ [τ0, ω) such that (x∗(t), t) /∈ K, for all t ∈ (tK , ω). In

particular, since (tK , ω) ∩ T ⊂ (tK , ω), we get

(x(t), t) = (x(t∗), t) = (x∗(t), t) /∈ K, for all t ∈ (tK , ω) ∩ T,

that is,

(x(t), t) /∈ K, for all t ∈ (tK , ω) ∩ T.

This completes the proof of the theorem.

Remark 4.31. Note that in the proof of the previous theorem, the set (tK , ω) ∩ T is

nonempty. Indeed, if ω < +∞, according to Theorem 4.27, ω ∈ T and ω is left-dense

and, therefore, (tK , ω) ∩ T 6= ∅. On the other hand, if ω = +∞, then (tK ,+∞) ∩ T 6= ∅,because supT = +∞.



(z0, s0) ∈ O × [t0,+∞)T, we have z0 + f(z0, s0)µ(s0) ∈ O. Suppose (x0, τ0) ∈ O × [t0,+∞)Tand x : [τ0, ω)T → Rn is the unique maximal solution of (4.45) with x(τ0) = x0. If x(t) ∈ Nfor all t ∈ [τ0, ω)T, where N is a closed subset of O, then, ω = +∞.

Proof. Let (x0, τ0) ∈ O× [t0,+∞)T and x : [τ0, ω)T → Rn be the unique maximal solution of

(4.45) with x(τ0) = x0. Note that the existence of such a solution is guaranteed by Theorem

4.27. Also, assume x(t) ∈ N for any t ∈ [τ0, ω)T, where N is a closed subset of O.

On the other hand, by Remark 4.29, the function x∗ : [τ0, ω)→ Rn is the unique maximal

solution of

y(t) = x0 +

∫ t

τ0

f ∗(y(s), s)dg(s), (4.52)

where f ∗ : O × [t0,+∞)T → Rn is given by f ∗(x, t) = f(x, t∗) for all (x, t) ∈ O × [t0,+∞)

and g(t) = t∗ for all t ∈ [t0,+∞).

Since x(t) ∈ N for all t ∈ [τ0,+∞)T, for each s ∈ [τ0, ω), we have

x∗(s) = x(s∗) ∈ N


because s∗ ∈ [τ0, ω)T. Thus

x∗(s) ∈ N, for all s ∈ [τ0, ω).

Now, N is compact (since N is closed and bounded in Rn). Moreover, by Theorem 4.26, f ∗

satisfies conditions (A2), (A3) and (A4) because f satisfies conditions (B1), (B2) and (B3),

and by Lemma 3.12, g is a nondecreasing and left-continuous function on (τ0,+∞). We can

prove, by the same arguments as in the proof of Theorem 4.27, that z0 +f ∗(z0, τ0)∆+g(τ0) ∈O, for all (z0, τ0) ∈ O × [t0,+∞). Thus, f ∗ : O × [t0,+∞) → Rn, g : [t0,+∞) → R and

x∗ : [τ0, ω) → Rn fulfill all the hypotheses of Corollary 4.20. Then, ω = +∞ and the proof

is complete.

Theorem 4.33. Let T be a time scale such that supT = +∞ and t0 ∈ T. Assume that

f : O × [t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3). Also, assume that for all

(z0, s0) ∈ O × [t0,+∞)T, we have z0 + f(z0, s0)µ(s0) ∈ O. Let (x0, τ0) ∈ O × [t0,+∞)T and

x : [τ0, ω)T → Rn be the maximal solution of the dynamic equation on time scales (4.46). If

ω < +∞, then the following conditions hold


x(t) exists;

(ii) (y, ω) ∈ ∂ΩT and (x(t), t)t→ω−→ (y, ω), where y := lim

t→ω−x(t) and ΩT = O × [t0,+∞)T.

Proof. Let (x0, τ0) ∈ O × [t0,+∞)T and x : [τ0, ω)T → Rn be the maximal solution of the

dynamic equation on time scales (4.46). Also, suppose ω < +∞.

Define a functions f ∗ : O × [t0,+∞) → Rn by f ∗(x, t) = f(x, t∗), for all (x, t) ∈ O ×[t0,+∞) and g : [t0,+∞) → R by g(t) = t∗, for all t ∈ [t0,+∞). Similarly as in the proof

of Theorem 4.27, we can prove that f ∗ satisfies conditions (A2), (A3), (A4), and g satisfies

condition (A1). Also, z0 + f ∗(z0, s0)∆+g(s0) ∈ O, for all (z0, s0) ∈ O × [t0,+∞).

On the other hand, by Remark 4.29, x∗ : [τ0, ω) → Rn is the maximal solution of the

integral form of a MDE of type

y(t) = x0 +

∫ t

τ0

f ∗(y(s), s)dg(s), t ≥ τ0.

Since all hypotheses from Theorem 4.21 are satisfied, the limit limt→ω−

x∗(t) exists and the

following conditions hold

(y, ω) ∈ ∂Ω and (x(t), t)t→ω−→ (y, ω), (4.53)

where y := limt→ω−

x(t) and Ω = O × [t0,+∞).


Note that by Theorem 4.27, ω ∈ T and ω is left-dense.

In order to prove item (i), let tkk∈N be a sequence in [τ0, ω)T such that tkn→+∞→ ω.

Clearly tk = t∗k, for all k ∈ N, and, therefore,

x(tk) = x(t∗k) = x∗(tk)k→+∞→ y,

that is, the limit limt→ω−

x(t) exists and

limt→ω−

x(t) = y. (4.54)

This proves (i).

Now, we will prove (ii). By (4.53), (y, ω) ∈ Ωc. But Ωc ⊂ ΩcT, because ΩT ⊂ Ω. Thus

(y, ω) ∈ ΩcT. (4.55)

On the other hand, since ω is left-dense, there exists a sequence skk∈N in [τ0, ω)T such

that skk→+∞→ ω. Thus, by item (4.54), we have

x(sk)k→+∞→ y.

Since (x(sk), sk)k∈N ∈ ΩT and (x(sk), sk)k→+∞→ (y, ω), we obtain

(y, ω) ∈ ΩT. (4.56)

Finally, item (ii) follows from (4.55) and (4.56), obtaining the desired result.


Rn × [t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3) (with O = Rn). Then, for

every (x0, τ0) ∈ Rn × [t0,+∞)T there exists a unique maximal solution of (4.45) defined in

[τ0,+∞)T such that x(τ0) = x0.

Proof. Let (x0, τ0) ∈ Rn × [t0,+∞)T (arbitrary, but fixed). Then (x0, τ0) ∈ Rn × [t0,+∞).

Define f ∗ : Rn× [t0,+∞)→ Rn by f ∗(x, t) = f(x, t∗) for all (x, t) ∈ Rn× [t0,+∞). Since

f satisfies conditions (B1), (B2) and (B3) (with O = Rn), by Theorem 4.26, f ∗ satisfies

conditions (A2), (A3) and (A4) (with O = Rn), and by Lemma 3.12, g is a nondecreasing

and left-continuous function on (t0,+∞). Hence f ∗ and g fulfill all the hypotheses of Corolary

4.22, Then there exists a unique maximal solution y : [τ0,+∞)→ Rn of

y(t) = x0 +

∫ t

τ0

f ∗(y(s), s)dg(s).


By Theorem 3.17, y must have the form y = x∗ : [τ0,+∞)→ Rn, where x : [τ0,+∞)T → Rn

is a solution of (4.45) with x(τ0) = x0. The result follows from Remark 4.29.

Chapter

5Boundedness of solutions

In this chapter, we present results concerning the boundedness of the solutions of gen-

eralized ordinary differential equations (generalized ODEs), measure differential equations

(MDEs) and dynamic equations on time scales.

It worths mentioning that all the results presented in Section 5.1 generalize the results

found in [2], and all the results presented in Sections 5.2 and 5.3 of this chapter improve

the results found in the literature (see, for instance, [4, 15, 36]) and the improvements are

obtained by using the correspondence between the solutions of the mentioned equations (see

Chapter 3). Such results are contained in [17].

5.1 Boundedness of solutions of generalized ODEs

In this section, our goal is to prove some results concerning the boundedness of the

solutions of generalized ODEs using Lyapunov functionals.

Throughout this section, consider X a Banach space with norm ‖·‖ and Ω = X×[t0,+∞)

where t0 ≥ 0. Let F : Ω → X be a function defined for every (x, t) ∈ Ω and taking values

in a Banach space X. Also, suppose F ∈ F(Ω, h), where the function h : [t0,+∞) → R is

left-continuous on (t0,+∞). Under these conditions, consider the following generalized ODE

dz

dτ= DF (z, t) (5.1)

with the initial condition

z(s0) = z0, (5.2)

79

80 Chapter 5 — Boundedness of solutions

where (z0, s0) ∈ Ω.

From now on, we assume that for every (z0, s0) ∈ Ω, there exists a unique (maximal)

solution x : [s0,+∞) → X of the generalized ODE (5.1) with x(s0) = z0. The existence of

such a solution is ensured by Theorem 4.13.

In what follows, for every (z0, s0) ∈ Ω, we denote by x(s, s0, z0) the unique maximal

solution of the generalized ODE (5.1) with x(s0) = z0.

Next, we present the concepts of uniform boundedness for generalized ODEs. The basic

references for this subject are [2] and [43].

Definition 5.1. We say that the generalized ODE (5.1) is

• Uniformly bounded: if for every α > 0, there exists M = M(α) > 0 such that, for

all s0 ∈ [t0,+∞) and all z0 ∈ X, with ‖z0‖ < α, we have

‖x(s, s0, z0)‖ < M, for all s ≥ s0,

where x(s, s0, z0) is the maximal solution of (5.1) with x(s0) = z0.

• Quasi-uniformly ultimately bounded: if there exists B > 0 such that for all α > 0,

there exists T = T (α) > 0, such that for all s0 ∈ [t0,+∞) and for all z0 ∈ X, with ‖z0‖ < α,

we have

‖x(s, s0, z0)‖ < B, for all s ≥ s0 + T,

where x(s, s0, z0) is the maximal solution of (5.1) with x(s0) = z0.

• Uniformly ultimately bounded: if it is uniformly bounded and quasi-uniformly

ultimately bounded.

The following result can be found in [38], Proposition 10.11. It will be essential to our

purposes.

Proposition 5.2. Suppose −∞ < a < b < +∞ and f, g : [a, b] → R are left-continuous

functions on (a, b]. If for every σ ∈ [a, b), there exists δ = δ(σ) > 0 such that for all η ∈ (0, δ),

the following inequality holds

f(σ + η)− f(σ) ≤ g(σ + η)− g(σ),

then

f(s)− f(a) ≤ g(s)− g(a),

for every s ∈ [a, b].

The next auxiliary result will be crucial to prove our main results.

5.1 Boundedness of solutions of generalized ODEs 81

Lemma 5.3. Let F ∈ F(Ω, h), where the function h : [t0,+∞) → R is nondecreasing and

left-continuous. Also, suppose V : [t0,+∞) × X → R is such that for each left-continuous

function z : [α, β] → X on (α, β], the function [α, β] 3 t 7→ V (t, z(t)) is left-continuous on

(α, β]. Moreover, suppose V satisfies the following conditions

(V1) For all functions x, y : [α, β] → X, [α, β] ⊂ [t0,+∞), of bounded variation in [α, β],

the following condition

|V (t, x(t))− V (t, y(t))− V (s, x(s)) + V (s, y(s))|

≤ (h1(t)− h1(s)) supξ∈[α,β]

‖x(ξ)− y(ξ)‖

holds for every α ≤ s < t ≤ β, where h1 : [t0, ,+∞) → R is a nondecreasing and

left-continuous function.

(V2) There exists a function Φ : X → R such that for every solution z : [s0,+∞) → X,

s0 ≥ t0, of (5.1), we have

V (s, z(s))− V (t, z(t)) ≤ (s− t)Φ(z(t)),

for every s0 ≤ t < s < +∞.

If x : [γ, v] → X, t0 ≤ γ < v < ∞, is left-continuous on (γ, v] and of bounded variation in

[γ, v], then

V (v, x(v))−V (γ, x(γ)) ≤ (h1(v)−h1(γ)) sups∈[γ,v]

∥∥∥∥x(s)− x(γ)−∫ s

γ

DF (x(τ), t)

∥∥∥∥+ (v− γ)K,

where K = sup Φ(x(t)) : t ∈ [γ, v].

Proof. Let x : [γ, v] → X, [γ, v] ⊂ [t0,+∞), be a left-continuous function on (γ, v] and of

bounded variation in [γ, v] ⊂ [t0,+∞) and K := sup Φ(x(t)) : t ∈ [γ, v] . If K = +∞, then

clearly the desired inequality is trivially satisfied. Therefore, the statement of the theorem

holds.

Now, let us assume that K < +∞. Note that Proposition 1.18 implies the existence of

the integral∫ vγDF (x(τ), t).

Take σ ∈ [γ, v]. Since (x(σ), σ) ∈ Ω = X × [t0,+∞), by Theorem 4.13, there exists

a unique maximal solution x : [σ,+∞) → X of the generalized ODE (5.1) on [σ,+∞),

satisfying the initial condition x(σ) = x(σ).

Let η1 > 0 be fixed. Then, x|[σ,σ+η1] is also a solution of the generalized ODE (5.1). Thus,

by Corollary 1.17 and Proposition 1.18, the integral∫ σ+η1σ

DF (x(τ), t) exists.


Consider η2 > 0 such that η2 ≤ η1 and σ + η2 ≤ v. Then, the integral∫ σ+η2σ

DF (x(τ), t)

exists and the integral∫ σ+η2σ

D[F (x(τ), t) − F (x(τ), t)] also exists by the property of inte-

grability on subintervals of the Kurzweil integral. Therefore, given ε > 0, there exists a

gauge δ of the interval [σ, σ + η2] corresponding to ε in the definition of the last integral.

