Measure functional differential equations and impulsive ... Measure functional differential equations

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Measure functional differential equations and impulsive functional dynamic equations on time

scales

Jaqueline Godoy Mesquita

!

Measure functional differential equations and impulsive

functional dynamic equations on time scales

Jaqueline Godoy Mesquita!

Advisor: Profa. Dra. Mrcia Cristina A. B. Federson! Co-advisor: Prof. Dr. Antonn Slavk

Doctoral dissertation submitted to the Instituto de Cincias Matemticas e de Computao - ICMC-USP, in partial fulfillment of the requirements for the degree of the Doctorate Program in Mathematics. EXAMINATION BOARD PRESENTATION COPY.

USP So Carlos July, 2012!

!

SERVIO DE PS-GRADUAO DO ICMC-USP Data de Depsito: Assinatura:______________________________

Ficha catalogrfica elaborada pela Biblioteca Prof. Achille Bassi e Seo Tcnica de Informtica, ICMC/USP,

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

G578mGodoy Mesquita, Jaqueline Measure functional differential equations andimpulsive functional dynamic equations on timescales / Jaqueline Godoy Mesquita; orientadoraMrcia Braz Federson; co-orientador Antonin Slavik. -- So Carlos, 2012. 216 p.

Tese (Doutorado - Programa de Ps-Graduao emMatemtica) -- Instituto de Cincias Matemticas ede Computao, Universidade de So Paulo, 2012.

1. Functional dynamic equations on time scales.2. Measure functional differential equations. 3.Generalized ODEs. 4. Impulsive equations. 5.Averaging. I. Braz Federson, Mrcia, orient. II.Slavik, Antonin, co-orient. III. Ttulo.

"There is no branch of mathemat-ics, however abstract, which maynot some day be applied to phe-nomena of the real world".(Nikolai Lobachevsky)

"Gosto de ser gente porque, ina-cabado, sei que sou um ser condi-cionado mas, consciente do ina-cabamento, sei que posso ir almdele. Esta a diferena profundaentre o ser condicionado e o serdeterminado".(Paulo Freire)

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Acknowledgment

First of all, I would like to thank Sri Sathya Sai Baba for everything in my life andfor giving me all the necessary force to continue my journey.

I am really grateful to Lus Ges Mesquita, my husband, who always believed in me.He is the best partner of the world. He is an essential and fundamental part of my life. Ido not have words to express his meaning in my life. He always gave me all the necessaryforce and support to deal with all the difficulties in my life. He always encourages me toimprove my knowledge and my career as a mathematician. He is the most responsible forall my professional success until now. I would like to thank him for being part of my lifeand for sharing with me a lot of unforgettable moments. Lus, I love you very much!

I am grateful to my dear parents, Maria Aparecida Bezerra and Gilson Godoy, forall their trust, encouragement and love. Without their support, I would not be here.They were my pillars during all these years, maintaining me strong to overcome all thedifficulties of my life. I do not have words to express their importance to me and howproud and glad I am to be their daughter.

I am grateful to my sister Cynthia for being one of my best friends and an essentialpart of my life. Also, I am grateful to my brothers, Victor, Louise and Vinicius, for theiraffection and friendship.

I would like to thank Elaine Godoy, Mrcio Zuany, Cleide Bezerra and Marcos Borgesfor all friendship, love, trust and encouragement.

I am grateful to my parents-in-law, Ana Maria Medeiros Ges and Alcione Mesquita,and to my brothers-in-law, Paulo Mesquita and Pedro Mesquita, for being so special andcaring with me. Their affection and trust were fundamental to bring me till here.

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I would like to thank to my dear friends: Pedro Lima, Renato Alejandro, IsmaelSobrinho, Laura Lobato, Roberta Pacheco, Ton Fortes, Paulo Mendes, Suzete Afonso,Lvia Gomes, Vinicus Fac, Patrcia Tacuri, Matheus Bortolan, Luana Mesquita andEduard Toon. Without their support, I would not be here. All of them shared with me alot of unforgettable moments. I am really glad to have all of them in my life. Thank youto be part of my life!

I would like to thank my advisor, professor Mrcia Federson, for all the opportunitiesthat she provided me during my PhD course, encouraging me to improve my career as aresearcher, showing a lot of opportunities. Under her supervision and knowledge, I wasable to improve myself as a mathematician, developing several works in collaboration withreputed researchers. I am really grateful to her for this and for her friendship.

