Integral equations in the sense of Kurzweil integral and applications. Rafael dos Santos Marques

Transcription

1 Integrl equtions in the sense of Kurzweil integrl nd pplictions Rfel dos Sntos Mrques

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3 SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP Dt de Depósito: Assintur: Rfel dos Sntos Mrques Integrl equtions in the sense of Kurzweil integrl nd pplictions Mster disserttion submitted to the Instituto de Ciêncis Mtemátics e de Computção - ICMC- USP, in prtil fulfillment of the requirements for the degree of the Mster Progrm in Mthemtics. EXAMINATION BOARD PRESENTATION COPY Concentrtion Are: Mthemtics Advisor: Prof. Dr. Márci Cristin Anderson Brz Federson USP São Crlos June 2016

4 Fich ctlográfic elbord pel Bibliotec Prof. Achille Bssi e Seção Técnic de Informátic, ICMC/USP, com os ddos fornecidos pelo() utor() M357i Mrques, Rfel dos Sntos Integrl equtions in the sense of Kurzweil integrl nd pplictions / Rfel dos Sntos Mrques; orientdor Márci Cristin Anderson Brz Federson. -- São Crlos, p. Dissertção (Mestrdo - Progrm de Pós-Grdução em Mtemátic) -- Instituto de Ciêncis Mtemátics e de Computção, Universidde de São Pulo, Integrção não bsolut. 2. Equções diferenciis ordináris generlizds. 3. Integrl Kurzweil-Henstock. 4. Equções integris de Volterr. I. Federson, Márci Cristin Anderson Brz, orient. II. Título.

5 Rfel dos Sntos Mrques Equções integris no sentido d integrl de Kurzweil e plicções Dissertção presentd o Instituto de Ciêncis Mtemátics e de Computção - ICMC-USP, como prte dos requisites pr obtenção do título de Mestre em Ciêncis Mtemátic. EXEMPLAR DE DEFESA Áre de Concentrção: Mtemátic Orientdor: Prof. Dr. Márci Cristin Anderson Brz Federson USP São Crlos Junho 2016

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7 Acknowledgment I m grteful for my prents. Since erly in childhood, they motivted me to study nd lwys be the best I cn be. Thnk you for supporting me throughout ll these yers nd, regrdless of my choices, for being lwys ble to sy I love you in the first plce. I m lso grteful for my sister, for her friendship nd for lwys being close to me. I m grteful for ll the techers I met in every stge of my life. I believe the work of ech one of them ws essentil to help me find out my pth in Mthemtics. I would like to thnk professor Mrcello Medeiros, my first dvisor while I ws n engineering student, for shring his knowledge with me nd for ll the fternoons we spent tlking bout cdemic stuff nd life during the time I decided to chnge the mjor course. I m very grteful for my dvisor, professor Márci Federson, for her importnt contribution to my formtion, for ll the opportunities tht she provided me nd for lwys believing in me throughout these yers. Thnk you very much for your ptience nd friendship. I m grteful for ll the people I met during my visit to Virgini Tech, in Blcksburg, USA. Everyone ws very nice to me nd mde me feel welcome. I would like to thnk professors Terry Herdmn, John Burns nd Lizette Zietsmn for their ptience, compny nd trust. I would lso like to thnk ll my friends for ll the time we spent together, the lughter, compnionship nd tlks. Whether we met before or fter I went to college, on the mt, 5

8 6 becuse of theter or poetry... You ll shre specil plce in my hert, thnk you very much. Finlly, I m grteful to FAPESP for the finncil support.

9 Abstrct Being prt of reserch group on functionl differentil equtions (FDEs, for short), due to my formtion in non-bsolute integrtion theory nd becuse certin kinds of FDEs cn be expressed s integrl equtions, I ws motivted to investigte the ltter. The purpose of this work, therefore, is to develop the theory of integrl equtions, when the integrls involved re in the sense of Kurzweil- Henstock or Kurzweil-Henstock-Stieltjes, through the correspondence between solutions of integrl equtions nd solutions of generlized ordinry differentil equtions (we write generlized ODEs, for short). In order to be ble to obtin results for integrl equtions, we propose extensions of both the Kurzweil integrl nd the generlized ODEs (found in [36]). We develop the fundmentl properties of this new generlized ODE, such s existence nd uniqueness of solutions results, nd we propose stbility concepts for the solutions of our new clss of equtions. We, then, pply these results to clss of nonliner Volterr integrl equtions of the second kind. Finlly, we consider model of popultion growth (found in [4]) tht cn be expressed s n integrl eqution tht belongs to this clss of nonliner Volterr integrl equtions. Keywords: integrl equtions, nonbsolute integrtion, generlized ordinry differentil equtions, Kurzweil-Henstock integrl. 7

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11 Resumo Sendo prte de um grupo de pesquis em equções diferenciis funcionis (escrevemos EDFs), por cus de minh formção em teori de integrção não bsolut e porque certos tipos de EDFs podem ser escrits como equções integris, decidi estudr esse último tipo de equções. O objetivo desse trblho, portnto, é desenvolver teori de equções integris, qundo s integris envolvids são no sentido de Kurzweil-Henstock ou Kurzweil-Henstock-Stieltjes, trvés d correspondênci entre soluções de equções integris e soluções de equções diferenciis ordináris generlizds (ou EDOs generlizds). A fim de obter resultdos pr ests equções integris, propomos extensões de mbs integrl de Kurzweil e s EDOs generlizds (encontrds em [36]). Desenvolvemos proprieddes fundmentis dess nov EDO generlizd, como resultdos de existênci e unicidde de solução, e propomos conceitos de estbilidde pr s soluções de noss nov clsse de equções. Nós, então, plicmos esses resultdos um clsse de equções integris de Volterr não lineres de segund espécie. Finlmente, considermos um modelo de crescimento de populções (encontrdo em [4]) que pode ser escrito como um equção integrl pertencente ess clsse de equções integris de Volterr não lineres. Plvrs-chve: equções integris, integrção não bsolut, equções diferenciis ordináris generlizds, integrl de Kurzweil-Henstock. 9