We can assume, without loss of generality, that η2 < δ(σ). By hypothesis (V2), there exists

a function Φ : X → R such that

V (s, x(s))− V (t, x(t)) ≤ (s− t)Φ(x(t)),

for every σ ≤ t < s < +∞. In particular for every 0 < η < η2, we have

V (σ + η, x(σ + η))− V (σ, x(σ)) ≤ ηΦ(x(σ)), (5.3)

where s = σ + η and t = σ. By Corollary 1.4, we have∥∥∥∥F (x(σ), s)− F (x(σ), σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥ < ηε

2(h1(σ + η)− h1(σ))(5.4)

and ∥∥∥∥F (x(σ), s)− F (x(σ), σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥ < ηε

2(h1(σ + η)− h1(σ)), (5.5)

for every s ∈ [σ, σ + η]. Notice that

sups∈[σ,σ+η]

∥∥∥∥∫ s

σ

D[F (x(τ), t)− F (x(τ), t)]

∥∥∥∥− sup

s∈[σ,σ+η]

‖F (x(σ), s)− F (x(σ), σ)− F (x(σ), s) + F (x(σ), σ)‖

≤ sups∈[σ,σ+η]

∥∥∥∥∫ s

σ

D[F (x(τ), t)− F (x(τ), t)]

−(F (x(σ), s)− F (x(σ), σ)− F (x(σ), s) + F (x(σ), σ))‖

≤ sups∈[σ,σ+η]

∥∥∥∥F (x(σ), s)− F (x(σ), σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥+ sup

s∈[σ,σ+η]

∥∥∥∥F (x(σ), s)− F (x(σ), σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥ . (5.6)

On the other hand,

sups∈[σ,σ+η]

‖F (x(σ), s)− F (x(σ), σ)− F (x(σ), s) + F (x(σ), σ)‖


≤ ‖x(σ)− x(σ)‖ sups∈[σ,σ+η]

|h(s)− h(σ)| = 0, (5.7)

where the first inequality follows from the fact that F ∈ F(Ω, h) and the second equality

follows since x(σ) = x(σ). Therefore, by (5.4), (5.5), (5.6) and (5.7), we have

sups∈[σ,σ+η]

∥∥∥∥∫ s

σ

D[F (x(τ), t)− F (x(τ), t)]

∥∥∥∥ ≤ ηε

(h1(σ + η)− h1(σ)). (5.8)

Since F ∈ F(Ω, h) and the function h is nondecreasing, by Corollary 1.17, x is of bounded

variation in [γ, v] and, therefore, in [σ, σ + η] ⊂ [γ, v]. Thus, by hypothesis (V1) and by the

relation x(σ) = x(σ), we get

V (σ + η, x(σ + η))− V (σ + η, x(σ + η))

= V (σ + η, x(σ + η))− V (σ + η, x(σ + η))− V (σ, x(σ)) + V (σ, x(σ))

≤ |V (σ + η, x(σ + η))− V (σ + η, x(σ + η))− V (σ, x(σ)) + V (σ, x(σ))|

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

‖x(s)− x(s)‖

= (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

‖x(s)− x(σ) + x(σ)− x(s)‖

= (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

‖x(s)− x(σ)−∫ s

σ

DF (x(τ), t)‖,

which implies

V (σ + η, x(σ + η))− V (σ + η, x(σ + η))

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

‖x(s)− x(σ)−∫ s

σ

DF (x(τ), t)‖. (5.9)

Therefore, by (5.3) and (5.9), we obtain

V (σ + η, x(σ + η))− V (σ, x(σ))

= V (σ + η, x(σ + η))− V (σ + η, x(σ + η)) + V (σ + η, x(σ + η))− V (σ, x(σ))

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

∥∥∥∥x(s)− x(σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥+ ηΦ(x(σ))

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

∥∥∥∥x(s)− x(σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥+ ηK

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

∥∥∥∥x(s)− x(σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥


+(h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

∥∥∥∥∫ s

σ

D[F (x(τ), t)− F (x(τ), t)]

∥∥∥∥+ ηK

≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

∥∥∥∥x(s)− x(σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥+ ηε+ ηK, (5.10)

where the last inequality follows from (5.8). Given s ∈ [γ, v], define

P (s) := x(s)−∫ s

γ

DF (x(τ), t).

Since x is a function of bounded variation in [γ, v] and (x(s), s) ∈ Ω for every s ∈ [γ, v],

then by Proposition 1.18, it follows that the Kurzweil integralv∫γ

DF (x(τ), t) exists and the

function s 7→s∫γ

DF (x(τ), t) is of bounded variation in [γ, v]. Hence, for each s ∈ [γ, v], the

Kurzweil integrals∫γ

DF (x(τ), t) also, exists by the property of integrability on subintervals

of the Kurzweil integral. Then the function P is well-defined and is of bounded variation

in [γ, v]. Moreover, by Lemma 1.15, P is left-continuous on (γ, v], since x and h are left-

continuous on (γ, v].

On the other hand, for s ∈ [γ, v], we have

P (s)− P (σ) = x(s)− x(σ)−∫ s

γ

DF (x(τ), t) +

∫ σ

γ

DF (x(τ), t)

= x(s)− x(σ)−∫ s

σ

DF (x(τ), t). (5.11)

Now, we define the function f : [γ, v]→ R by

f(t) =

(h1(t)− h1(σ)) sup

s∈[γ,t]

‖P (s)− P (σ)‖+ εt+Kt, t ∈ [γ, σ]

(h1(t)− h1(σ)) sups∈[σ,t]

‖P (s)− P (σ)‖+ εt+Kt, t ∈ [σ, v].

Clearly, f is well-defined. Moreover, by the left-continuity of the functions h1 and P, f is

left-continuous on (γ, v]. Also, since x : [γ, v] → X is left-continuous, it follows from the

hypotheses that the function [γ, v] 3 t 7→ V (t, x(t)) is left-continuous on (γ, v].

On the other hand, by (5.10) and (5.11), we have

V (σ + η, x(σ + η))− V (σ, x(σ)) ≤ (h1(σ + η)− h1(σ)) sups∈[σ,σ+η]

‖P (s)− P (σ)‖+ ηε+ ηK

= f(σ + η)− f(σ).


Thus, the functions [γ, v] 3 t 7→ V (t, x(t)) and [γ, v] 3 t 7→ f(t) satisfy all the hypotheses of

Proposition 5.2. Hence

V (v, x(v))− V (γ, x(γ)) ≤ f(v)− f(γ)

= (h1(v)− h1(σ)) sups∈[σ,v]

‖P (s)− P (σ)‖+ εv +Kv

− (h1(γ)− h1(σ)) sups∈[γ,γ]

‖P (s)− P (σ)‖ − εγ −Kγ

= (h1(v)− h1(σ)) sups∈[σ,v]

‖P (s)− P (σ)‖+ εv +Kv

+ (h1(σ)− h1(γ)) sups∈[γ,γ]

‖P (s)− P (σ)‖ − εγ −Kγ

≤ (h1(v)− h1(σ)) sups∈[γ,v]

‖P (s)− P (σ)‖+ εv +Kv

+ (h1(σ)− h1(γ)) sups∈[γ,v]

‖P (s)− P (σ)‖ − εγ −Kγ

= (h1(v)− h1(γ)) sups∈[γ,v]

‖P (s)− P (σ)‖+ ε(v − γ) +K(v − γ)

= (h1(v)− h1(γ)) sups∈[γ,v]

∥∥∥∥x(s)− x(σ)−∫ s

σ

DF (x(τ), t)

∥∥∥∥+ K(v − γ) + ε(v − γ).

Since ε > 0 is arbitrary, the result follows.

In the sequel, we present a result which ensures that the generalized ODE (5.1) is uni-

formly bounded. Our result generalizes the result found in [2].

Theorem 5.4. Let V : [t0,+∞)×X → R be a function such that, for each left-continuous

function z : [α, β] → X on (α, β], the function [α, β] 3 t 7→ V (t, z(t)) is left-continuous on

(α, β]. Moreover, suppose V satisfies the following conditions

(i) There are two monotone increasing functions p, b : R+ → R+ such that p(0) = b(0) =

0,

lims→+∞

b(s) = +∞ (5.12)

and

b(‖z‖) ≤ V (t, z) ≤ p(‖z‖), (5.13)

for every pair (t, z) ∈ [t0,+∞)×X.

(ii) For every solution of type z : [s0,+∞) → X, s0 ≥ t0, of the generalized ODE (5.1),

we have

V (s, z(s))− V (t, z(t)) ≤ 0,

for every s0 ≤ t < s < +∞


Then the generalized ODE (5.1) is uniformly bounded.

Proof. Let α > 0 be fixed. Since p(α) > 0, by (5.12), there exists M = M(α) > 0 such that

p(α) < b(s), for all s ≥M.

In particular, for s = M, we obtain

p(α) < b(M). (5.14)

Now, let s0 ∈ [t0,+∞), z0 ∈ X and x(·) = x(·, s0, z0) : [s0,+∞) → X be the solution of

the generalized ODE (5.1) with initial condition x(s0) = z0, where ‖z0‖ < α. We will show

that

‖x(t)‖ < M, for all s ≥ s0.

Indeed, by hypothesis (ii) and condition (5.13), for each s ≥ s0, we have

V (s, x(s)) ≤ V (s0, x(s0)) = V (s0, z0) (5.15)

≤ p(‖z0‖)≤ p(α)

< b(M),

that is,

V (s, x(s)) < b(M), for all s ≥ s0. (5.16)

Finally, we will show that ‖x(s, s0, z0)‖ = ‖x(s)‖ < M, for all s ≥ s0. Suppose the

contrary, that is, suppose there exists s ∈ [s0,+∞) such that ‖x(s)‖ ≥ M . Then, by

hypothesis (5.13) and using the fact that b is an increasing function, we have

V (s, x(s)) ≥ b(‖x(s)‖) ≥ b(M),

which contradicts (5.16). Therefore ‖x(s)‖ < M, for all s ≥ s0, and the result follows.

The next result establishes conditions which guarantee that the generalized ODE (5.1)

is uniformly ultimately bounded. Our result generalizes a result found in [2].

Theorem 5.5. Let V : [t0,∞) × X → R be a function such that for each left-continuous

function z : [α, β] → X on (α, β], the function [α, β] 3 t 7→ V (t, z(t)) is left-continuous

on (α, β] and satisfies condition (i) from Theorem 5.4. Moreover, suppose V satisfies the

following conditions


(V1) For every x, y : [α, β]→ X, [α, β] ⊂ [t0,+∞), of bounded variation in [α, β], we have

|V (t, x(t)− V (t, y(t))− V (s, x(s)) + V (s, y(s))|

≤ (h1(t)− h1(s)) supξ∈[α,β]

‖x(ξ)− y(ξ)‖ ,

for every α ≤ s < t ≤ β, where h1 : [t0, ,+∞) → R is a nondecreasing and left-

continuous function.

(V2) There exists a continuous function Φ : X −→ R, with Φ(0) = 0 and Φ(x) > 0, x 6= 0,

such that for every solution z : [s0,+∞)→ X, s0 ≥ t0, of (5.1), we have

V (s, z(s))− V (t, z(t)) ≤ (s− t)(− Φ(z(t))

),


Then the generalized ODE (5.1) is uniformly ultimately bounded.

Proof. At first, notice that, by hypothesis (V 2),

V (s, z(s))− V (t, z(t)) ≤ (s− t)(− Φ(z(t))

)≤ 0,

for every solution z : [s0,+∞)→ X, s0 ∈ [t0,+∞), of the generalized ODE (5.1), with s0 ≤t < s < +∞ Hence, all the hypotheses of the Theorem 5.4 are satisfied and, consequently,

the generalized ODE (5.1) is uniformly bounded. It remains to show that equation (5.1) is

quasi-uniformly ultimately bounded.

By the uniform boundedness of the generalized ODE (5.1), there exists B = B(t0 +1) > 0

such that, for every t ∈ [t0,+∞) and for every x ∈ X with ‖x‖ < t0 + 1, we have∥∥x(s, t, x)∥∥ < B, for all s ≥ t, (5.17)

where x(s, t, x) is the maximal solution of the generalized ODE (5.1) with x(t) = x(t, t, x) =

x. Without loss of generality, we can take B ∈ (t0 + 1,+∞) (because otherwise we can take

B > B′ such that B′ ∈ (t0 + 1,+∞)) and we have∥∥x(s, t, x)∥∥ < B, for all s ≥ t.

Let α > 0, s0 ∈ [t0,+∞), z0 ∈ X and x(·) = x(·, s0, z0) : [s0,+∞) → X be the

solution of the generalized ODE (5.1) with initial condition (5.2), where ‖z0‖ < α. Since

(5.1) is uniformly bounded, there exists a positive number M1 = M1(α) (we can take M1 >


max α, t0 + 1) such that

‖x(s, s0, z0)‖ < M1, for all s ≥ s0.

On the other hand, using the same argument as in (5.14) from the proof of Theorem 5.4,

there exists M2 = M2(α) > 0 such that p(α) < b(M2).

Now, let M = M(α) := max M1(α),M2(α). Notice that

‖x(s, s0, z0)‖ < M, for all s ≥ s0. (5.18)

and

p(α) < b(M). (5.19)

Define

N := sup −Φ(z) : t0 + 1 ≤ ‖z‖ < M < 0

and

T (α) := −2b(M)

N> 0.

We want to show that ‖x(s, s0, z0)‖ < B, for all s ≥ s0 + T (α). Suppose the contrary, that

is, there exists s > s0 + T (α) such that

‖x(s, s0, z0)‖ ≥ B > t0 + 1. (5.20)

Assertion 1. The following inequality holds

‖x(s, s0, z0)‖ ≥ t0 + 1, for all s ∈ [s0, s].

Suppose the assertion is false, that is, there exists t ∈ [s0, s] such that∥∥x(t, s0, z0)∥∥ < t0 + 1.

On the other hand, by (5.17) (with x = x(t, s0, z0)) that∥∥x(s, t, x)∥∥ < B, for all s ≥ t. (5.21)

Also, we know that x(s, t, x), s ∈ [t,+∞), is the unique solution of the initial value

problem dz

dτ= DF (z, t)

z(t) = x = x(t, s0, z0).(5.22)


But x(·, s0, z0)|[t,+∞) is also a solution of the generalized ODE (5.22). Therefore, we have

x(s, s0, z0) = x(s, t, x), for all s ∈ [t,+∞).

In particular, since s ∈ [t,+∞), we obtain

x(s, s0, z0) = x(s, t, x). (5.23)

Thus, (5.21) and (5.23) imply that ‖x(s, s0, z0)‖ < B, which contradicts (5.20). Thus,

Assertion 1 follows.

By Assertion 1, we have

‖x(s, s0, z0)‖ ≥ t0 + 1, for all s ∈[s0 +

T (α)

2, s0 + T (α)

]. (5.24)

since [s0 + T (α)2, s0 + T (α)] ⊂ [s0, s].