I am grateful to my co-advisor, professor Antonn Slavk, who gave me all the necessarysupport during my stay in Prague. He taught me a lot and shared with me his knowledge.This was really important to me. Also, he shared with me a lot of good moments in Prague.I appreciated a lot his company during our trips and meals. I am really grateful for hisfriendship, patience and trust.

I would like to thank professor Milan Tvrd, who was part of my family in Prague.Without him, I would not be able to finish my sandwich doctorate. He helped me a lotwith my visa, and I do not have words to describe how I am grateful to him. Also, I amgrateful for his company, patience and advises which were really important to me.

I would like to thank professor Maria do Carmo Carbinatto for her friendship duringmy doctorate. She is much more than a professor to me, she is also a friend.

I am grateful to professor Alexandre Nolasco de Carvalho for everything. He was myfirst professor at the Universidade de So Paulo and he always encourages me. He taughtme a lot during my master and PhD courses. I admire him as a mathematician and heinspires me to improve my knowledge and my career as a researcher.

I appreciate the opportunity to learn a lot with the professors Martin Bohner, IstvnGyri, Plcido Tboas, Miguel Frasson, Everaldo Bonotto and Dan Frankov during mydoctorate and to collaborate with them. It was a pleasure to me.

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

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Abstract

The aim of this work is to investigate and develop the theory of impulsive functionaldynamic equations on time scales. We prove that these equations represent a special caseof impulsive measure functional differential equations. Moreover, we present a relationbetween these equations and measure functional differential equations and, also, a corre-spondence between them and generalized ordinary differential equations. Also, we clarifythe relation between measure functional differential equations and functional dynamicequations on time scales.

We obtain results on the existence and uniqueness of solutions, continuous dependenceon parameters, non-periodic and periodic averaging principles and stability results forall these types of equations. Moreover, we prove some properties concerning regulatedfunctions and equiregulated sets in a Banach space which were essential to our purposes.

The new results presented in this work are contained in 7 papers, two of which havealready been published and one accepted. See [16], [32], [34], [36], [37], [38] and [84].

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Resumo

O objetivo deste trabalho investigar e desenvolver a teoria de equaes dinmicasfuncionais impulsivas em escalas temporais. Mostramos que estas equaes representamum caso especial de equaes diferenciais funcionais em medida impulsivas. Tambm,apresentamos uma relao entre estas equaes e as equaes diferenciais funcionais emmedida e, ainda, mostramos uma relao entre elas e as equaes diferenciais ordinriasgeneralizadas. Relacionamos, tambm, as equaes diferenciais funcionais em medida eas equaes dinmicas funcionais em escalas temporais.

Obtemos resultados sobre existncia e unicidade de solues, dependncia contnua,mtodo da mdia peridico e no-peridico bem como resultados de estabilidade paratodos os tipos de equaes descritos anteriormente. Tambm, provamos algumas pro-priedades relativas s funes regradas e aos conjuntos equiregrados em espaos de Ba-nach, que foram esssenciais para os nossos propsitos.

Os resultados novos apresentados neste trabalho esto contidos em 7 artigos, dos quaisdois j foram publicados e um aceito. Veja [16], [32], [34], [36], [37], [38] e [84].

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Contents

Introduction 1

1 Regulated functions 9

1.1 Characterizations of compact sets in G([a, b],Rn) . . . . . . . . . . . . . . 14

1.2 Characterizations of compact sets in G([a, b], X) . . . . . . . . . . . . . . . 17

2 Generalized ODEs 23

2.1 The Kurzweil integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Generalized ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Dynamic equations on time scales 33

3.1 Time scales calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Darboux and Riemann delta-integrals . . . . . . . . . . . . . . . . . . . . . 40

3.4 Kurzweil-Henstock delta integrals . . . . . . . . . . . . . . . . . . . . . . . 48

4 Correspondences between equations 51

4.1 Measure FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Measure FDEs and generalized ODEs . . . . . . . . . . . . . . . . . . . . . 53

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xii CONTENTS

4.3 Measure FDEs and functional dynamic equations on time scales . . . . . . 67

4.4 Impulsive measure FDEs and measure FDEs . . . . . . . . . . . . . . . . . 75

4.5 Impulsive functional differential and dynamic equations on time scales . . . 82

5 Existence and uniq