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13 Contents 1 Generlized Ordinry Differentil Equtions The Kurzweil integrl Bsic concepts bout generlized ODEs Locl existence nd uniqueness of solution result Qulittive Properties of Solutions of Generlized ODEs Stbility concepts for generlized ODEs Lypunov-type theorems for generlized ODEs Volterr Integrl Equtions Integrl equtions s generlized ODEs A globl existence nd uniqueness result Stbility concepts for Volterr integrl equtions Applictions Existence nd uniqueness of solutions Stbility results

14 Appendices 89 A The Kurzweil Integrl 91 B Generlized Ordinry Differentil Equtions 95

15 Introduction Mny problems which pper in Physics, Engineering, Biology, Economy nd other sciences cn be described s integrl equtions. On science s relm some lws re written s differentil equtions s, for instnce, on Newton s dynmics theory. Such differentil equtions denote n instntneous blnce or locl equilibrium, whose solutions re ssocited to dequte initil or boundry conditions obtined from experiments or observtions. Thus nturl phenomen cn be understood when compred with experimentl dt nd process in science cn be used in order to coincide with clculted dt. However, when nturl lws re not sufficiently estblished, s n instntneous blnce or locl equilibrium nd, therefore, cnnot be regrded s differentil eqution, or when the experimentl methods or observtion conditions cnnot be nlysed seprtely from the object being studied, so tht it cnnot be expressed with initil or boundry condition, then the problem is usully modelled s n integrl eqution. Moreover, problems with singulr boundry conditions nd some types of differentil equtions cn be more esily studied when written s integrl equtions. In short, problems formulted s integrl equtions stisfy, utomticlly, initil or boundry conditions within its integrl formultion. Furthermore, presenting the solution s n integrl eqution is often more useful for pproximtions. Integrl equtions hve been studied in Kurzweil-Henstock s sense, which coincide with the Perron integrl nd which integrtes highly oscillting functions (i.e. of unbounded 13

16 14 CONTENTS vrition) s well s functions with mny discontinuities. The ppers [1], [10], [11], [12], [13], [14], [15], [16], [17], [27] nd [31] re some references in which the uthors investigte different types of differentil equtions using their integrl forms, where the integrl is considered in the Kurzweil-Henstock s sense, nd obtin results through the correspondence between the solutions of these differentil equtions nd generlized ODEs. As n exmple, we cn cite nonliner Volterr integrl eqution, which rises in the study of popultion growth, considered in [4]. We ssume the growth rte, g : [0, ) R, depends only on the size of the popultion nd the probbility of deth, P : [0, ) [0, 1], depends only on the ge of the individuls. These ssumptions led to Volterr integrl eqution of the form x(t) = f(t) + 0 g(x(t s))p (s)ds, 0 t <, (0.1) where f : [0, ) R represents the number of survivors t time t from the originl popultion f(0). This model is useful to studying popultions fflicted with disese, provided people who recover from the disese re not immune to it (thus, the spred of the disese depends only on the number of people with the disese) nd there is no incubtion period (thus, no dely terms pper in the integrl). The ppers [7] nd [8] derive nd use this model to study the spred of gonorrhe. Finlly, this model is lso used to describe economic models, where the solution x describes the totl vlue of cpitl in the instnt t, the production of new cpitl within the economy depends only on x(t) nd the rte of production is g(x(t)), P (t) is the probbility of survivl of unit of equipment to ge t lest t (considering this equipment retins its full vlue until brekdown) nd f(t) mens the mount of cpitl left from the originl investment f(0).

17 CONTENTS 15 Note tht the model (0.1) is prticulr cse of the nonliner Volterr integrl eqution of the second kind x(t) = f(t) + 0 K(t, x(s), s)du(s), 0 t <, (0.2) where the integrl is considered in the Kurzweil-Henstock-Stieltjes sense. Thus, in order to pply the theory of generlized ordinry differentil equtions (generlized ODEs, for short) to integrl equtions of the form (0.2), it ws necessry to crete n extension of the Kurzweil integrl nd of the stndrd generlized ODE (both found in [36]). The first chpter of this work is dedicted to defining this new integrl nd the lrger clss of generlized ODE, proving fundmentl properties expected from these mthemticl objects, including locl existence nd uniqueness of solutions result. The second chpter proposes concepts of stbility for the solutions of our generlized ODE nd we prove Lypunov-type theorems for these stbility concepts. The pper [15] ws the reference for the min results of this chpter. In Chpter 3, we investigte integrl equtions of the form (0.2) nd show correspondence result between the solutions of these integrl equtions nd the solutions of the new generlized ODE. We obtin globl existence nd uniqueness of solution result to nonliner integrl equtions bsed on ides from the theory of liner Stieltjes integrl equtions, found in the pper [37]. Finlly, we propose stbility concepts for Volterr integrl equtions using the stbility concepts defined for generlized ODEs in Chpter 2. In the lst chpter, we pply the results obtined to the popultion model (0.1) nd we lso replicte some of the results from [4] ssuming different hypotheses. We lso include two ppendices: Appendix A contins the fundmentl properties of the Kurzweil integrl, considered in [36], tht we use in this work, nd Appendix B contins informtion bout generlized ODEs nd concepts of stbility of their solutions, both found in [36].

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19 Chpter Generlized Ordinry Differentil Equtions 1 In the present text, we propose n extension of the integrl introduced by J. Kurzweil in 1957, see [27]. The im of this extension is to propose better frmework to del with integrl equtions. This frmework is in the sense of the theory of generlized ordinry differentil equtions (generlized ODEs, for short) proposed by J. Kurzweil. As mtter of fct, our extension of the Kurzweil integrl genertes lrger clss of generlized ODEs. This chpter introduces this new integrl (we cll it the K-integrl) nd its min properties. We then use the K-integrl to define lrger clss of generlized ODEs nd we close the chpter with result on the existence nd uniqueness of solution of the new clss of generlized ODEs. 1.1 The Kurzweil integrl At first, let us provide some definitions nd nottion. A guge on [, b] R is ny function δ : [, b] (0, ). 17