However, since x(·) := x(·, s0, z0)∣∣∣[s0+

T (α)2,s0+T (α)]

is a solution of the generalized ODE

(5.1) and F ∈ F(Ω, h), where the function h is left-continuous and nondecreasing, then

the function x(·, s0, z0)∣∣∣[s0+

T (α)2,s0+T (α)]

is left-continuous on (s0 + T (α)/2, s0 + T (α)] and of

bounded variation in Iα := [s0 + T (α)2, s0 + T (α)] by Lemma 1.16 and Corollary 1.17. Thus,

by Lemma 5.3, it follows that

V (s0 + T (α), x(s0 + T (α))) ≤ V

(s0 +

T (α)

2, x

(s0 +

T (α)

2

))+

+

(h1(s0 + T (α))− h1

(s0 +

T (α)

2

))sups∈Iα

∥∥∥∥∥∥∥∥x(s)− x(s0 +

T (α)

2

)−

s∫s0+

T (α)2

DF (x(τ, s0, z0), t)

∥∥∥∥∥∥∥∥︸︷︷︸zero

+

+T (α)

2· sup −Φ(x(s)) : s ∈ [s0 + T (α)/2, s0 + T (α)]


which implies

V (s0 + T (α), x(s0 + T (α))) ≤ V

(s0 +

T (α)

2, x

(s0 +

T (α)

2

))+

T (α)

2. sup

−Φ(x(s)) : s ∈

[s0 +

T (α)

2, s0 + T (α)

]≤ V

(s0 +

T (α)

2, x

(s0 +

T (α)

2

))+

T (α)

2. sup−Φ(z) : t0 + 1 ≤ ‖z‖ < M, (5.25)

where the last inequality follows from the relation

t0 + 1 ≤ ‖x(s)‖ = ‖x(s, s0, z0)‖ < M

for all s ∈ [s0 + T (α)2, s0 + T (α)]. Also, by (5.19) and using the same argument as in (5.15)

from the proof of Theorem 5.4, we get

V

(s0 +

T (α)

2, x

(s0 +

T (α)

2

))< b(M).

Therefore, by (5.25), we have

V (s0 + T (α), x(s0 + T (α))) < b(M) +T (α)

2. sup−Φ(z) : t0 + 1 ≤ ‖z‖ < M

= b(M) +T (α)

2.N

= b(M)− 2b(M)

2N.N

= 0,

which implies that

V (s0 + T (α), x(s0 + T (α))) < 0. (5.26)

On the other hand, by condition (5.13) of the present theorem and by (5.24), we have

V (s0 + T (α), x(t0 + T (α))) ≥ b(‖x(s0 + T (α))‖) ≥ b(t0 + 1) > 0,

which contradicts (5.26). Therefore ‖x(s, s0, z0)‖ < B, for all s ≥ s0 + T (α). The the

generalized ODE (5.1) is quasi-uniformly ultimately bounded and the proof is complete.

5.2 Boundedness of solutions of measure differential equations 91

5.2 Boundedness of solutions of measure differential equa-

tions

In this section, our goal is to present results concerning boundedness of solutions of

measure differential equations.

Let Rn be the n-dimensional Euclidean space with norm ‖ · ‖ .

Consider the integral form of a measure differential equation of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (5.27)

where τ0 ≥ t0, f : Rn × [t0,+∞)→ Rn and g : [t0,+∞)→ R.

From now on, we assume that for every (z0, s0) ∈ Rn × [t0,+∞), there exists a unique

(maximal) solution x : [s0,+∞) → Rn of the measure differential equation (5.27) with

x(s0) = z0. The Corollary 4.13, ensures the existence and uniqueness of such a solution.

Also, assume f satisfies conditions (A2), (A3) and (A4), and g satisfies condition (A1)

defined in Chapter 3.

In what follows, for every (z0, s0) ∈ Rn × [t0,+∞), we denote by x(s, s0, z0) the unique

maximal solution of the measure differential equation (5.27) with x(s0) = z0.

Now, we present some concepts related to uniform boundedness for measure differential

equations.

Definition 5.6. We say that the measure differential equation (5.27) is


all s0 ∈ [t0,+∞) and all z0 ∈ Rn, where ‖z0‖ < α, we have

‖x(s, s0, z0)‖ < M, for all s ≥ s0.

• Quasi-uniformly ultimately bounded: if there exists B > 0 such that for every

α > 0, there exists T = T (α) > 0, such that for all s0 ∈ [t0,+∞) and all z0 ∈ Rn, where

‖z0‖ < α, we have

‖x(s, s0, z0)‖ < B, for all s ≥ s0 + T.


ultimately bounded.

The next result ensures that the measure differential equation (5.27) is uniformly bounded.


Theorem 5.7. Assume f : Rn × [t0,+∞) → Rn satisfies conditions (A2), (A3) and (A4)

and the function g : [t0,+∞) → R satisfies condition (A1). Let U : [t0,+∞) × Rn → Rbe a function such that for each left-continuous function z : [α, β] → Rn on (α, β], the

function [α, β] 3 t 7→ U(t, z(t)) is left-continuous on (α, β]. Moreover, suppose U satisfies

the following conditions

(i) There are two monotone increasing functions p, b : R+ → R+ such that p(0) = b(0) = 0,

lims→+∞

b(s) = +∞

and

b(‖z‖) ≤ U(t, z) ≤ p(‖z‖)

for every pair (t, z) ∈ [t0,+∞)× Rn.

(ii) For every solution z : [s0,+∞) → Rn, s0 ≥ t0, of the measure differential equation

(5.27), we have

U(s, z(s))− U(t, z(t)) ≤ 0,


Then the measure differential equation (5.27) is uniformly bounded.

Proof. Define a function F : Rn × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s) (5.28)

for all (x, t) ∈ Rn× [t0,+∞). Since f satisfies conditions (A2), (A3) and (A4) and g satisfies

condition (A1), by Theorem 3.2, F ∈ F(Ω, h), where the function h : [t0,+∞)→ R is given

by

h(t) =

∫ t

t0

(M(s) + L(s))dg(s).

By Remark 3.3, h : [t0,+∞) → R is left-continuous on (t0,+∞). Also, by Theorem 3.8

and the hypotheses of the present theorem, it is not difficult to see that U : [t0,+∞)×Rn → Rsatisfies all the hypotheses from Theorem 5.4. Hence the generalized ODE (5.1) is uniformly

bounded, where F is given by (5.28).

Again, by Theorem 3.8, it follows that the measure differential equation (5.27) is also

uniformly bounded, obtaining the desired result.

Finally, we present the last result of this section. Such result ensures that the measure

differential equation (5.27) is uniformly ultimately bounded.

5.2 Boundedness of solutions of measure differential equations 93

Theorem 5.8. Assume f : Rn × [t0,+∞) → Rn satisfies conditions (A2), (A3) and (A4)

and the function g : [t0,+∞) → R satisfies condition (A1). Let U : [t0,+∞) × Rn → R be

a function such that for each left-continuous function z : [α, β]→ Rn on (α, β], the function

[α, β] 3 t 7→ U(t, z(t)) is left-continuous on (α, β] and satisfies condition (i) from Theorem

5.7. Moreover, suppose U satisfies the following conditions

(U1) For every x, y : [α, β] → Rn, [α, β] ⊂ [t0,+∞), of bounded variation in [α, β], we

have

|U(t, x(t))− U(t, y(t))− U(s, x(s)) + U(s, y(s))| ≤(∫ t

s

K(τ)du(τ)

)supξ∈[α,β]

‖x(ξ)− y(ξ)‖,

for every α ≤ s < t ≤ β, where u : [t0,+∞) → R is a nondecreasing and left-

continuous function and K : [t0,+∞) → R is a locally Kurzweil-Henstock-Stieltjes

integrable function with respect to u.

(U2) There exists a continuous function φ : Rn → R, with φ(0) = 0 and φ(x) > 0, x 6= 0,

such that for every solution z : [s0,+∞) → Rn, s0 ≥ t0, of the measure differential

equation (5.27), we have

U(s, z(s))− U(t, z(t)) ≤ (s− t)(− φ(z(t))

),


Then, the measure differential equation (5.27) is uniformly ultimately bounded.

Proof. Define a function F : Rn × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s), (5.29)

for all (x, t) ∈ Rn× [t0,+∞). Since f satisfies conditions (A2), (A3) and (A4) and g satisfies


by

h(t) =

∫ t

t0

(M(s) + L(s))dg(s).

Notice that the function h : [t0,+∞)→ R is left-continuous by Remark 3.3.

By Theorem 3.8, U satisfies hypothesis (i) from Theorem 5.4. Also, defining the function

h1 : [t0,+∞)→ R by

h1(t) =

∫ t

t0

K(τ)du(τ),


for every t ∈ [t0,+∞), it follows that h1 is a nondecreasing and left-continuous function.

Moreover, by (U1), U satisfies the following condition

|U(t, x(t))− U(t, y(t))− U(s, x(s)) + U(s, y(s))| ≤ (h1(t)− h1(s)) supξ∈[α,β]

‖x(ξ)− y(ξ)‖,

for every α ≤ s < t ≤ β, and for every x, y : [α, β] → Rn, [α, β] ⊂ [t0,+∞), of bounded

variation in [α, β].

Also, by Theorem 3.8 and by hypothesis (U2), it is clear that U satisfies the hypothesis

(V 2) from Theorem 5.5. Therefore, all the hypotheses from Theorem 5.5 are fulfilled and

the generalized ODE (5.1) is uniformly ultimately bounded. By Theorem 3.8, the measure

differential equation (5.27) is also uniformly ultimately bounded, obtaining the desired result.

5.3 Boundedness of solutions of dynamic equations on

time scales

In this section, our goal is to prove the results concerning boundedness of solutions of

dynamic equations on time scales.

Consider the dynamic equation on time scales given by

x∆(t) = f(x∗, t), (5.30)

where f : Rn × [t0,+∞)→ Rn satisfies the following conditions

(B1) The Kurzweil-Henstock ∆-integral

∫ t2

t1

f(y(t), t)∆t exists, for all y ∈ G([t0,+∞)T,Rn)

and all t1, t2 ∈ [t0,+∞)T.

(B2) There exists a locally Kurzweil-Henstock ∆-integrable function M : [t0,+∞)T → Rsuch that ∣∣∣∣∫ t2

t1

f(y(t), t)∆t

∣∣∣∣ ≤ ∫ t2

t1

M(t)∆t,

for all y ∈ G([t0,+∞)T,Rn) and all t1, t2 ∈ [t0,+∞)T, t1 ≤ t2.

(B3) There exists a locally Kurzweil-Henstock ∆-integrable function L : [t0,+∞)T → R such

that ∣∣∣∣∫ t2

t1

[f(y(t), t)− f(w(t), t)]∆t

∣∣∣∣ ≤ ‖y − w‖[t0,+∞)T

∫ t2

t1

L(t)∆t,

for all y, w ∈ G0([t0,+∞)T,Rn) and all t1, t2 ∈ [t0,+∞)T, t1 ≤ t2.

5.3 Boundedness of solutions of dynamic equations on time scales 95

Moreover, assume that for every (z0, s0) ∈ Rn×[t0,+∞)T, there exists a unique (maximal)

solution x : [s0,+∞)T → Rn of the dynamic equation on time scales (5.30) with x(s0) = z0.

The existence of such a solution is ensured by Theorem 4.34.

In what follows, for every (z0, s0) ∈ Rn × [t0,+∞)T, we denote by x(s, s0, z0) the unique

(maximal) solution of the dynamic equation on time scales (5.30) with x(s0) = z0 defined on

[s0,+∞)T.

In the sequel, we introduce the concepts concerning the boundedness of the solutions of

the dynamic equation on time scales (5.30).

Definition 5.9. Let T be a time scale such that supT = +∞. We say that the dynamic

equation on time scales (5.30) is


all s0 ∈ [t0,+∞)T and all z0 ∈ Rn, where ‖z0‖ < α, we have

‖x(s, s0, z0)‖ < M, for all s ∈ [s0,+∞)T,

where x is the maximal solution of (5.30) with x(s0) = z0.

• Quasi-uniformly ultimately bounded: if there exists a B > 0 such that for every

α > 0, there exists a T = T (α) > 0, such that for all s0 ∈ [t0,+∞)T and for all z0 ∈ Rn,

where ‖z0‖ < α, we have:

‖x(s, s0, z0)‖ < B, for all s ∈ [s0 + T,+∞) ∩ T,

where x is the maximal solution of (5.30) with x(s0) = z0.


ultimately bounded.

In what follows, we will prove a result which ensures that the dynamic equation on time

scales (5.30) is uniformly bounded.

Theorem 5.10. Let T be a time scale such that supT = +∞ and [t0,+∞)T be a time

scale interval. Suppose f : Rn × [t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3),

and U : [t0,+∞)T × Rn → R be a function such that for each left-continuous function

z : [α, β]T → Rn on (α, β]T, the function [α, β]T 3 t 7→ U(t, z(t)) is left-continuous on

(α, β]T. Moreover, suppose the following conditions concerning U are satisfied

(i) There are two monotone increasing functions p, b : R+ → R+ such that b(0) = 0,

lims→+∞

b(s) = +∞


and

b(‖z‖) ≤ U(t, z) ≤ p(‖z‖),

for every pair (t, z) ∈ [t0,+∞)T × Rn.

(ii) For every solution z : [s0,+∞) ∩ T → Rn, s0 ≥ t0, of the dynamic equation on time

scales (5.30), we have

U(s, z(s))− U(t, z(t)) ≤ 0,

for every s, t ∈ [s0,+∞) ∩ T with t ≤ s.

Then, the dynamic equation on time scales (5.30) is uniformly bounded.

Proof. Since f satisfies conditions (B1), (B2) and (B3), by Theorem 4.26, f ∗ satisfies condi-

tions (A2), (A3) and (A4). Define a function U∗ : [t0,+∞)× Rn → R by

U∗(t, x) = U(t∗, x), t ∈ [t0,+∞), x ∈ Rn.

Then, by hypothesis (i), there are two monotone increasing functions p, b : R+ → R+ such

that p(0) = b(0) = 0,

lims→+∞

b(s) = +∞

and

b(‖z‖) ≤ U∗(t, z)︸︷︷︸=U(t∗,z)

≤ p(‖z‖)

for every pair (t, z) ∈ [t0,+∞)× Rn, which implies that U∗ satisfies the hypothesis (i) from

Theorem 5.7.

Now, let y : [s0,+∞) → Rn, s0 ≥ t0, be a solution of the measure differential equation

(5.27). By Corollary 3.18, y : [s0,+∞) → Rn must have the form y = z∗, where z :

[s0,+∞) ∩ T → Rn is a solution of the dynamic equation on time scales (5.30). Hence, for

each s0 ≤ t < s < +∞, we have

U∗(s, y(s))− U∗(t, y(t)) = U∗(s, z∗(s))− U∗(t, z∗(t))

= U(s∗, z∗(s))− U(t∗, z∗(t)) = U(s∗, z(s∗))− U(t∗, z(t∗)) ≤ 0,

by hypothesis (ii). Therefore all the hypotheses from Theorem 5.7 are satisfied. Then the

measure differential equation (5.27) is uniformly bounded. By Corollary 3.18, the dynamic

equation on time scales (5.30) is uniformly bounded, obtaining the desired result.

Finally, we present the last result of our chapter. Our result ensures that the dynamic

equation on time scales (5.30) is uniformly ultimately bounded.