20 18 Chpter 1 Generlized Ordinry Differentil Equtions A division of [, b] is ny finite collection of points D = {s 0, s 1,, s D } such tht = s 0 < s 1 < < s D 1 < s D = b, where D denotes the number of subintervls into which [, b] is divided. For ech integer i, the numbers s i re clled nodes of the division D. A tgged division of [, b] is ny finite collection of point-intervl pirs D = {(τ i, [s i 1, s i ]) i = 1,, D } such tht = s 0 < s 1 < < s D 1 < s D = b nd τ i [s i 1, s i ] for every integer i between 1 nd D. For every integer i, the number τ i is clled tg of the intervl [s i 1, s 1 ]. A tgged division D = {τ i, [s i 1, s i ]} D i=1 of [, b] is δ-fine, whenever [s i 1, s i ] (τ i δ(τ i ), τ i + δ(τ i )) for every integer i between 1 nd D. Definition 1.1. Let X be Bnch spce. A function F : [, b] [, b] X is Kurzweil integrble over the intervl [, b], if there is n element I X, denoted by b DF (τ, s), such tht for every ε > 0, there is guge δ on [, b] such tht D [F (τ i, s i ) F (τ i, s i 1 )] I < ε i=1 for ll δ-fine tgged divisions D = {τ i, [s i 1, s i ]} D i=1 of [, b]. We denote the set of ll Kurzweil integrble functions over [, b] by K([, b]). The reder my wnt to consult the Appendix A in the end of this text for the bsic results regrding the Kurzweil integrl nd used in this work. It is importnt to discuss whether the Kurzweil integrl is well-defined or not, which mens verifying tht for ny guge δ on [, b], there is t lest one δ-fine tgged division. This is the content of the following lemm, known s Cousin Lemm, whose proof cn be found in [21], Theorem 4.1. Lemm 1.2. Given guge δ on [, b], there is δ-fine tgged division D of [, b]. By L(X) we men the Bnch spce of ll liner opertors T : X X with norm T = sup x T (x), where the supremum is tken over ll x X with x 1. When

21 1.1 The Kurzweil integrl 19 F : [, b] [, b] X in Definition 1.1 is such tht F (τ, t) = f(τ)g(t), with f : [, b] L(X) nd g : [, b] X, we obtin the well-known Perron-Stieltjes integrl, which is equivlent to the Kurzweil-Henstock-Stieltjes integrl, when the functions f nd g re rel vlued. In prticulr, if g(t) = t for ll t [, b], then we obtin the Perron integrl, lso known s generlized Riemnn integrl. The book [19] is gret reference for the generlized Riemnn integrl when X = R. Reference [36] is detiled tretise of the theory of Generlized Ordinry Differentil Equtions, which we will del with lter. The first chpter of such book is dedicted to studying the Kurzweil integrl, s presented in Definition 1.1, but when X = R n. We lso include the recent book of J. Kurzeil on the sucject, see [28]. A subset H of K([, b]) is sid to be equi-integrble, if for every ε > 0, there is guge δ on [, b] such tht Definition 1.1 is stisfied for every function F in H. Next we use the concept of equi-integrbility to define our extension of the Kurzweil integrl. Definition 1.3. A function F : [, b] [, b] [, b] X is K-integrble over [, b], if for ech t [, b], the function F t : [, b] [, b] X given by F t (τ, s) = F (t, τ, s) is Kurzweil integrble over [, b] nd, moreover, the fmily of functions {F t ; t [, b]} is equi-integrble. We denote the set of ll K-integrble functions over [, b] by K([, b]). Ech F K([, b]) cn be identified with n element F K([, b]) by setting F (t, τ, s) = F (τ, s). Hence the set of Kurzweil integrble functions is contined in the set of K-integrble functions, tht is, K([, b]) K([, b]). However the other inclusion is not vlid, becuse the Kurzweil integrl of K-integrble functions is not well-defined. Indeed, the Kurzweil integrl is defined for functions whose domin is contined in R 2, while the K-integrl is defined for functions whose domin is subset of R 3. The definition bove cn be extended to unbounded intervls, tht is, intervls where t lest one of its endpoints is or. The usul pproch in elementry clculus is to define these types of integrls s limits of integrls over bounded intervls. Insted of deling

22 20 Chpter 1 Generlized Ordinry Differentil Equtions with limit, we impose F (t, τ, s 2 ) F (t, τ, s 1 ) = 0, t / {, }, whenever s 1 =, s 2 = or both. When t {, }, we set F (t, τ, s) = 0 for ll τ, s where F is defined. Thus, it does not mtter the vlue of F (t, τ, s) when τ {, }, becuse only n unbounded intervl cn hve or s tg. The reference [30] contins more detils bout integrtion over unbounded intervls for the cse where F (t, τ, s) = f(τ)s nd f is rel vlued. Also, [24] is gret reference, in Portuguese, for the Henstock-Kurzweil integrl over bounded nd unbounded intervls. In order to simplify the nottion, S(F, t, D) is going to represent the Riemnn sum corresponding to function F : [, b] [, b] [, b] X, point t [, b] nd tgged division D = {τ i, [s i 1, s i ]} D i=1 of [, b]: D S(F, t, D) = [F (t, τ i, s i ) F (t, τ i, s i 1 )]. i=1 Likewise, if J = [t 1, t 2 ] is subset of [, b], then S(F, J, D) = S(F, t 2, D) S(F, t 1, D) = D = [F (t 2, τ i, s i ) F (t 2, τ i, s i 1 ) F (t 1, τ i, s i ) + F (t 1, τ i, s i 1 )]. i=1 As expected, linerity of the integrl is vlid becuse of the following identity: S(c 1 F + c 2 G, t, D) = c 1 S(F, t, D) + c 2 S(G, t, D) for ny Riemnn sum of functions F, G : [, b] [, b] [, b] X, constnts c 1, c 2 R, tgged division D of [, b] nd t [, b]. Now, we discuss integrbility on subintervls for the K-integrl.

23 1.1 The Kurzweil integrl 21 Theorem 1.4. If F K([, b]), then for every [c, d] [, b], we hve F K([c, d]). Proof. A function F : [, b] [, b] [, b] X belongs to K([c, d]), if the Kurzweil integrl over [c, d] of F t : [, b] [, b] X, given by F t (τ, s) = F (t, τ, s), exists for every t [c, d] nd, moreover, the set {F t ; t [c, d]} is equi-integrble. Since F K([, b]), the Kurzweil integrl of F t over [, b] exists for every t [, b] nd, thus, we cn pply Theorem A.3 (see Appendix A) to conclude F t is Kurzweil integrble over [c, d] for ll t [, b]. Hence, F t is Kurzweil integrble over [c, d] for ll t [c, d], becuse [c, d] [, b]. The equi-integrbility of {F t ; t [c, d]} follows from the definition of the K-integrl nd becuse F K([, b]). Therefore F K([c, d]). Most of the proofs of the results of this section follow the sme line of thinking: we need to prove property for the K-integrl, we use its nlogue vilble for the Kurzweil integrl nd finish with n rgument similr to the one used in the end of the proof of Theorem 1.4. Theorem 1.5 (Theorem A.4). Let c (, b). F K([, b]). If F K([, c]) nd F K([c, b]), then The next result, known s Sks-Henstock Lemm, is very useful tool, though technicl in nture. Lemm 1.6 (Theorem A.5). Let F : [, b] [, b] [, b] X be K-integrble over [, b]. Given ɛ > 0, ssume δ is the guge on [, b] such tht b S(F, t, D) D s F (t, τ, s) < ε for every δ-fine tgged division D of [, b] nd t [, b]. If β 1 ξ 1 γ 1 β 2 ξ 2 γ 2 β m ξ m γ m b stisfies ξ j [β j, γ j ] (ξ j δ(ξ j ), ξ j + δ(ξ j ))

24 22 Chpter 1 Generlized Ordinry Differentil Equtions for every integer j between 1 nd m, then t [, b]. m [F (t, ξ j, γ j ) F (t, ξ j, β j ) j=1 γj β j D s F (t, τ, s)] < ε (1.1) The K-integrl contins ll its Cuchy extensions, mening tht it contins its improper integrls. This property is known s Hke-type theorem. Theorem 1.7 (Theorem A.6). Let F : [, b] [, b] [, b] X belong to K([, c]) for ll c [, b) nd suppose there is I = I(t) X, with t [, b], such tht [ c ] lim c b D s F (t, τ, s) F (t, b, c) + F (t, b, b) = I(t). Then F K([, b]) nd b D sf (t, τ, s) = I(t). The reder my wnt to consult the Appendix A for more informtion bout the left endpoint nlogue of Theorem 1.7. Such result cn be crried out with similr proof. Let Ω be n open subset of Bnch spce X nd F : [, b] Ω [, b] X be function. As consequence of Theorem 1.7, we cn prove tht the K-integrl b D sf (t, x(τ), s) exists, whenever x : [, b] Ω is step function. We recll function f : [, b] X is clled regulted, if its one-sided limits exist t every point of the domin. The spce of regulted functions from [, b] to X is denoted by G([, b], X). The book [25] nd the pper [18] re gret references for the theory of regulted functions. Corollry 1.8. Let the function F : [, b] Ω [, b] X be regulted on its third vrible, i.e. F (t, τ, ) is regulted for every t [, b] nd τ Ω, nd x : [, b] Ω be step function. Then the integrl b D sf (t, x(τ), s) exists.

25 1.1 The Kurzweil integrl 23 Proof. Let t [, b]. Then we cn pply Corollry B.5 (see Appendix B) nd conclude the Kurzweil integrl b DF (t, x(τ), s) exists nd b DF (t, x(τ), s) = k [F (t, c j, s j ) F (t, c j, s j 1 +)]+ j=1 + k [F (t, x(s j 1 ), s j 1 +) F (t, x(s j 1 ), s j 1 ) F (t, x(s j ), s j ) + F (t, x(s j ), s j )], (1.2) j=1 where = s 0 < s 1 < < s k = b is division of [, b] such tht x(s) = c j X for ll s (s j 1, s j ) nd the division depends only on the step function x. Note tht the bove limits re well-defined, becuse we ssumed the function F (t, τ, ) is regulted for every t [, b] nd τ Ω. Since t ws chosen rbitrrily in [, b] nd the bove division does not rely on the point t, we cn conclude the K-integrl b D sf (t, x(τ), s) exists. Given F K([, b]), for ech t [, b], the function F t (τ, s) = F (t, τ, s) dmits n indefinite integrl, in Kurzweil s sense, given by ξ [, b] ξ DF t (τ, s) = ξ DF (t, τ, s), (1.3) whose continuity is discussed in [36], Theorem 1.16 (see Theorem A.8 in Appendix A in the sequel) nd Remrk 1.17 from [36]. Lemm 1.9. The indefinite K-integrl, given by ξ [, b] ξ D s F (t, τ, s), is continuous t ξ [, b], if nd only if, the function F tξ : [, b] X given by F tξ (s) = F (t, ξ, s) is continuous t ξ, where t [, b]. Proof. Let ε > 0 be given nd let δ be guge on [, b] which corresponds to ε in the definition of the K-integrbility of F (Definition 1.3). Then, for ech t [, b] nd for ech

26 24 Chpter 1 Generlized Ordinry Differentil Equtions σ [ξ δ(ξ), ξ + δ(ξ)] [, b], we cn pply Sks-Henstock Lemm (Lemm 1.6) to obtin tht is, The bove limit yields the result. σ F (t, ξ, σ) F (t, ξ, ξ) D s F (t, τ, s) < ε, (1.4) ξ σ lim σ ξ ξ D s F (t, τ, s) = lim[f (t, ξ, σ) F (t, ξ, ξ)]. σ ξ (1.5) We re lso interested in determining when the following limit is vlid: b lim σ t D s (σ, τ, s) = b D s F (t, τ, s). At first, we need version of convergence theorem for the K-integrl. Lemm 1.10 (Theorem A.9). Let functions F, F n : [, b] [, b] [, b] X be given, where F n K([, b]) for n N. Assume there is guge ω on [, b] such tht lim [F n(t, τ, s 2 ) F n (t, τ, s 1 )] = F (t, τ, s 2 ) F (t, τ, s 1 ) n for t [, b] nd ll s 1, s 2 [, b] such tht s 1 τ s 2, [s 1, s 2 ] (τ ω(τ), τ + ω(τ)). Assume, further, tht the sequence {F n } n N is equi-integrble. Then F K([, b]) nd b lim n D s F n (t, τ, s) = b D s F (t, τ, s). Corollry Let F : [, b] [, b] [, b] X be function. If, for every τ, s [, b], F (, τ, s) is continuous nd F K([, b]), then b lim σ t D s F (σ, τ, s) = b D s F (t, τ, s). Proof. We will prove the bove limit is true when σ t+. nlogously. The cse σ t follows