5.3 Boundedness of solutions of dynamic equations on time scales 97

Theorem 5.11. Let T be a time scale such that supT = +∞, t0 ∈ T and t0 ≥ 0. Suppose

f : Rn× [t0,+∞)T → Rn satisfies conditions (B1), (B2) and (B3), and U : [t0,+∞)T×Rn →R be a function such that for each left-continuous function z : [α, β]T → Rn on (α, β]T, the

function [α, β]T 3 t 7→ U(t, z(t)) is left-continuous on (α, β]T and satisfies condition (i) from

Theorem 5.10. Moreover, suppose U satisfies the following conditions

(I) For all x, y, z, w ∈ Rn and for all α, β ∈ [t0,+∞)T with α ≤ β, we have

|U(β, x)− U(β, y)− U(α, z) + U(α,w)| ≤(∫ β

α

K(τ)∆τ

)max ‖x− y‖ , ‖z − w‖ ,

where K : [t0,+∞)T → R is a locally Kurzweil-Henstock ∆-integrable function.

(II) There exists a continuous function φ : Rn → R, with φ(0) = 0 and φ(x) > 0, x 6= 0

such that for every solution z : [s0,+∞)T ∩ T→ Rn, s0 ≥ t0, of the dynamic equation

on time scales (5.30), we have

U(s∗, z(s∗))− U(t∗, z(t∗)) ≤ (s− t)∗(− φ(z(t∗))

),

for every t, s ∈ [s0,+∞) ∩ T with t ≤ s.

Then, the dynamic equation on time scales (5.30) is uniformly ultimately bounded.

Proof. Since f satisfies conditions (B1), (B2) and (B3), by Theorem 4.26, f ∗ satisfies con-

ditions (A2), (A3) and (A4), and by Lemma 3.12, g is a nondecreasing and left-continuous

function.

Also, define the functional U∗ : [t0,+∞)T × Rn → R by

U∗(t, x) = U(t∗, x),

for every pair (t, x) ∈ [t0,+∞)× Rn.

Since K is a locally Kurzweil-Henstock ∆-integrable function on [t0,+∞)T, it follows from

Theorem 3.13 that the function K∗ : [t0,+∞) → R is locally Kurzweil-Henstock integrable

with respect to the nondecreasing function u : T∗ → R, given by u(t) = t∗ and∫ β

α

K(τ)∆τ =

∫ β

α

K∗(τ)du(τ),


for all α, β ∈ [t0,+∞)T. Thus, for all x, y : [υ, γ] → Rn, [υ, γ] ⊂ [t0,+∞), of bounded

variation in [υ, γ] and all υ ≤ s < t ≤ γ, we have

|U∗(t, x(t))− U∗(t, y(t))− U∗(s, x(s)) + U∗(s, y(s))|

= |U(t∗, x(t))− U(t∗, y(t))− U(s∗, x(s)) + U(s∗, y(s))|

≤(∫ t∗

s∗K(τ)∆τ

)max ‖x(t)− y(t)‖ , ‖x(s)− y(s)‖

=

(∫ t∗

s∗K∗(τ)du(τ)

)max ‖x(t)− y(t)‖ , ‖x(s)− y(s)‖

≤(∫ t∗

s∗K∗(τ)du(τ)

)supξ∈[υ,γ]

‖x(ξ)− (ξ)‖

=

(∫ t

s

K∗(τ)du(τ)

)supξ∈[υ,γ]

‖x(ξ)− (ξ)‖,

where Theorem 3.14 is used to establish the last equality. Thus, the hypothesis (U1) from

Theorem 5.8 is fulfilled.

Now, let y : [s0,+∞) → Rn, s0 ≥ t0, be a solution of the measure differential equation

(5.27). Then By Corollary 3.18, y : [s0,+∞) → Rn must have the form y = z∗, where

z : [s0,+∞) ∩ T→ Rn is a solution of the dynamic equation on time scales (5.30). Hence

U∗(s, y(s))− U∗(t, y(t)) = U∗(s, z∗(s))− U∗(t, z∗(t))

= U(s∗, z∗(s))− U(t∗, z∗(t)) = U(s∗, z(s∗))− U(t∗, z(t∗))

≤ (s− t)∗(−φ(z(t∗))) = (s− t)∗(−φ(y(t)))

≤ (s− t)(−φ(y(t))),

for every s0 ≤ t < s < +∞. Thus, condition (U2) from the Theorem 5.8 is fullfiled. There-

fore, all the hypotheses from Theorem 5.8 are fulfilled and the measure differential equation

(5.27) is uniformly ultimately bounded. By Corollary 3.18, it follows that the dynamic equa-

tion on time scales (5.30) is uniformly ultimately bounded, obtaining the desired result.

Chapter

6Lyapunov stability

In this chapter, our goal is to investigate Lyapunov-type stability results for generalized

ODEs and, then, to obtain Lyapunov-type results for measure differential equations. Using

relations between the solutions of measure differential equations and the solutions of dynamic

equations on time scales, we also present Lyapunov-type results for these last equations.

The results presented in this chapter are new and they are contained in the paper [18].

6.1 Lyapunov stability for generalized ODEs

In this section, we will present Lyapunov-type stability results for generalized ODEs.

Let us assume that X is a Banach space with the norm ‖ · ‖ and set Ω = Bc × [t0,+∞),

where Bc = x ∈ X : ‖x‖ < c, c > 0 and t0 ≥ 0.

Consider the generalized ODE given by

dx

dτ= DF (x, t). (6.1)

We assume that F ∈ F(Ω, h), where the function h is nondecreasing and left-continuous,

and F (0, t)− F (0, s) = 0, for t, s ≥ t0. Then for every [a, b] ⊂ [t0,+∞), we have∫ b

a

DF (0, t) = F (0, b)− F (0, a) = 0,

which implies that x ≡ 0 is a solution of the generalized ODE (6.1) on [t0,+∞).

99

100 Chapter 6 — Lyapunov stability

We also assume that Ω = ΩF , that is, Ω = (x, t) ∈ Ω : x+ F (x, t+)− F (x, t) ∈ Bc .Then, by Theorems 4.8 and 4.9, for every (x0, s0) ∈ Ω, there exists a unique maximal

solution x : [s0, ω(s0, x0)) → X of the generalized ODE (6.1) such that x(s0) = x0. Here,

we will denote by x(t) = x(t, s0, x0), t ∈ [s0, ω(s0, x0)), the unique maximal solution of the

generalized ODE (6.1) with x(s0) = x0.

Remark 6.1. For simplicity of notations, when it is clear, we write only ω instead of

ω(s0, x0).

The following Lyapunov-type stability concepts of the trivial solution of a generalized

ODE extend the concepts presented in [44, Definition 4.1].

Definition 6.2. The trivial solution x ≡ 0 of the generalized ODE (6.1) is said to be

• Stable, if for every s0 ≥ t0, ε > 0, there exists δ = δ(ε, s0) > 0 such that if x0 ∈ Bc

with

‖x0‖ < δ,

then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω), where x is the maximal solution of (6.1) with x(s0) = x0.

• Uniformly stable, if it is stable with δ independent of s0.

• Uniformly asymptotically stable, if there exists δ0 > 0 and for every ε > 0, there

exist T = T (ε) ≥ 0 such that if s0 ≥ t0 and x0 ∈ Bc with

‖x0‖ < δ0,

then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω)∩[s0+T,+∞), where x is the maximal solution of (6.1) with x(s0) = x0.

In the sequel, we present a concept of Lyapunov functional for generalized ODEs.

Definition 6.3. We say that V : [t0,+∞)×Bρ → R is a Lyapunov functional with respect

to the generalized ODE (6.1), 0 < ρ < c, if the following conditions are satisfied:

(i) V : [t0,+∞)×Bρ → R is left-continuous with respect to the first variable on (t0,+∞),

for all x ∈ Bρ;

6.1 Lyapunov stability for generalized ODEs 101

(ii) There exists a continuous increasing function b : R+ → R+ satisfying b(0) = 0 (we say

that such a function is of Hahn class) such that

V (t, x) ≥ b(‖x‖),

for every (t, x) ∈ [t0,+∞)×Bρ;

(iii) The function [s0, ω) 3 t 7→ V (t, x(t)) is nonincreasing along every maximal solution

x(t) = x(t, s0, x0), (x0, s0) ∈ Bρ× [t0,+∞), of the generalized ODE (6.1) with x(s0) =

x0.

In what follows, we present a result which ensures that the trivial solution of the general-

ized ODE (6.1) is uniformly stable. Such result does not require any Lipschitz-type condition

concerning the Lyapunov functional, improving, then, the results found in the literature. See,

for instance, [3, 22, 24, 38].

Theorem 6.4. Let F ∈ F(Ω, h), where h : [t0,+∞) → R is a left-continuous and nonde-

creasing function, and let V : [t0,+∞) × Bρ → R, 0 < ρ < c, be a Lyapunov functional.

Assume that V satisfies the following condition

(H1) There exists a continuous increasing function a : R+ → R+ satisfying a(0) = 0, such

that for every solution x : I → Bρ, I ⊂ [t0,+∞), of the generalized ODE (6.1), we have

V (t, x(t)) ≤ a(‖x(t)‖)

for all t ∈ I.

Then the trivial solution x ≡ 0 of the generalized ODE (6.1) is uniformly stable.

Proof. Since V is a Lyapunov functional, there exists a function of Hahn class b : R+ → R+

such that

V (t, x) ≥ b(‖x‖),

for every (t, x) ∈ [t0,+∞)×Bρ.

Let s0 ≥ t0 and ε > 0. Since a(0) = 0, a is increasing and a|[0,ε] is uniformly continuous,

there exists δ = δ(ε), 0 < δ < ε, such that a(δ) < b(ε).

Suppose x0 ∈ Bρ and the maximal solution x(t) = x(t, s0, x0) of the generalized ODE

(6.1) satisfies

‖x(s0)‖ = ‖x0‖ < δ.

We want to prove that

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω).


Since V is a Lyapunov functional, by (iii) from Definition 6.3, we have

V (t, x(t)) ≤ V (s0, x(s0)), for all t ∈ [s0, ω).

Therefore, for every t ∈ [s0, ω),

b(‖x(t)‖) ≤ V (t, x(t)) ≤ V (s0, x(s0)) ≤ a(‖x(s0)‖) < a(δ) < b(ε).

Since b is an increasing function, we obtain

‖x(t)‖ < ε,

for all t ∈ [s0, ω), which completes the proof.

The next result establishes sufficient conditions so that the trivial solution of the gener-

alized ODE (6.1) is uniformly asymptotically stable.

Theorem 6.5. Suppose F ∈ F(Ω, h), where h : [t0,+∞) → R is left-continuous and non-

decreasing, V : [t0,+∞) × Bρ → R, 0 < ρ < c, satisfies conditions (i), (ii) from Defini-

tion 6.3 and (H1) from Theorem 6.4. Moreover, suppose there exists a continuous function

Φ : X → R satisfying Φ(0) = 0 and Φ(x) > 0 for x 6= 0, such that for every maximal solution

x(t) = x(t, s0, x0), (x0, s0) ∈ Bρ × [t0,+∞), of the generalized ODE (6.1),

V (s, x(s))− V (t, x(t)) ≤ (s− t)(− Φ(x(t))

), (6.2)

for all t, s ∈ [s0, ω), with t ≤ s. Then the trivial solution x ≡ 0 of the generalized ODE (6.1)

is uniformly asymptotically stable.

Proof. From the inequality (6.2), it is clear that the function [s0,+∞) 3 t 7→ V (t, x(t)) is

nonincreasing along every maximal solution x(t) = x(t, s0, x0), (x0, s0) ∈ Ω, of the generalized

ODE (6.1). Since all the hypotheses from Theorem 6.4 are satisfied, the trivial solution x ≡ 0

of (6.1) is uniformly stable.

Let δ0 :=ρ

2and ε > 0. By the uniform stability of the solution x ≡ 0 of the generalized

ODE (6.1), there exists δ = δ(ε) > 0 (we can take δ < ρ) such that, if τ0 ≥ t0 and y0 ∈ Bρ

satisfies

‖y0‖ < δ,

then

‖x(t, τ0, y0)‖ < ε, for all t ∈ [τ0, ω(τ0, y0)). (6.3)

Let

N := sup −Φ(ϑ) : δ(ε) ≤ ‖ϑ‖ < ρ < 0


and

T (ε) := −a(δ0)

N> 0.

Suppose s0 ≥ t0 and x0 ∈ Bρ and consider the maximal solution x(·) = x(·, s0, x0) :

[s0, ω+(s0, x0))→ Bρ of (6.1) such that

‖x(s0)‖ = ‖x0‖ < δ0. (6.4)

We want to prove that

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε, for all t ∈ [s0, ω(s0, x0)) ∩ [s0 + T (ε),+∞). (6.5)

At first, assume that ω(s0, x0) < +∞. Then let us consider two cases.

Case 1. Suppose T (ε) ≥ ω(s0, x0)− s0, that is, ω(s0, x0) ≤ s0 + T (ε). Therefore,

[s0, ω(s0, x0)) ∩ [s0 + T (ε),+∞) = ∅.

Under these conditions, (6.5) holds trivially.

Case 2. Suppose T (ε) < ω(s0, x0)− s0, that is, s0 + T (ε) < ω(s0, x0). Therefore,

[s0, ω(s0, x0)) ∩ [s0 + T (ε),+∞) = [s0 + T (ε), ω(s0, x0)).

Now, let us prove that there exists t ∈ [s0, s0 + T (ε)] such that∥∥x(t, s0, x0)

∥∥ =∥∥x(t)

∥∥ <δ(ε). Suppose the contrary, that is, ‖x(s, s0, x0)‖ = ‖x(s)‖ ≥ δ(ε), for every s ∈ [s0, s0+T (ε)].

Thus,

δ(ε) ≤ ‖x(s)‖ < ρ, for all s ∈ [s0, s0 + T (ε)]. (6.6)

By (H1), (6.2), (6.4) and (6.6), we obtain

V (s0 + T (ε), x(s0 + T (ε))) ≤ V (s0, x(s0)) + T (ε)(− Φ(x(s0))

)≤ V (s0, x(s0)) + T (ε) sup

s∈[s0,s0+T (ε)]

−Φ(x(s))

≤ a(‖x(s0)‖) + T (ε) sup −Φ(ϑ) : δ(ε) ≤ ‖ϑ‖ < ρ︸︷︷︸N

< a(δ0) + T (ε)N = a(δ0) +(− a(δ0)

N

)N = 0,

that is,

V (s0 + T (ε), x(s0 + T (ε))) < 0,


which contradicts the following inequality

V (s0 + T (ε), x(s0 + T (ε))) ≥ b(‖x(s0 + T (ε))‖) ≥ 0.