27 1.2 Bsic concepts bout generlized ODEs 25 For ech n N, let us define F n : [, b] [, b] [, b] X by F n (t, τ, s) = F (t + 1n, τ, s ). Then, for every t, τ, s 1, s 2 [, b] such tht τ [s 1, s 2 ] [, b], we hve lim [F n(t, τ, s 2 ) F n (t, τ, s 1 )] = n [ ( = lim F t + 1 ) ( n n, τ, s 2 F t + 1 )] n, τ, s 1 = F (t, τ, s 2 ) F (t, τ, s 1 ) since F is continuous on the first vrible. From the definition of the K-integrl (Definition 1.3), it is cler the sequence {F n } n N is equi-integrble. Hence we cn pply Lemm 1.10 to conclude b lim σ t+ nd the proof is complete. D s F (σ, τ, s) = lim n b D s F n (t, τ, s) = b D s F (t, τ, s) 1.2 Bsic concepts bout generlized ODEs This section introduces new kind of generlized ODE using the K-integrl. We recll our min reference for generlized ODEs, s introduced by J. Kurzweil, is [36] nd the results we used from this book re collected in Appendix B. Let Ω be n open set of n rbitrry Bnch spce X. We re going to consider functions F : G X, where G = [, b] Ω [, b], for some, b R. Definition A function x : [α, β] Ω is clled solution of the generlized ordinry differentil eqution dx dτ = D sf (t, x, s) (1.6)

28 26 Chpter 1 Generlized Ordinry Differentil Equtions on the intervl [α, β] [, b], if x(t) = holds for every t [α, β]. β α D s F (t, x(τ), s) Our first exmple points out (1.6) is nottion only, whose purpose is to describe forml eqution-like object defined by its solutions. Moreover, note tht the symbol dx dτ does not stnd for differentibility. Exmple Let r : [0, 1] R be continuous function for which its derivtive does not exist t ny point of the intervl [0, 1] nd F (t, x, s) = r(s)χ [0,t] (s), where χ [0,t] : [0, 1] R is the chrcteristic function of [0, t] on [0, 1], tht is, 1 s [0, t] χ [0,t] (s) = 0 s (t, 1]. Then x : [0, 1] R, defined by x(t) = r(t) r(0), is solution of the generlized differentil eqution dx dτ = D sf (t, x, s) on [0, 1] becuse, by the definition of the integrl, we hve 1 D s F (t, x, s) = 0 0 D s r(s) = r(t) r(0) = x(t) though the function x hs no derivtive t ny point of [0, 1]. Now, we would like to investigte how the stndrd generlized ODE, introduced by J. Kurzweil, fits in this new definition. As consequence of Proposition B.2, ny function x : [α, β] Ω is solution of the generlized ODE dx dτ = DU(x, t)

29 1.2 Bsic concepts bout generlized ODEs 27 on [α, β], where U : Ω [, b] X, Ω X is open nd [α, β] [, b], provided x(t) = x(α) + α DU(x(τ), s) t [α, β]. Let us define F : [α, β] Ω [α, β] X by Then F (t, τ, s) = U(τ, s)χ [α,t] (s) + x(α) β α s. β α D s F (t, x(τ), s) = α DU(x(τ), s) + x(α) = x(t) x(α) + x(α) = x(t) which mens tht x : [α, β] Ω is solution of dx dτ = D sf (t, x, s) on [α, β]. Thus, we proved tht ny solution of generlized ODE, in the clssicl sense, is solution of generlized ODE s presented in Definition In order to obtin qulittive properties of the solutions of our generlized ODE, we re going to consider specil clss of functions given by the definition below. Definition Let h : [, b] R be nondecresing function. A function F : G X belongs to the clss F(G, h), if the following inequlities nd b b D s [F (t 2, x(τ), s) F (t 1, x(τ), s)] h(t 2) h(t 1 ) (1.7) D s [F (t 2, x(τ), s) F (t 1, x(τ), s) F (t 2, y(τ), s) + F (t 1, y(τ), s)] x y h(t 2) h(t 1 ) (1.8) hold for ll t 1, t 2 [, b] nd x, y E, where E is the set of ll step functions x : [, b] Ω.

30 28 Chpter 1 Generlized Ordinry Differentil Equtions For the reminder of this section, we re going to present some bsic properties expected for the solutions of generlized ODE whose right-hnd side F belongs to F(G, h) for some nondecresing rel function h. Now, we would like to describe the discontinuities of the solutions of generlized ODE. If x : [α, β] X is solution of the generlized ODE (1.6) nd F is continuous on the first vrible, then we cn pply Corollry 1.11 nd conclude every solution of (1.6) is continuous. Lemm Assume F : Ω X stisfies condition (1.7). If [α, β] [, b] nd x : [α, β] X is solution of (1.6), then for every t 1, t 2 [α, β] the inequlity x(t 2 ) x(t 1 ) h(t 2 ) h(t 1 ) holds. Proof. From Definition 1.12 nd condition (1.7), we hve x(t 2 ) x(t 1 ) = β α D s [F (t 2, x(τ), s) F (t 1, x(τ), s)] h(t 2) h(t 1 ) nd this concludes our proof. Hence, if x : [α, β] X is solution of (1.6), where F stisfies condition (1.7), then x is continuous on t [, b], whenever h is continuous on t. There is lso prticulr cse of Definition 1.12 which is worth considering. Let F, F : G X be such tht F (t, τ, s) = F (t, τ, s)χ [,t] (s). (1.9) If x : [α, β] Ω, [α, β] [, b] is solution of dx dτ = D s F (t, x, s) (1.10) then x(t) = β α D s F (t, x(τ), s) = D s F (t, x(τ), s). α