Then we conclude that there exists t ∈ [s0, s0+T (ε)] such that∥∥x(t, s0, x0)

∥∥ =∥∥x(t)

∥∥ < δ(ε).

Therefore, by (6.3) (with τ0 = t and y0 = x(t) = x(t, s0, x0) =: y), we obtain∥∥x(t, t, y)∥∥ < ε, for all t ∈ [t, ω(t, y)). (6.7)

Since

x(·, t, y) : [t, ω(t, y))→ Bρ

and the restriction

x|[t,ω(s0,x0))(·, s0, x0) : [t, ω(s0, x0))→ Bρ

is a solution of the generalized ODEdx

dτ= DF (x, t)

x(t) = y = x(t, s0, x0),

by Lemma 4.7, it follows that

x(t, t, y) = x|[t,ω(s0,x0))(t, s0, x0), for all t ∈ [t, ω(t, y)) ∩ [t, ω(s0, x0)),

that is,

x(t, t, y) = x(t, s0, x0), for all t ∈ [t, ω(t, y)) ∩ [t, ω(s0, x0)). (6.8)

Now, we want to prove that ω(t, y) = ω(s0, x0). Without loss of generality, suppose

ω(t, y) < ω(s0, x0). Thus, by (6.8), we have

x(t, t, y) = x(t, s0, x0), for all t ∈ [t, ω(t, y)). (6.9)

On the other hand, since ω(t, y) < +∞ (because ω(t, y) < ω(s0, x0)), by Proposition

4.12, the limit limt→ω(t,y)−

x(t, t, y) exists. Thus, by (6.9) and using the fact that x(·, s0, x0) is a

left-continuous function at the point ω(t, y), we obtain

x(ω(t, y), s0, x0) = limt→ω(t,y)−

x(t, s0, x0) = limt→ω(t,y)−

x(t, t, y),

that is,

limt→ω(t,y)−

x(t, t, y) = x(ω(t, y), s0, x0).


Therefore, by Lemma 4.1, the function z1 : [t, ω(s0, x0))→ X given by the relation

z1(t) =

x(t, t, y), t ∈ [t, ω(t, y))

x(t, s0, x0), t ∈ [ω(t, y), ω(s0, x0)),

is a solution of the generalized ODE (6.1) with z1(t) = x(t, t, y) = y. Notice that z1 is a proper

right prolongation of x : [t, ω(t, y))→ Bρ, which contradicts the fact that x : [t, ω(t, y))→ Bρ

is the maximal solution of the generalized ODE (6.1) with x(t) = y.

Analogously, if ω(s0, x0) < ω(t, y) (note that ω(s0, x0) < +∞ in the case we are consid-

ering), then the function

z2(t) =

x(t, s0, x0), t ∈ [s0, ω(s0, x0)),

x(t, t, y), t ∈ [ω(s0, x0), ω(t, y))

is a proper right prolongation of x : [s0, ω(s0, x0)) → Bρ, which contradicts the fact that

x : [s0, ω(s0, x0))→ Bρ is the maximal solution of the generalized ODE (6.1) with x(s0) = x0.

Then,

ω(t, y) = ω(s0, x0). (6.10)

Thus, by (6.7), (6.8) and (6.10), we obtain

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε, for all t ∈ [t, ω(s0, x0)).

In particular,

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε, for all t ∈ [s0, ω(s0, x0)) ∩ [s0 + T (ε),+∞)︸︷︷︸[s0+T (ε),ω(s0,x0))

,

since [s0 + T (ε), ω(s0, x0)) ⊂ [t, ω(s0, x0)) (because t ≤ s0 + T (ε)). Hence the trivial solution

x ≡ 0 of the generalized ODE (6.1) is uniformly asymptotically stable.

Now, let us consider ω(s0, x0) = +∞. Then, we have

[s0, ω(s0, x0)︸︷︷︸+∞

) ∩ [s0 + T (ε),+∞) = [s0 + T (ε),+∞).

We have only to consider the case where s0 + T (ε) < +∞. Using the same arguments as in

the first part of proof of Case 2, there exists t ∈ [s0, s0 + T (ε)] such that∥∥x(t, s0, x0)∥∥ =

∥∥x(t)∥∥ < δ(ε).


Then, by (6.3) (with τ0 = t and y0 = x(t) = x(t, s0, x0) =: y), we have∥∥x(t, t, y)∥∥ < ε, for all t ∈ [t, ω(t, y)). (6.11)

Since

x(·, t, y) : [t, ω(t, y))→ Bρ

and

x|[t,+∞)(·, s0, x0) : [t,+∞)→ Bρ

are solutions of dx

dτ= DF (x, t)

x(t) = y = x(t, s0, x0),

by Lemma 4.7, we obtain

x(t, t, y) = x|[t,+∞)(t, s0, x0), for all t ∈ [t, ω(t, y)) ∩ [t,+∞), (6.12)

that is,

x(t, t, y) = x(t, s0, x0), for all t ∈ [t, ω(t, y)) ∩ [t,+∞). (6.13)

Finally, we will prove that ω(t, y) = +∞. Suppose the contrary, that is, ω(t, y) < +∞.Then using the same argument as before for z1, the function z3 : [t,+∞) → Bρ, given by

the relation

z3(t) =

x(t, t, y), t ∈ [t, ω(t, y))

x(t, s0, x0), t ∈ [ω(t, y),+∞),

is a proper right prolongation of x : [t, ω(t, y)) → Bρ, which contradicts the fact that x :

[t, ω(t, y)) → Bρ is the maximal solution of the generalized ODE (6.1) with x(t) = y.

Therefore,

ω(t, y) = +∞. (6.14)

Thus, by (6.11), (6.13) and (6.14), we get

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε, for all t ∈ [t,+∞).

In particular

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε, for all t ∈ [s0,+∞) ∩ [s0 + T (ε),+∞)︸︷︷︸[s0+T (ε),+∞)

,

because [s0 + T (ε),+∞) ⊂ [t,+∞) (since t ≤ s0 + T (ε)). The proof is then completed.

6.2 Lyapunov stability for measure differential equations 107

6.2 Lyapunov stability for measure differential equations

In this section, our goal is to prove stability results for measure differential equations using

Lyapunov functionals. Throughout this section, let us consider that Rn is the n-dimensional

Euclidean space with norm ‖ · ‖ , and set

Bc = x ∈ Rn : ‖x‖ < c ,

with c > 0.

We consider the integral form of a measure differential equations of type

x(t) = x(τ0) +

∫ t

τ0

f(x(s), s)dg(s), t ≥ τ0, (6.15)

where τ0 ≥ t0, f : Bc × [t0,+∞)→ Rn and g : [t0,+∞)→ R.

Throughout this section, we assume that the function f : Bc × [t0,+∞) → Rn satis-

fies conditions (A2), (A3), (A4) and g : [t0,+∞) → R satisfies condition (A1) (all these

conditions presented in Chapter 3).

We also assume that f(0, t) = 0, for every t ∈ [t0,+∞). This condition implies that x ≡ 0

is a solution of (6.15) on [t0,+∞) and the function F, given by F (x, t) =∫ tt0f(x, s)dg(s),

satisfies F (0, s) − F (0, t) = 0, for all t, s ∈ [t0,+∞). Notice that, under this condition, it

makes sense to prove the stability of the trivial solution of the measure differential equation

(6.15).

Moreover, we assume that x0 + f(x0, s0)∆+g(s0) ∈ Bc, for all s0 ∈ [t0,+∞) and x0 ∈ Bc.

This condition ensures us that for all (x0, s0) ∈ Bc× [t0,+∞), there exists a unique maximal

solution x : [s0, ω) → Rn of the measure differential equation (6.15) with x(s0) = x0. Note

that the existence of such a solution is guaranteed by Theorems 4.8 and 4.9. In what follows,

for all (x0, s0) ∈ Bc × [t0,+∞), we denote by x(t, s0, x0) the unique maximal solution of the

measure differential equation (6.15) with x(s0) = x0 defined on [s0,+∞).

In the sequel, we present some concepts of stability of the trivial solution of the measure

differential equations (6.15).

Definition 6.6. The trivial solution x ≡ 0 of the measure differential equation (6.15) is said

to be

• Stable, if for every s0 ≥ t0 and ε > 0, there exists δ = δ(ε, s0) > 0 such that if x0 ∈ Bc

with

‖x0‖ < δ,


then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω+(s0, x0)), where x is the maximal solution of (6.15) with x(s0) = x0.

• Uniformly stable, if it is stable and δ is independent of s0.

• Uniformly asymptotically stable, if there exists δ0 > 0 and for every ε > 0, there

exists T = T (ε) ≥ 0 such that if s0 ≥ t0 and x0 ∈ Bc with

‖x0‖ < δ0,

then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω+(s0, x0)) ∩ [s0 + T,+∞), where x is the maximal solution of (6.15)

with x(s0) = x0.

In what follows, we introduce a concept of Lyapunov functional with respect to the

measure differential equation (6.15).

Definition 6.7. We say that U : [t0,+∞) × Bρ → R, 0 < ρ < c, is a Lyapunov functional

with respect to the measure differential equation (6.15), if the following conditions are satisfied

(I) U(·, x) : [t0,+∞)→ R is left-continuous on (t0,+∞) for all x ∈ Bρ;

(II) There exists a continuous increasing function b : R+ → R+ satisfying b(0) = 0 (we say

that such function is of Hahn class) such that

U(t, x) ≥ b(‖x‖),

for every (t, x) ∈ [t0,+∞)×Bρ;

(III) The function [s0, ω) 3 t 7→ U(t, x(t)) is a nonincreasing function along every maximal

solution x(t) = x(t, s0, x0), (x0, s0) ∈ Bρ×[t0,+∞), of the measure differential equation

(6.15).

The next result ensures us that the trivial solution x ≡ 0 of the measure differential

equation (6.15) is uniformly stable.

Theorem 6.8. Suppose f : Bc × [t0,+∞) → Rn satisfies conditions (A2), (A3) and (A4),

g : [t0,+∞) → R satisfies condition (A1) and f(0, t) = 0 for every t ∈ [t0,+∞). Also, let

U : [t0,+∞) × Bρ → R, 0 < ρ < c, be a Lyapunov functional with respect to the measure

differential equation (6.15). Assume that U satisfies the condition

6.2 Lyapunov stability for measure differential equations 109

(H2) There exists a continuous increasing function a : R+ → R+ such that a(0) = 0 and for

all solutions x : I → Bρ, I ⊂ [t0,+∞) of (6.15), we have

U(t, x(t)) ≤ a(‖x(t)‖)

for all t ∈ I.Then the trivial solution x ≡ 0 of the measure differential equation (6.15) is uniformly stable.

Proof. Define a function F : Bc × [t0,+∞)→ Rn by

F (x, t) :=

∫ t

t0

f(x, s)dg(s) (6.16)

for all (x, t) ∈ Bc × [t0,+∞). Then, by Theorem 3.2, it follows that F ∈ F(Ω, h), where the

function h : [t0,+∞)→ R is given by

h(t) =

∫ t

t0

(M(s) + L(s))dg(s),

since the function f : Bc × [t0,+∞)→ Rn satisfies conditions (A2), (A3) and (A4) and the

function g : [t0,+∞)→ R satisfies condition (A1).

By Theorem 3.8, it is clear that U : [t0,+∞) × Bρ → R is a Lyapunov functional with

respect to the generalized ODE (6.1), where F is given by (6.16). Moreover, by condition

(H2), it is clear that U also satisfies condition (H1) from Theorem 6.4. Therefore, the trivial

solution x ≡ 0 of the generalized ODE (6.1) is unifomly stable. By Theorem 3.8, it follows

that the trivial solution x ≡ 0 of the measure differential equation (6.15) is also uniformly

stable, obtaining the desired result.

In the sequel, we present our last result of this section, which ensures that the trivial

solution x ≡ 0 of the measure differential equation (6.15) is uniformly asymptotically stable.

Theorem 6.9. Suppose f : Bc × [t0,+∞) → Rn satisfies conditions (A2), (A3) and (A4),

g : [t0,+∞) → R satisfies condition (A1) and f(0, t) = 0 for every t ∈ [t0,+∞). Suppose

U : [t0,+∞) × Bρ → R, 0 < ρ < c, satisfies conditions (I) and (II) from Definition 6.7

and condition (H2) from Theorem 6.8. Moreover, suppose there exists a continuous function

Φ : X → R satisfying Φ(0) = 0 and Φ(z) > 0 for z 6= 0, such that for every maximal solution

x(t) = x(t, s0, x0), (x0, s0) ∈ Bρ × [t0,+∞), of the measure differential equation (6.15), we

have

U(s, x(s))− U(t, x(t)) ≤ (s− t)(− Φ(x(t))

), (6.17)

for all t, s ∈ [s0, ω) with t ≤ s. Then the trivial solution x ≡ 0 of (6.15) is uniformly

asymptotically stable.


Proof. Define a function F : Bc × [t0,+∞)→ Rn by

F (x, t) =

∫ t

t0

f(x, s)dg(s)

for all (x, t) ∈ Bc× [t0,+∞). Since f satisfies conditions (A2), (A3) and (A4) and g satisfies


by

h(t) =

∫ t

t0

(M(s) + L(s))dg(s).

By Theorem 3.8, it is easy to check that U : [t0,+∞)×Bρ → R satisfies all the hypotheses

of Theorem 6.5. Therefore, by Theorem 6.5, the trivial solution x ≡ 0 of the generalized ODE

(6.1) is uniformly asymptotically stable. Applying Theorem 3.8 again, the trivial solution

x ≡ 0 of the measure differential equation (6.15) is uniformly asymptotically stable and we

obtain the desired result.

6.3 Lyapunov stability for dynamic equations on time

scales

In this section, our goal is to prove results concerning stability of solutions of dynamic

equations on time scales using Lyapunov functionals.

Consider the dynamic equation on time scale given by

x∆(t) = f(x∗, t), t ∈ [t0,+∞)T, (6.18)

where f : Bc × [t0,+∞)T → Rn. We recall the reader that x∗ denotes the extension of x

defined in Chapter 3.

Throughout this section, we assume that the fucntion f satisfies conditions (B1), (B2)

and (B3) presented in Chapter 4.

From now on, let us assume that f(0, t) = 0 for every t ∈ [t0,+∞)T. This condition

implies that x ≡ 0 is a solution of the dynamic equation on time scales (6.18).

We also assume that for all (z0, τ0) ∈ Bc × [t0,+∞)T, we have z0 + f(z0, τ0)µ(τ0) ∈ Bc.