31 1.2 Bsic concepts bout generlized ODEs 29 In the next result, we ssume the functions F nd F stisfy (1.9). Lemm If x : [α, β] Ω is solution of the generlized ODE (1.10) nd F : G X, given by (1.9), stisfies condition (1.7), then x(r+) x(r) = lim x(σ) x(r) = σ r+ ( r ) = lim σ r+ α D s [F (σ, x(τ), s) F (r, x(τ), s)] + F (r+, x(r), r+) F (r+, x(r), r) (1.11) for r [α, β) nd x(r) x(r ) = x(r) lim σ r ( σ ) = lim σ r α D s [F (r, x(τ), s) F (σ, x(τ), s)] + F (r, x(r), r) F (r, x(r), r ) (1.12) for r (α, β], where F (r+, x(r), r+) F (r+, x(r), r) = lim σ r+ (F (σ, x(r), σ) F (σ, x(r), r)) for r [α, β) nd F (r, x(r), r ) = lim σ r F (r, x(r), σ) for r (α, β]. Proof. At first, note tht ll the bove limits involving the function F exist becuse of the estimte in condition (1.7). If σ is sufficiently close to r nd σ > r, then for ny ε > 0, we cn write r x(σ) x(r) = = r β β D s F (σ, x(τ), s) D s F (r, x(τ), s) = α α D s [F (σ, x(τ), s) F (r, x(τ), s)] + σ r D s F (σ, x(τ), s) D s [F (σ, x(τ), s) F (r, x(τ), s)] + ε + F (σ, x(r), σ) F (σ, x(r), r) where the third inequlity follows from (1.4) in Lemm 1.9. Since ε ws tken rbitrrily, the equlity (1.11) holds. The reltion (1.12) cn be proved nlogously.

32 30 Chpter 1 Generlized Ordinry Differentil Equtions Corollry If x : [α, β] Ω is solution of (1.10), F : G X stisfies condition (1.7) nd F is continuous on the first vrible, then x(r+) x(r) = F (r+, x(r), r+) F (r+, x(r), r) = F (r, x(r), r+) F (r, x(r), r) for r [α, β) nd x(r) x(r ) = F (r, x(r), r) F (r, x(r), r ) for r (α, β]. Proof. It follows from Lemm 1.16 nd Corollry Note tht Lemm 1.15 implies tht ny solution of the generlized ODE (1.6) is of bounded vrition in its domin, whenever F stisfies condition (1.7). It is well-known tht every function of bounded vrition is regulted (for proof, check [37], Proposition 1.5 on pge 437). Hence it is resonble to restrict our discussion to functions F whose integrls b D sf (t, x(τ), s) exist for regulted functions x. Theorem Given F F(G, h), let x : [, b] Ω be the pointwise limit of functions x k : [, b] Ω, k N, nd ssume the integrl b D sf (t, x k (τ), s) exists for every k N nd t [, b]. Then the integrl b D sf (t, x(τ), s) exists nd b D s F (t, x(τ), s) = lim k b D s F (t, x k (τ), s) uniformly in t [, b]. Proof. Let X be the set of ll functions x : [, b] Ω. Then there is subset S of X contining ll the functions x such tht the K-integrl b D sf (t, x(τ), s) exists. Note tht E S, where E is the set of ll step functions from [, b] to Ω (Corollry 1.8). We re going to consider sequence of functions ξ k : [, b] S given by ξ k (τ) = x k, where x k is the sequence we hve ssumed pointwise convergent. Then ξ k converges pointwisely to ξ : [, b] X given by ξ(τ) = x.

33 1.2 Bsic concepts bout generlized ODEs 31 Let us set Φ : S [, b] X by Φ(x, t) = Then Φ(x, t 2 ) Φ(x, t 1 ) = b b D s F (t, x(τ), s). (1.13) D s [F (t 2, x(τ), s) F (t 1, x(τ), s)] h(t 2) h(t 1 ) nd = b Φ(x, t 2 ) Φ(x, t 1 ) Φ(y, t 2 ) + Φ(y, t 1 ) = D s [F (t 2, x(τ), s) F (t 1, x(τ), s) F (t 2, y(τ), s) + F (t 1, y(τ), s)] x y h(t 2 ) h(t 1 ) for ll t 1, t 2 [, b] nd x, y S. Hence, the function Φ belongs to F 0 (C, h). The clss of functions F 0 is introduced in Definition B.3. Let us fix t [, b]. Then, for ny tgged division D = {(σ j, [t j 1, t j ]) j = 1,, D } of [, t], we hve D j=1 (Φ(ξ k (σ j ), t j ) Φ(ξ k (σ j ), t j 1 )) = Φ(x k, t) Φ(x k, ). Hence the Kurzweil integrl DΦ(ξ k(τ), s) exists, for every k N nd t [, b], nd it is given by DΦ(ξ k (τ), s) = b D s [F (t, x k (τ), s) F (, x k (τ), s)] which llows us to pply Theorem B.4 to conclude tht the Kurzweil integrl DΦ(ξ(τ), s) exists nd, hence, b D s [F (t, x(τ), s) F (, x(τ), s)] = lim k b D s [F (t, x k (τ), s) F (, x(τ), s)]. (1.14)

34 32 Chpter 1 Generlized Ordinry Differentil Equtions If we fix t = in (1.13), then it is cler we cn pply Theorem B.4 to get the existence of the Kurzweil integrl Φ(x, ) = b DF (, x(τ), s) nd b DF (, x(τ), s) = lim k b DF (, x k (τ), s). Therefore, it follows from (1.14) tht the integrl b D sf (t, x(τ), s) exists nd b D s F (t, x(τ), s) = lim k b D s F (t, x k (τ), s). This concludes our proof. Since the K-integrl b D sf (t, x(τ), s) exists, whenever x : [, b] X is step function, we cn pply the bove result nd conclude the K-integrl b D sf (t, x(τ), s) exists, whenever x : [, b] X is regulted function, becuse every regulted function is the uniform limit of step functions (check [25], Theorem 3.1 on pge 16). 1.3 Locl existence nd uniqueness of solution result This section is devoted to estblishing locl existence nd uniqueness result for the generlized ODE when F belongs to F(G, h) nd it stisfies dx dτ = D sf (t, x, s), F (t, τ, s) = F (t, τ, s)χ [,t] (s) for some F : G X. We re lso going to ssume the nondecrsing rel function h is continuous from the left on (, b], becuse it simplifies the proof nd the generl cse, where h is only nondecrsing rel function, cn be treted with the sme resoning.