Then, by Theorem 4.27, for all (x0, s0) ∈ Bc × [t0,+∞)T, there exists a unique maximal

solution x : [s0, ω)T → Rn, ω ≤ +∞ of the dynamic equation on time scales (6.18) with

x(s0) = x0. Here, we will denote by x(t, s0, x0), t ∈ [s0, ω(s0, x0))T, the unique maximal

solution of the dynamic equation on time scales (6.18) with x(s0) = x0.

6.3 Lyapunov stability for dynamic equations on time scales 111

In the sequel, let us present the concept of stability of the trivial solution of the dynamic

equations on time scales (6.18).

Definition 6.10. Let T be a time scale such that supT = +∞. The trivial solution x ≡ 0

of the dynamic equation on time scales (6.18) is said to be

(i) Stable, if for every s0 ∈ T, with s0 ≥ t0 and ε > 0, there exists δ = δ(ε, s0) > 0 such

that if x0 ∈ Bc with

‖x0‖ < δ,

then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω(s0, x0))T, where x is the maximal solution of (6.18) with x(s0) = x0.

(ii) Uniformly stable, if it is stable with δ independent of s0.

(iii) Uniformly asymptotically stable, if there exists δ0 > 0 and for every ε > 0, there

exists T = T (ε) ≥ 0 such that if s0 ∈ T, s0 ≥ t0 and x0 ∈ Bc with

‖x0‖ < δ0,

then

‖x(t, s0, x0)‖ = ‖x(t)‖ < ε,

for all t ∈ [s0, ω(s0, x0)) ∩ [s0 + T,+∞) ∩ T, where x is the maximal solution of (6.18)

with x(s0) = x0.

In what follows, we will present the definition of a Lyapunov functional with respect to

the dynamic equation on time scales (6.18).

Definition 6.11. Let t0 ∈ T and c > 0. We say that U : [t0,+∞)T × Bρ → R, 0 < ρ < c,

is a Lyapunov functional with respect to the dynamic equation on time scales (6.18), if the

following conditions are satisfied

(i) U(·, x) : [t0,+∞)T → R is left-continuous on (t0,+∞)T for all x ∈ Bρ;

(ii) There exists a continuous increasing function b : R+ → R+ satisfying b(0) = 0 (we say

that such function is of Hahn class) such that

U(t, x) ≥ b(‖x‖),

for every (t, x) ∈ [t0,+∞)T ×Bρ;


(iii) The function [τ0, ω)T 3 t 7→ U(t, z(t)) is nonincreasing along every maximal solution

z(t) = z(t, τ0, x0), (x0, τ0) ∈ Bρ × [t0,+∞), of the dynamic equation on time scales

(6.18).

We recall the reader that the function g : [t0,+∞) → R given by g(t) = t∗, for all

t ∈ [t0,+∞), is nondecreasing and lef-continuous on (t0,+∞).

The following results ensure us the uniform stability and uniform asymptotically stability

of the trivial solution of the dynamic equation on time scales (6.18) via Lyapunov functionals.


scale interval. Assume that f : Bc × [t0,+∞)T → Rn satisfies conditions (B1), (B2), (B3)

and f(0, t) = 0 for every t ∈ [t0,+∞)T. Let U : [t0,+∞)T × Bρ → R, 0 < ρ < c, be a

Lyapunov functional with respect to the dynamic equation on time scales (6.18). Assume

that U satisfies the following condition

(H3) There exists a continuous increasing function a : R+ → R+ such that a(0) = 0 and for

all (t, x) ∈ [t0,+∞)T ×Bc, we have

U(t, x) ≤ a(‖x‖). (6.19)

Then the trivial solution x ≡ 0 of the dynamic equation on time scales (6.18) is uniformly

stable.


ditions (A2), (A3) and (A4). Also, since f(0, t) = 0, for every t ∈ [t0,+∞)T, clearly

f ∗(0, t) = 0, for every t ∈ [t0,+∞).

Define

U∗(t, x) := U(t∗, x),

for every (t, x) ∈ [t0,+∞) × Bρ. We affirm that U∗ : [t0,+∞) × Bρ → R is a Lyapunov

functional with respect to the measure differential equation

y(t) = y(τ0) +

∫ t

τ0

f ∗(y(s), s)dg(s), for t, τ0 ∈ [t0,+∞), (6.20)

where g(t) = t∗. Indeed, by condition (i) from Definition 6.11, U(·, x) : [t0,+∞)T → R is left-

continuous on (t0,+∞)T for all x ∈ Bρ. Since U∗(t, x) = U(t∗, x) = U(g(t), x) and g(t) = t∗

is a left-continuous function on (t0,+∞), we have U∗(·, x) : [t0,+∞)→ R is left-continuous

on (t0,+∞) for all x ∈ Bρ, that is, U∗ satisfies condition (I) from Definition 6.7.


On the other hand, by condition (ii) from Definition 6.11, there exists a continuous and

increasing function b : R+ → R+ satisfying b(0) = 0 such that

U(s, x) ≥ b(‖x‖),

for every (s, x) ∈ [t0,+∞)T ×Bρ. Then, for every (t, x) ∈ [t0,+∞)×Bρ, we obtain

U∗(t, x) = U(t∗, x) ≥ b(‖x‖),

that is, U∗ satisfies condition (II) from Definition 6.7.

Finally, we will prove that U∗ satisfies condition (III) from Definition 6.7. Indeed, let

(x0, s0) ∈ Bρ× [t0,+∞) and y = y(·, s0, x0) : [s0, ω)→ Bρ be the unique maximal solution of

measure differential equation (6.20). Since y(t) ∈ Bρ ⊂ Bc, for t ∈ [s0, ω), by Corollary 4.20

(with N = Bρ), ω = +∞. Now, since y : [s0,+∞)→ Bρ is a solution of (6.20), we obtain

y(τ) = y(s0) +

∫ τ

s0

f(y(s), s∗)dg(s), τ ∈ [s0,+∞),

where g(ξ) = ξ∗.

We consider two cases.

Case 1. Suppose s0 ∈ [t0,+∞)\T. Then g is constant on [s0, s∗0] which implies

ξ∗ = s∗0, for all ξ ∈ [s0, s∗0]. (6.21)

Therefore,

∫ τ

s0

f(y(s), s∗)dg(s) = 0, for all τ ∈ [s0, s∗0], that is, y is constant on [s0, s

∗0]. In

particular,

y(τ) = y(s∗0), for all τ ∈ [s0, s∗0]. (6.22)

On the other hand, by Corollary 3.18, y|[s∗0,+∞) must have the form

y|[s∗0,+∞) = x∗, (6.23)

where x : [s∗0,+∞)T → Rn is a solution of the dynamic equation on time scales (6.18). Notice

that x is also the maximal solution of the dynamic equation on time scales (6.18) through

(x(s∗0), s∗0) and y(s∗0) = x∗(s∗0) = x(s∗0). Hence the following holds true

• If s0 ≤ t < s ≤ s∗0, then, by (6.21), (6.22) and the fact y(s∗0) = x(s∗0), we get

U∗(s, y(s))− U∗(t, y(t)) = U(s∗, y(s))− U(t∗, y(t))

= U(s∗, x(s∗))− U(t∗, x(t∗)) = 0


• If s0 ≤ t ≤ s∗0 < s, then, by (6.23), (6.21) and (6.22), we have


= U(s∗, x∗(s))− U(s∗0, y(s∗0))

= U(s∗, x(s∗))− U(s∗0, x(s∗0))

= U(s∗, x(s∗))− U(t∗, x(t∗)) ≤ 0.

• If s∗0 ≤ t < s, then, by (6.23), we obtain


= U(s∗, x∗(s))− U(t∗, x∗(t))

= U(s∗, x(s∗))− U(t∗, x(t∗)) ≤ 0.

Thus, U∗(s, y(s)) − U∗(t, y(t)) ≤ 0 for all s0 ≤ t < s < +∞, that is, the function

t 7→ U∗(t, y(t)) is nonincreasing and this completes the proof of condition (III) in the situation

of case 1.

Case 2. Now, assume that s0 ∈ T. Then s0 = s∗0 ∈ T and, therefore, [s0,+∞)T = [s∗0,+∞)T.

By Corollary 3.18, y : [s0,+∞)→ Rn must have the form y = x∗, where x : [s0,+∞)T → Rn

is a solution (more precisely, the maximal solution) of the dynamic equation on time scales

(6.18) through (y(s0), s0). Hence

U∗(s, y(s))− U∗(t, y(t)) = U∗(s, x∗(s))− U∗(t, x∗(t))= U(s∗, x∗(s))− U(t∗, x∗(t))

= U(s∗, x(s∗))− U(t∗, x(t∗)) ≤ 0,

for every s0 ≤ t < s < +∞ and, therefore, the function t 7→ U∗(t, y(t)) is nonincreasing.

This completes the proof of condition (III) in case 2.

Conditions (I), (II) and (III) proved previously imply U∗ is a Lyapunov functional with

respect to the measure differential equation (3.20).

Now, by hypothesis (H3), there exists a continuous and increasing function a : R+ → R+

such that a(0) = 0 which satisfies condition (6.19). Then, for all solutions y : I → Bρ,

I ⊂ [t0,+∞) of the measure differential equation (6.20), we have

U∗(t, y(t)) = U(t∗, y(t)) ≤ a(‖y(t)‖)

for all t ∈ I. Therefore all the conditions of Theorem 6.8 are satisfied and, hence, the trivial

solution x∗ ≡ 0 of the measure differential equation (6.20) is uniformly stable. By Corolary


3.18, the trivial solution x ≡ 0 of the dynamic equation on time scales (6.18) is uniformly

stable and we obtain the desired result.


scale interval. Assume that f : Bc × [t0,+∞)T → Rn satisfies conditions (B1), (B2), (B3)

and f(0, t) = 0 for every t ∈ [t0,+∞)T. Suppose U : [t0,+∞)T × Bρ → R, 0 < ρ < c,

satisfies conditions (i) and (ii) from Definition 6.11 and condition (H3) from Theorem 6.12.

Moreover, suppose there exists a continuous function φ : X → R satisfying φ(0) = 0 and

φ(υ) > 0 for υ 6= 0 such that for every maximal solution x(t) = x(t, s0, x0), (x0, s0) ∈Bρ × [t0,+∞)T, of the dynamic equation on time scales (6.18),

U(s∗, x(s∗))− U(t∗, x(t∗)) ≤ (s− t)∗(−φ(x(t∗))) (6.24)

for all t, s ∈ [s0, ω)T with t ≤ s. Then the trivial solution x ≡ 0 of the dynamic equation on

time scales (6.18) is uniformly asymptotically stable.


ditions (A2), (A3) and (A4). Furthermore, f(0, t) = 0, for every t ∈ [t0,+∞)T, hence

f ∗(0, t) = 0, for every t ∈ [t0,+∞).

Define

U∗(t, x) := U(t∗, x),

for all (t, x) ∈ [t0,+∞)×Bρ. Then U∗ : [t0,+∞)×Bρ → R satisfies conditions (I) and (II)

from Definition 6.7, using the same arguments as in the proof of Theorem 6.12.

By hypothesis (H3) from Theorem 6.12, U∗ satisfies the hypothesis (H2) from Theorem

6.8, using the same arguments as in the proof of Theorem 6.12.

Finally, we will prove that U∗ satisfies the hypothesis (6.17). Indeed, let (x0, s0) ∈Bρ × [t0,+∞) and y = y(·, s0, x0) : [s0, ω) → Bρ be the unique maximal solution of the

measure differential equation (6.20). Since y(t) ∈ Bρ ⊂ Bc, for all t ∈ [s0, ω), by Corollary

4.20 (with N = Bρ), we have ω = +∞.

We consider two cases.

Case 1. Suppose s0 ∈ [t0,+∞)\T. Using the same argument as in Case 1 from the proof of

Theorem 6.12, we can prove that y|[s∗0,+∞) must have the form

y|[s∗0,+∞) = x∗,

where x : [s∗0,+∞)T → Rn is the maximal solution of the dynamic equation on time scales

(6.18) through (x(s∗0), s∗0). Hence


• If s0 ≤ t < s ≤ s∗0, we get


= U(s∗, x(s∗))− U(t∗, x(t∗)) = 0 (6.25)

On the other hand, by (6.24),

U(s∗, x(s∗))− U(t∗, x(t∗)) ≤ (s− t)∗(−φ(x(t∗))) ≤ 0. (6.26)

Then, by (6.25) and (6.26), we have (−φ(x(t∗))) = 0 and, therefore, x(t∗) = 0, that is,

y(t) = 0 (because x(t∗)

(6.21)

↓= x(s∗0) = y(s∗0)

(6.22)

↓= y(t)). Thus, we obtain

U∗(s, y(s))− U∗(t, y(t)) = (s− t)(−φ(y(t))) = 0.

• If s0 ≤ t ≤ s∗0 < s, we have

U∗(s, y(s))− U∗(t, y(t)) = U(s∗, x(s∗))− U(t∗, x(t∗))

≤ (s− t)∗(−φ(x(t∗)))

= (s− t)∗(−φ(x∗(t)))

= (s− t)∗(−φ(y(t))) ≤ (s− t)(−φ(y(t))).

• If s∗0 ≤ t < s, we obtain

U∗(s, y(s))− U∗(t, y(t)) = U(s∗, x(s∗))− U(t∗, x(t∗))

≤ (s− t)∗(−φ(x(t∗)))

= (s− t)∗(−φ(x∗(t)))

= (s− t)∗(−φ(y(t))) ≤ (s− t)(−φ(y(t))).

Thus, U∗(s, y(s))− U∗(t, y(t)) ≤ (s− t)(−φ(y(t))) for all s0 ≤ t < s < +∞.

Case 2. Now, assume that s0 ∈ T. Then s0 = s∗0 ∈ T and, therefore, [s0,+∞)T = [s∗0,+∞)T.

By Corollary 3.18, y : [s0,+∞)→ Rn must have the form y = x∗, where x : [s0,+∞)T → Rn

is a solution (more precisely, the maximal solution) of the dynamic equation on time scales


(6.18) through (x(s0), s0). Hence

U∗(s, y(s))− U∗(t, y(t)) = U∗(s, x∗(s))− U∗(t, x∗(t))= U(s∗, x∗(s))− U(t∗, x∗(t))

= U(s∗, x(s∗))− U(t∗, x(t∗))

≤ (s− t)∗(−φ(x(t∗)))

= (s− t)∗(−φ(y(t))) ≤ (s− t)(−φ(y(t))).

Thus U∗(s, y(s))− U∗(t, y(t)) ≤ (s− t)(−φ(y(t))), for all s0 ≤ t < s < +∞.

Therefore all the hypotheses of Theorem 6.9 are satisfied. Then the trivial solution x∗ ≡ 0

of the measure differential equation (6.20) is uniformly asymptotically stable. By Corollary

3.18 the trivial solution x ≡ 0 of the dynamic equation on time scales (6.18) is uniformly

asymptotically stable, obtaining the desired result.