35 1.3 Locl existence nd uniqueness of solution result 33 From the results of the previous section, we know tht the cndidtes for solution of (1.6) must be of bounded vrition, continuous from the left nd its discontinuities from the right stisfy (1.11). Thus, given n initil condition x Ω, we need to ssume x+ Ω in order to be possible to continue the solution for t > α. Applying Lemm 1.16 for r = α, we obtin x+ = x + F (α+, x, α+) F (α+, x, α). (1.15) The locl existence nd uniqueness result we re going to present follows the ides of the proof of Theorem 2.15 from [16], which relies on the Bnch Fixed Point Theorem (check [26], for proof of the ltter). Theorem Let F : G X be function belonging to the clss F(G, h) nd (α, x+, α) G, where x+ is given by (1.15). Then there is > 0 such tht the generlized ODE (1.6) dmits unique solution x : [α, α + ] X with initil condition x(α) = x. Proof. At first, we ssume the function h is continuous t α. Let > 0 be such tht [α, α+ ] [, b], h(α+ ) h(α) < 1 2 implies x Ω for ll t [α, α + ]. nd x x h(t) h(α) We denote by BV ([α, α + ], X) the spce of ll functions z : [α, α + ] X of bounded vrition. When equipped with the vrition norm given by z BV = z(α) + vr α+ α z, where vr α+ α D z = sup z(σ j ) z(σ j 1 ) j=1 nd the supremum is tken over ll finite divisions D of [α, α + ], with α = σ 0 < σ 1 < < σ D = α +. The set BV ([α, α + ], X) is Bnch spce (check [23], pge 187). Note tht the supremum norm nd the bounded vrition norm re relted by the inequlity

36 34 Chpter 1 Generlized Ordinry Differentil Equtions z := sup z(t) z BV. (1.16) t [α,α+ ] Let Q BV ([α, α + ], X) be the set of functions z : [α, α + ] X such tht z(t) x h(t) h(α) for ll t [α, α + ]. We re going to show tht Q is closed. Indeed, let {z n } n N be convergent sequence in Q nd let z BV ([α, α + ], X) be its limit. Then z n z uniformly becuse of (1.16). Hence, given ε > 0, tke n 0 N from the uniform convergence of {z n } n N. Then for ll n > n 0, we hve z (t) x z (t) z n (t) + z n (t) x ε + h(t) h(α) for ll t [α, α + ] nd we cn conclude tht z Q. Consequently, Q equipped with the vrition norm is lso Bnch spce. Now, we re redy to define the opertor which we will use s the contrction for the Bnch Fixed Point Theorem. Let T : Q BV ([α, α + ], X) be given by T z(t) := x + α D s F (t, z(τ), s) for ll t [α, α + ]. This opertor is well-defined becuse the integrl bove lwys exists when z is regulted (every function of bounded vrition is regulted) nd F F(Ω, h). Furthermore, from condition (1.7) of functions in F(Ω, h), we hve T z(t) x = Therefore T z Q nd, hence, T : Q Q. Now, we will prove T is contrction. α D s F (t, z(τ), s) h(t) h(α).

37 1.3 Locl existence nd uniqueness of solution result 35 Let z 1, z 2 Q, t 1, t 2 [α, α + ] with t 1 < t 2 nd ε > 0. Then = 2 T z 2 (t 2 ) T z 1 (t 2 ) T z 2 (t 1 ) + T z 1 (t 1 ) = D s [F (t 2, z 2 (τ), s) F (t 2, z 1 (τ), s] 1 α α z 2 z 1 [h(t 2 ) h(t 1 )]. D s [F (t 1, z 2 (τ), s) F (t 1, z 1 (τ), s] Note tht we used the definition of the opertor T nd condition (1.8) from the clss F(G, h). Also, it is cler tht ll z Q stisfy z(α) = x. Since t 1, t 2 [α, α + ] re rbitrry nd (1.16) is vlid, we hve T z 2 T z 1 BV z 2 z 1 BV [h(α + ) h(α)] < 1 2 z 2 z 1 BV. Therefore, by the Bnch Fixed Point Theorem, the opertor T dmits unique fixed point nd we cn conclude Theorem 1.19 is vlid when h is continuous t α. Now, we ssume h is not countinuous t α. Consider function h : [, b] R given by h(t) = h(t), if t α, h(t) (h(α+) h(α)), if t α. Then h is non-decresing function continuous t α. Moreover, define F : G X by F (t, x, s), if t α, F (t, x, s) = F (t, x, s) (F (α+, x, s) F (α, x, α)), if t > α. Then F F(G, h). Therefore we cn follow the steps of the previous cse nd conclude the generlized ODE dx dτ = D s F (t, x(τ), s),

38 36 Chpter 1 Generlized Ordinry Differentil Equtions with initil condition x(α) = x+, dmits unique solution z. Hence x : [α, α + ] X given by x, if t = α, x(t) = z(t), if t > α, is the solution of dx dτ = D sf (t, x(τ), s) with x(α) = x. Thus the proof is complete.

39 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs Let G = [0, ) B c [0, ), where B c = {x X; x < c, c > 0}. Throughout this chpter we ssume F : G X belongs to the clss F(G, h), stisfies F (t, 0, s 2 ) F (t, 0, s 1 ) = 0 for s 1, s 2, t [0, ) nd F (t, τ, s) = F (t, τ, s)χ [0,t] (s) for some F : G X. Thus x 0 is solution on [0, ) of the generlized ODE dx dτ = D sf (t, x, s), (2.1) becuse for every t [0, ). 0 D s F (t, 0, s) = F (t, 0, t) F (t, 0, 0) = 0 This chpter proposes stbility concepts for the solutions of generlized ODEs nd pplies the method of Lypunov functionls for these stbility concepts. We use the pper [15] s 37