Chapter

7Remarks on autonomous generalized

ODEs

In this section, our goal is to prove that autonomous generalized ODEs do not enlarge

the class of classical autonomous ODEs. More precisely, we will prove that if a function

H : Ω → X belongs to the class F(Ω, h, ω) (or to one of the class F(Ω, h, ω), F(Ω∞, h, ω)

or F(Ω, h, ω, E) defined in this chapter) and H(x, t) = F (x)t, for all (x, t) ∈ Ω, then F is a

continuous function and this fact implies that the classes of autonomous generalized ODEsdx

dτ= D[F (x)t] and of classic autonomous ODE

dx

dt= F (x(t)) coincide.

The results presented here are new and are contained in [19].

7.1 Autonomous generalized ODEs

We start this section by recalling the concept of an autonomous generalized ODE intro-

duced in [38] and used in [44].

Definition 7.1. Let X be a Banach space and O ⊂ X be open. An autonomous generalized

ODE is a generalized ODE of the form

dx

dτ= D[H(x, t)], (7.1)

where H : Ω→ X, Ω = O × [α, β], is given by

H(x, t) = F (x)t

119

120 Chapter 7 — Remarks on autonomous generalized ODEs

with F : O → X and t ∈ [α, β].

At this moment, the reader may find awkward the name “autonomous” given after an

equation of typedx

dτ= D[F (x)t], (7.2)

with t“appearing” on its right-hand side. Recall that a solution of the generalized ODE (7.2)

is any function x : [σ, η]→ X, [σ, η] ⊂ [α, β], such that x(t) ∈ O for all t ∈ [σ, η] and

x(v)− x(γ) =

∫ v

γ

D[F (x(τ))t] = (K)

∫ v

γ

F (x(t))dt (7.3)

for all γ, v ∈ [σ, η], where the last integral is precisely the Kurzweil integral (the prefix (K)

will be used to distinguish this integral from others). Thus, the integral form of (7.2) is given

by

x(t) = x(σ) + (K)

∫ t

σ

F (x(s))ds. (7.4)

The next result ensures that all solutions of the autonomous generalized ODE (7.2) are

continuous.

Lemma 7.2. If x : [σ, η] → X, [σ, η] ⊂ [α, β], is a solution of the autonomous generalized

ordinary differential equation (7.2) on [σ, η], then x is a continuous function on [σ, η].

Proof. Suppose x : [σ, η] → X, [σ, η] ⊂ [α, β], is a solution of the autonomous generalized

ODE (7.2) on [σ, η]. Let a ∈ [σ, η]. We have

x(s) = x(a) +

s∫a

D[F (x(τ))t], s ∈ [σ, η].

Now, define U : [σ, η]× [σ, η]→ X by U(τ, t) := F (x(τ))t, for every (τ, t) ∈ [σ, η]× [σ, η].

Notice that the function

U(a, ·) : [σ, η]→ X

t 7→ U(a, t) = F (x(a))t

is continuous at the point t = a. Thus, by Remark 1.7, the function [σ, η] 3 s 7→s∫a

D[F (x(τ))t]

is continuous at the point s = a and, therefore, x is continuous at a.

7.1 Autonomous generalized ODEs 121

In the sequel, we recall an important class of right-hand sides of generalized ODEs. The

reader may want to consult [38]. Assume that h : [α, β] → R is a nondecreasing function

and that ω : [0,+∞)→ R is a continuous and increasing function with ω(0) = 0.

Definition 7.3. We say that a function H : Ω→ X belongs to the class F(Ω, h, ω), if

‖H(x, s2)−H(x, s1)‖ ≤ |h(s2)− h(s1)| (7.5)

for all (x, s2), (x, s1) ∈ Ω and

‖H(x, s2)−H(x, s1)−H(y, s2) +H(y, s1)‖ ≤ ω(‖x− y‖)|h(s2)− h(s1)| (7.6)

for all (x, s2), (x, s1), (y, s2), (y, s1) ∈ Ω.

Remark 7.4. When the function ω : [0,+∞)→ R is the identity, we will denote this class

simply by F(Ω, h).

We recall the reader that if F : O → X is a continuous function and x0 ∈ O, then

x : I → X is a solution of the autonomous ODE given by

dx

dt= F (x(t)). (7.7)

on I with x(t0) = x0 if, and only if, x is a continuous function on I, x(t) ∈ O for every t ∈ Iand

x(t) = x(t0) + (R)

∫ t

t0

F (x(s))ds,

where the last integral is the Riemann integral on Banach spaces.

The following result ensures us that if F : O → X is a continuous function, then the

autonomous generalized ODE (7.2) and the autonomous ODE (7.7) coincide. Thus all the

theory in the literature concerning autonomous ODE remains true also for autonomous

generalized ODEs.

Proposition 7.5. Let F : O → X be a function. If F is a continuous function, then the

autonomous generalized ODE (7.2) and the autonomous ODE (7.7) coincide.

Proof. Suppose x : [σ, η] → X satisfies the autonomous generalized ODE (7.2). Then, by

Lemma 7.2, x is a continuous function on [σ, η]. Thus, the function [σ, η] 3 ξ 7→ F (x(ξ))

is a continuous function on [σ, η] and, therefore, the Kurzweil integral∫ ξαD[F (x(τ))t] =

(K)∫ ξαF (x(s))ds and the Riemann integral (R)

∫ ξαF (x(s))ds coincide (i.e are equal). Then

x(ξ) = x(σ) + (K)

∫ ξ

σ

F (x(s))ds = x(σ) + (R)

∫ ξ

σ

F (x(s))ds, ξ ∈ [σ, η], (7.8)


that is, x satisfies the autonomous ODE (7.7).

Analogously, if x : [σ, η]→ X satisfies the autonomous ODE (7.7), by the equality (7.8),

x satisfies the autonomous generalized ODE (7.2).

Proposition 7.6. Let H ∈ F(Ω, h, ω) be such that

H(x, t) = F (x)t, (7.9)

for all (x, t) ∈ Ω, where F : O → X. Then

‖F (x)− F (y)‖ ≤ Cω(‖x− y‖), (7.10)

for every x, y ∈ O, with C = |h(β)−h(α)|β−α . In particular, F is a continuous function.

Proof. let x, y ∈ O. Then, by condition (7.6) from Definition 7.3, we have

‖F (x)(β − α)− F (y)(β − α)‖ = ‖H(x, β)−H(x, α)−H(y, β) +H(y, α)‖≤ (h(β)− h(α))ω(‖x− y‖),

that is,

‖F (x)− F (y)‖ ≤ Cω(‖x− y‖),

where C = h(β)−h(α)β−α . Since Cω(‖x− y‖)→ 0 as x→ y, we get ‖F (x)− F (y)‖ → 0 as x→ y.

Thus F is a continuous function. The result follows now from Proposition 7.5.

Corollary 7.7. Let H ∈ F(Ω, h, ω) be such that

H(x, t) = F (x)t, (7.11)

for all (x, t) ∈ Ω, where F : O → X. Then the autonomous generalized ODE (7.2) and the

autonomous ODE (7.7) coincide.

Proof. The result follows from Propositions 7.5 and 7.6.

7.2 Correspondence between F(Ω, h, ω) and F(Ω, h, ω)

In this section, we prove that if H ∈ F(Ω, h, ω) with H(x, t) = F (x)t, then the au-

tonomous generalized ODE (7.2) and the autonomous ODE (7.7) are coincide. Also, we will

prove that the class F(Ω, h, ω) and the class F(Ω, h, ω) coincide.

The next definition is a slightly modified version of [44, Definition 1.7] (which is concerned

with the special case ω(t) = t)

7.2 Correspondence between F(Ω, h, ω) and F(Ω, h, ω) 123

Definition 7.8. A function H : Ω → X belongs to the class F(Ω, h, ω), if the integralβ∫α

DH(z(τ), t) exists for all z ∈ G([α, β], O) and the following conditions

∥∥∥∥∥∥s2∫s1

DH(z(τ), t)

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)| (7.12)

and ∥∥∥∥∥∥s2∫s1

D[H(z(τ), t)−H(x(τ), t)]

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)|ω(‖z − w‖∞) (7.13)

hold, for all z, x ∈ G([α, β], O) and all s2, s1 ∈ [α, β].

Note that Proposition 1.18 ensures us the existence of the integrals in (7.12) and (7.13).

The following statement is a slightly modified version of [44, Proposition 1.8] (which is

concerned with the special case ω(t) = t). The proof from [44] can be carried over without

any changes and we reproduce it here.

Proposition 7.9. F(Ω, h, ω) ⊂ F(Ω, h, ω).

Proof. Suppose H ∈ F(Ω, h, ω) and let z, w ∈ G([α, β], O) be given. By Proposition 1.18

and Lemma 1.15, we have (7.12). Thus it remains to prove that (7.13) also holds.

By the existence of the integrals∫ βαDH(z(τ), s) and

∫ βαDH(w(τ), s), and by the inte-

grability on subintervals, given s1, s2 ∈ [α, β], with s1 < s2, and given ε > 0, there is a gauge

δ of [s1, s2] such that, for every δ-fine tagged division D = (τi, [ti−1, ti]) of [s1, s2], we have∥∥∥∥∥∥∫ s2

s1

DH(z(τ), t)−|D|∑i=1

[H(z(τi), ti)−H(z(τi), ti−1)]

∥∥∥∥∥∥ < ε

and ∥∥∥∥∥∥∫ s2

s1

DH(w(τ), t)−|D|∑i=1

[H(w(τi), ti)−H(w(τi), ti−1)]

∥∥∥∥∥∥ < ε.


Thus, ∥∥∥∥∫ s2

s1

[DH(z(τ), t)−DH(w(τ), t)]

∥∥∥∥≤

∥∥∥∥∥∥∫ s2

s1

DH(z(τ), t)−|D|∑i=1

[H(z(τi), ti)−H(z(τi), ti−1)]

∥∥∥∥∥∥+

∥∥∥∥∥∥∫ s2

s1

DH(w(τ), t)−|D|∑i=1

[H(w(τi), ti)−H(w(τi), ti−1)]

∥∥∥∥∥∥+

|D|∑i=1

‖H(z(τi), ti)−H(z(τi), ti−1)−H(w(τi), ti) +H(w(τi), ti−1)‖

< 2ε+

|D|∑i=1

ω(‖z(τi)− w(τi)‖)[h(ti)− h(ti−1)] ≤ 2ε+ ω(‖z − w‖∞)[h(s2)− h(s1)].

Hence, since ε > 0 is arbitrarily small, we obtain∥∥∥∥∫ s2

s1

[DH(z(τ), t)−DH(w(τ), t)]

∥∥∥∥ ≤ ω(‖z − w‖∞)[h(s2)− h(s1)]

which completes the proof.

Proposition 7.10. Let H ∈ F(Ω, h, ω) be such that

H(x, t) = F (x)t, (7.14)

for all (x, t) ∈ Ω, where F : O → X. Then

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖), (7.15)

for every d, b ∈ O, with K = |h(β)−h(α)|β−α . In particular, F is a continuous function.

Proof. Let d, b ∈ O. Define the following functions

z : [α, β]→ O (7.16)

s 7→ z(s) = d,

and

w : [α, β]→ O (7.17)

s 7→ w(s) = b.

7.2 Correspondence between F(Ω, h, ω) and F(Ω, h, ω) 125

Notice that z, w ∈ G([α, β], O). Thus, by condition (7.13) from Definition 7.8, we have

∥∥∥∥∥∥β∫α

D[H(z(τ), t)−H(w(τ), t)]

∥∥∥∥∥∥ ≤ |h(β)− h(α)|ω(‖z − w‖∞)︸︷︷︸=ω(‖b−d‖)

. (7.18)

We assert that

β∫α

DH(z(τ), t) = F (d)(β − α) and

β∫α

DH(w(τ), t) = F (b)(β − α). (7.19)

We will only prove the first equality holds. The second equality follows analogously. Let

U(τ, t) = H(z(τ), t), for all (τ, t) ∈ [α, β]× [α, β]. Notice that

U(τ, t) = H(z(τ), t)

(7.16)

↓= H(d, t)

(7.14)

↓= F (d)t,

that is,

U(τ, t) = F (d)t, for every (τ, t) ∈ [α, β]× [α, β]. (7.20)

Let D =(τk, [tk−1, tk]

)be an arbitrary tagged division of the interval [α, β]. Then,

S(U,D) =

|D|∑k=1

[U(τk, tk)− U(τk, tk−1)]

(7.20)

↓=

|D|∑k=1

[F (d)tk − F (d)tk−1]︸︷︷︸telescopic sum

= F (d)t|D| − F (d)t0 = F (d)(t|D| − t0) = F (d)(β − α).

Hence, we obtainβ∫α

DH(z(τ), t) = F (d)(β − α),

which completes the proof of the assertion.

By (7.18) and our (7.19), we obtain

‖F (b)(β − α)− F (d)(β − α)‖ ≤ |h(β)− h(α)|ω(‖b− d‖),

that is,

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖),

where K = |h(β)−h(α)|β−α . This completes the proof of the proposition.


As an immediate consequence of Proposition 7.10, we have the following result.

Corollary 7.11. Let H ∈ F(Ω, h, ω) be such that

H(x, t) = F (x)t, (7.21)



Proof. The proof is a consequence of Propositions 7.5 and 7.10.

Now, let us prove the inclusion F(Ω, h, ω) ⊂ F(Ω, h, ω).

Proposition 7.12. F(Ω, h, ω) ⊂ F(Ω, h, ω).

Proof. Suppose H ∈ F(Ω, h, ω). Let x, y ∈ O and s2, s1 ∈ [α, β] be given. Define the

following functions

z : [α, β]→ O (7.22)

s 7→ z(s) = x,

and

w : [α, β]→ O (7.23)

s 7→ w(s) = y.

Using the same arguments as in the proof of Theorem 7.10, we obtain

s2∫s1

DH(z(τ), t) = H(x, s2)−H(x, s1) and

s2∫s1

DH(w(τ), t) = H(y, s2)−H(y, s1).

Thus, since H ∈ F(Ω, h, ω), we get

‖H(x, s2)−H(x, s2)‖ =

∥∥∥∥∫ s2

s1

DH(z(τ), t)

∥∥∥∥ ≤ |h(s2)− h(s1)|

and

‖H(y, s2)−H(y, s1)−H(x, s2) +H(x, s2)‖ =

∥∥∥∥∫ s2

s1

DH(w(τ), t)−∫ s2

s1

DH(z(τ), t)

∥∥∥∥≤ |h(s2)− h(s1)|ω(‖w − z‖∞)

= |h(s2)− h(s1)|ω(‖y − x‖),

7.3 The classes F(Ω∞, h, ω) and F(Ω∞, h, ω, E) 127

that is, H ∈ F(Ω, h, ω), obtaining the desired result.

Corollary 7.13. The class F(Ω, h, ω) and the class F(Ω, h, ω) coincide.

Proof. The result is a consequence of Propositions 7.9 and 7.12.

7.3 The classes F(Ω∞, h, ω) and F(Ω∞, h, ω, E)

In this section, let us show that if H ∈ F(Ω∞, h, ω) with H(x, t) = F (x)t, then the

autonomous generalized ODE (7.2) and autonomous ODE (7.7) are coincide. Moreover, we

will prove a surprising consequence when H ∈ F(Ω∞, h, ω, E).

Let us assume that Ω∞ = O × [t0,+∞), where O ⊂ Rn is open and t0 ∈ R.

We recall the reader that G([t0,+∞),Rn) is the vector space of functions ϕ : [t0,+∞)→Rn such that ϕ|[u,v] belongs to the space G([u, v],Rn) for all [u, v] ⊂ [t0,+∞). For every

compact interval [u, v] ⊂ [t0,+∞), we can define a seminorm on G([t0,+∞),Rn)

‖ · ‖∞,[u,v] : G([t0,+∞),Rn)→ R

by

‖ψ‖∞,[u,v] := supt∈[u,v]

∣∣ψ|[u,v](t)∣∣ = sup

t∈[u,v]

|ψ(t)| ,

for all ψ ∈ G([t0,+∞),Rn). The topology induced by the family of seminorms‖ · ‖∞,K

K∈Γ

on G([t0,+∞),Rn), where Γ := [u, v] : [u, v] ⊂ [t0,+∞) is called the topology of locally

uniform convergence on G([t0,+∞),Rn). We will use the notation x ∈ G([t0,+∞), O) for a

function x ∈ G([t0,+∞),Rn) such that x(s) ∈ O, for all s ∈ [t0,+∞).

Assume that h : [t0,+∞) → R is a nondecreasing function defined on [t0,+∞) and

ω : [0,+∞)→ R is a continuous and increasing function with ω(0) = 0.

Definition 7.14. A function H : Ω∞ → X belongs to the class F(Ω∞, h, ω), if the integralv∫u

DH(z(τ), t) exists for every z ∈ G([t0,+∞), O) and, for all [u, v] ⊂ [t0,+∞), we have

∥∥∥∥∥∥s2∫s1

DH(z(τ), t)

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)| (7.24)

and ∥∥∥∥∥∥s2∫s1

D[H(z(τ), t)−H(w(τ), t)]

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)|ω(‖z − w‖∞,[s1,s2]), (7.25)


for all z, w ∈ G([t0,+∞), O) and for all s2, s1 ∈ [t0,+∞).

Remark 7.15. When the function ω is the identity, we write F(Ω∞, h) instead of F(Ω∞, h, ω).

In the sequel, we present a surprising result which follows from the definition of the class

F(Ω∞, h, ω).

Proposition 7.16. Let H ∈ F(Ω∞, h) be such that

H(x, t) = F (x)t, (7.26)

for all (x, t) ∈ Ω∞, where F : O → X. Then

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖), (7.27)

for all d, b ∈ O, where K = |h(t0 + 1)− h(t0)| . In particular, F is a continuous function.

Proof. Let d, b ∈ O. Define the following functions

z : [t0,+∞)→ O (7.28)

s 7→ z(s) = d,

and

w : [t0,+∞)→ O (7.29)

s 7→ w(s) = b.

Notice that z, w ∈ G([t0,+∞), O). Thus, by condition (7.25) of Definition 7.14, we have∥∥∥∥∥∥t0+1∫t0

D[H(z(τ), t)−H(w(τ), t)]

∥∥∥∥∥∥ ≤ |h(t0 + 1)− h(t0)|ω(‖z − w‖∞,[t0,t0+1])︸︷︷︸=ω(‖b−d‖)

. (7.30)

Now, using the same arguments as in the proof of Proposition 7.10, we have

t0+1∫t0

DH(z(τ), t) = F (d)(t0+1−t0) = F (d) and

t0+1∫t0

DH(w(τ), t) = F (b)(t0+1−t0) = F (b).

Then, (7.30) implies

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖),

where K = |h(t0 + 1)− h(t0)| and we complete the proof.

7.3 The classes F(Ω∞, h, ω) and F(Ω∞, h, ω, E) 129

The next result is an immediate consequence of Propositions 7.5 and 7.16.

Corollary 7.17. Let H ∈ F(Ω∞, h, ω) be such that

H(x, t) = F (x)t, (7.31)



Note that the main idea behind the proofs of Propositions 7.10, 7.12 and 7.16 is to define

constant functions z and w. But, now, let us define a class which does not contain such

constant functions and let us show that, even in this class, our result remains true.

Consider the following set

E := ϕ ∈ G([t0,+∞), O) : ϕ is constant .

Definition 7.18. Let h : [t0,+∞) → R be a nondecreasing function defined on [t0,+∞)

and Ω∞ = O × [t0,+∞). A function H : Ω∞ → X belongs to the class F(Ω∞, h, ω, E),

if the integralβ∫α

DH(z(τ), t) exists for every z ∈ G([t0,+∞), O)\E and for every interval

[α, β] ⊂ [t0,+∞), we have ∥∥∥∥∥∥s2∫s1

DH(z(τ), t)

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)| (7.32)

and ∥∥∥∥∥∥s2∫s1

D[H(z(τ), t)−H(w(τ), t)]

∥∥∥∥∥∥ ≤ |h(s2)− h(s1)|ω(‖z − w‖∞,[s1,s2]), (7.33)

for all z, w ∈ G([t0,+∞), O)\E and all s2, s1 ∈ [t0,+∞).

Proposition 7.19. Let H ∈ F(Ω∞, h, E) be such that

H(x, t) = F (x)t, (7.34)

for all (x, t) ∈ Ω∞, where F : O → X. Then

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖), (7.35)

for every d, b ∈ O, with K = |h(t0 + 1)− h(t0)| . In particular, F is a continuous function.

Proof. Let d, b ∈ O. If d = b, then (7.35) holds trivially. We suppose d 6= b.


Define functions z, w : [t0,+∞)→ O by

z(t) =

d, t ∈ [t0, t0 + 1]

b, t ∈ (t0 + 1,+∞)

and

w(t) =

b, t ∈ [t0, t0 + 1]

d, t ∈ (t0 + 1,+∞).

Notice that, z, w ∈ G([t0,+∞), O)\E. Therefore, by condition (7.33) of Definition 7.18, we

obtain∥∥∥∥∥∥t0+1∫t0

D[H(z(τ), t)−H(w(τ), t)]

∥∥∥∥∥∥ ≤ |h(t0 + 1)− h(t0)|ω(‖z − w‖∞,[t0,t0+1])︸︷︷︸=ω(‖b−d‖)

. (7.36)

Now, using the same arguments as in the proof of Proposition 7.10, we have

t0+1∫t0

DH(z(τ), t) = F (d)(t0+1−t0) = F (d) and

t0+1∫t0

DH(w(τ), t) = F (b)(t0+1−t0) = F (b).

Therefore, (7.36) can be transformed into

‖F (b)− F (d)‖ ≤ Kω(‖b− d‖),

with K = |h(t0 + 1)− h(t0)| and the proof is complete.

As an immediate consequence of Proposition 7.19, we have the following result.

Corollary 7.20. Let H ∈ F(Ω∞, h, ω, E) be such that

H(x, t) = F (x)t, (7.37)



Proof. The proof is a consequence of Propositions 7.5 and 7.19.

Bibliography

[1] S. M. Afonso, Equacoes diferenciais funcionais com retardamento e impulsos em tempo

variavel via equacoes diferenciais ordinarias generalizadas, PhD thesis. Universidade De

Sao Paulo, ICMC-USP, 2011.

[2] S. M. Afonso, E. M. Bonotto, M. Federson and L. P. Gimenes, Boundednessof solutions

of retarded functional differential equations with variable impulses via generalized ordi-

nary differential equations, Mathematische Nachrichten, v.285, no 5-6, 545-561, 2012.

[3] S. M. Afonso, E. Bonotto, M. Federson and L. P. Gimenes, Stability of functional

differential equations with variable impulsive perturbations via generalized ordinary

differential equations, Bulletin des Sciences Mathematiques. (Paris, 1885), v. 13, 189-

214, 2013.

[4] E. Akin-Bohner and Y. Raffoul, Boundedness in functional dynamic equations on time

scales, Advances in Difference Equations, p. 1-18, 2006.

[5] Z. Artstein, Topological dynamics of an ordinary differential equations and Kurzweil

equations, J. Differential Equations, 23, 224-243, 1977.

[6] S. S. Bellale and B. C. Dhage, Abstract measure integro-differential equations, Global

J. Math. Anal. 1, 91-108, 2007.

[7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with

Applications, Birkhauser, Boston, 2001.

[8] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales,

Birkhauser, Boston, 2003.

[9] D. N. Chate, B. C. Dhage and S.K. Ntouyas, Abstract measure differential equations,

Dynam, System Appl. 13, No. 1, 1005-1017, 2004.

131

132 Bibliography

[10] P. C. Das and R. R. Sharma, Existence and stability of measure differential equations,

Czech. Math. Journal. 22(97), 145-158, 1972.

[11] S. G. Deo and S. R. Joshi, On abstract measure delay differential equations, An. Stiint.

Univ. Al I. Cuza Iasi XXVI, 327-335, 1980.

[12] S. G. Deo and S. G. Pandit, Differential Systems Involving Impulses., Springer-Verlag,

Berlin-New York, 954, 1982.

[13] B. C. Dhage, On system of abstract measure integro-differential inequalities and appli-

cations, Bull. Inst. Math. Acad. Sinica 18, 65-75, 1989.

[14] B. C. Dhage and J. R. Graef, On stability of abstract measure delay integro-differential

equations, Dynam. Systems Appl. 19, 323-334, 2010.

[15] J. Diblık, M. Ruzickova and B. Vaclavıkova, Bounded solutions of dynamic equations on

time scales, International Journal of Difference Equations, v.3 number 1, 61-69, 2008.

[16] M. Federson, R. Grau and J. G. Mesquita, Prolongation of solutions of measure differ-

ential equations and dynamical equations on time scales, submitted.

[17] M. Federson, R. Grau, J. G. Mesquita and E.Toon, Boundedness of solutions of measure

differential equations and dynamic equations on time scales, submitted.

[18] M. Federson, R. Grau, J. G. Mesquita and E.Toon, Lyapunov stability for measure

differential equations and dynamic equations on time scales, submitted.

[19] M. Federson, R. Grau, J. G. Mesquita and E.Toon, Remark on autonomous generalized

ordinary differential equations, submitted.

[20] M. Federson, J. G. Mesquita and A. Slavık, Measure functional differential equations

and functional dynamic equations on time scales, J. Differential Equations. 252, no. 6,

3816-3847, 2012.

[21] M. Federson, J. G. Mesquita and A. Slavık, Basic results for functional differential and

dynamic equations involving impulses, Math. Nachr. 286, no. 2-3, 181-204, 2013.

[22] M. Federson, J. G. Mesquita and E. Toon, Lyapunov theorems for measure functional

differential equations via kurzweil equations, Mathematische Nachrichten. v. 218, p.

1487-1511, 2015.

[23] M. Federson and S. Schwabik, Generalized ODEs approach to impulsive retarded dif-

ferential equations, Diff. integral equations. 19(11), 1201-1234, 2006.

Bibliography 133

[24] M. Federson and S. Schwabik, A new approach to impulsive retarded differential equa-

tions: stability results, Functional Differential Equations. 16(4), 583-607, 2009.

[25] D. Frankova, Regulated functions. Math. Bohem. 116, no. 1, 20-59, 1991.

[26] R. Henstock, Lectures in the Theory of Integration., World Scientific, Singapore, 1998.

[27] S. Hilger, Ein Maβkettenkalkul mit Anwendung auf zentrum smanning faltig keiten,

PhD thesis. Universitat Wurzburg, 1988.

[28] Y. Hino, S. Murakami and T. Naito, Functional differential Equations with Infinite

Delay, Springer- Verlag, 1991.

[29] C. S. Honig, Volterra Stieltjes Integral Equations., North Holland Publ. Comp. Amster-

dam, 1975.

[30] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on

a parameter, Czechoslovak Math. J. 7(82), 418-448, 1957.

[31] J. Kurzweil, Generalized ordinary differential equations, Czechecoslovak Math. J. 8(83),

360-388, 1958.

[32] S. Leela, Stability of measure differential equations, Pacific Journal of Mathematics.

Vol. 55. No. 2, 1974.

[33] F. Oliva and Z. Vorel, Functional equations and generalized ordinary differential equa-

tions, Bol. Soc. Mat. Mexicana, 11, 40-46, 1966.

[34] S. G. Pandit, Differential systems with impulsive perturbations, Pacific J. Math 86,

553-560, 1980.

[35] A. Peterson and B. Thompson, Henstock-Kurzweil delta and nabla integrals, J. Math.

Anal. Appl. 323, 162-178, 2006.

[36] A. Peterson and C. Tisdell, Boundedness and uniqueness of solutions to dynamic equa-

tions on time scales, Journal of Difference Equations and Applications, v.10, 13-15,

1295-1306, 2004.

[37] W. W. Schmaedeke, Optimal control theory for nonlinear vector differential equations

containing measures, SIAM J., Ser. A, Control 3, 231–280, 1965.

[38] S. Schwabik, Generalized Ordinary Differential Equations, Series in Real Analysis, vol.

5, World Scientific, Singapore, 1992.

134 Bibliography

[39] P. C. Das, R. R. Sharma, On optimal controls for measure delay-differential equations,

J. SIAM Control 9, 43-61, 1971.

[40] R. R. Sharma, An abstract measure differential equations, Proc. Amer. Math. Soc., 32,

503-510, 1972.

[41] R. R. Sharma, A measure differential inequality with applications, Proc. Amer. Math.

Soc., 48, 87-97, 1975.

[42] A. Slavık, Dynamic equations on time scales and generalized ordinary diffrerential equa-

tions, J. Math. Anal. Appl., 385, 534-550, 2012.

[43] I. M. Stamova, Boundedness of impulsive functional differential equations with variable

impulsive perturbations, Bull Austral Math. Soc., 77, 331-345, 2008.

[44] E. Toon, Equacoes diferenciais ordinarias generalizadas e aplicacoes as equacoes difer-

enciais classicas, PhD thesis. Universidade De Sao Paulo, ICMC-USP. 2012.

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Rogelio Grau Acuña - USP...Acknowledgment First of all, I want to thank God for giving me the wisdom and strength to make this a reality project. I would like to thank my advisor,