40 38 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs the min reference nd mke the necessry dpttions in order to pply its results to the generlized ODE we defined in Chpter Stbility concepts for generlized ODEs We strt this section by defining ll the stbility concepts we re going to use. Definition 2.1. The trivil solution x 0 of (2.1) is clled regulrly stble, if for every ε > 0, there exists δ = δ(ε) > 0 such tht if x : [α, β] B c, with 0 α < β <, is regulted function which stisfies x(α) < δ nd sup t [α,β] x(t) x(α) D s F (t, x(τ), s) < δ, α then x(t) < ε, for t [α, β]. Definition 2.2. The trivil solution x 0 of (2.1) is clled regulrly ttrcting, if there exists δ 0 > 0 nd for every ε > 0, there exist T = T (ε) 0 nd ρ = ρ(ε) > 0 such tht if x : [α, β] B c, with 0 α < β <, is regulted function stisfying x(α) < δ 0 nd sup t [α,β] x(t) x(α) D s F (t, x(τ), s) < ρ, α then x(t) < ε, for t [α, β] [α + T, ). Definition 2.3. The trivil solution x 0 of (2.1) is clled regulrly symptoticlly stble, if it is regulrly stble nd regulrly ttrcting. Besides the generlized ODE (2.1), we consider the perturbed generlized ODE dx dτ = D s[f (t, x, s) + P (s)χ [0,t] (s)], (2.2)

41 2.1 Stbility concepts for generlized ODEs 39 where P : [0, ) X. functions from [α, β] to X. Let G ([α, β], X) denote the spce of left-continuous regulted Definition 2.4. The trivil solution x 0 of (2.1) is clled regulrly stble with respect to pertubtions, if for every ε > 0, there exists δ = δ(ε) > 0 such tht if x 0 < δ nd P G ([α, β], X) with sup P (t) P (α) < δ, t [α,β] then x(t, α, x 0 ) < ε, for t [α, β], where x(t, α, x 0 ) is solution of the perturbed generlized ODE (2.2), with initil condition x(α, α, x 0 ) = x 0 nd [α, β] [0, ). Definition 2.5. The trivil solution x 0 of (2.1) is clled regulrly ttrcting with respect to perturbtions, if there exists δ 0 > 0 nd for every ε > 0, there exist T = T (ε) 0 nd ρ = ρ(ε) > 0 such tht if x 0 < δ 0 nd with P G ([α, β], X), then sup P (t) P (α) < ρ, t [α,β] x(t, α, x 0 ) < ε, for ll t α + T, t [α, β], where x(t, α, x 0 ) is solution of the perturbed generlized ODE (2.2), with initil condition x(α, α, x 0 ) = x 0 nd [α, β] [0, ). Definition 2.6. The trivil solution x 0 of (2.1) is clled regulrly symptoticlly stble with respect to perturbtions, if it is both stble nd ttrcting with respect to perturbtions. Before proceeding, we would like to mke some comments bout the nture of these stbility concepts. Regulr stbility mens tht if x, defined on [α, β] [0, ), hs initil

42 40 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs vlue x(α) sufficiently smll nd is such tht x(t) x(α) α D s F (t, x(τ), s), (2.3) where t [α, β], is smll enough when mesured with the supremum norm, then x on [α, β] is s close to zero s we wish. Note tht when x is solution of the generlized ODE (2.1), then (2.3) equls zero. On the other hnd, regulr stbility with respect to perturbtions mens tht ny solution of the perturbed generlized ODE (2.2), defined on certin [α, β], is close to zero whenever the initil vlue x(α) nd the perturbing term, re smll enough. sup P (t) P (α), t [α,β] In prticulr, when X = R n nd P (t) = p(s)ds, t α, with p α L 1 ([α, β], R n ), then P is bsolutely continuous nd the derivtive P (s) = dp ds everywhere in [α, β] nd β β sup P (t) P (α) vr β αp = P (s) ds = p(s) ds. t [α,β] α α exists lmost Thus we cn esily see tht these regulr stbility concepts of the trivil solution of (2.1) re more generl thn Hlny s concepts of integrl stbility (see [20] nd [17]). In the book [36], the uthor proposes similr stbility concepts for generlized ODEs which we hve included in Appendix B. Insted of using the supremum norm, he uses the vrition of the functions x(t) x(α) α D s F (t, x(τ), s) nd P (t) P (α).

43 2.1 Stbility concepts for generlized ODEs 41 This seems nturl choice, since the solutions of generlized ODEs re functions of bounded vrition (check [36], Lemms 3.9 nd 3.10). However, we choose regulr stbility concepts, becuse they re more generl thn the ones found in [36] nd mde it possible to connect stbility theory for generlized ODEs with integrl stbility for mesure functionl differentil equtions, s described in [15]. The next result shows us how ll the previous concepts of regulr stbility cn be relted nd it follows the line of thinking of Theorem 4.7 from [15]. Theorem 2.7. The following sttements hold. (i) The trivil solution x 0 of (2.1) is regulrly stble if, nd only if, it is regulrly stble with respect to perturbtions. (ii) The trivil solution x 0 of (2.1) is regulrly ttrcting if, nd only if, it is regulrly ttrcting with respect to perturbtions. (iii) The trivil solution x 0 of (2.1) is regulrly symptoticlly stble if, nd only if, it is regulrly symptoticlly stble with respect to perturbtions. Sketch of the proof. The lst item in the sttement bove clerly follows from (i) nd (ii); the proofs of (i) nd (ii) re very similr. In order to illustrte the techniques, we re going to write down the min ides of the proof of (i). If x 0 is regulrly stble, then given ε > 0, we pick up the δ given by the definition of regulr stbility (Definition 2.1). Any solution x on [α, β] of the perturbed generlized ODE (2.2) is regulted nd stisfies x(t) x(α) α D s F (t, x(τ), s) = P (t) P (α). Hence, ssuming x(α) < δ nd sup t [α,β] P (t) P (α) < δ, we obtin sup t [α,β] x(t) x(α) D s F (t, x(τ), s) < δ. α

44 42 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs This mens x(t) < ε, for t [α, β], becuse x 0 is regulrly stble. Therefore, the trivil solution of (2.1) is regulrly stble with respect to perturbtions. Conversely, given ε > 0, we pick up the δ from the definition of regulr stbility with respect to perturbtions (Definition 2.4). Any regulted function x : [α, β] Ω is the solution of perturbed generlized ODE (2.2), where P (t) = P (α) + x(t) x(α) nd t [α, β]. Hence, ssuming x(α) < δ nd sup t [α,β] x(t) x(α) D s F (t, x(τ), s) < δ, α α D s F (t, x(τ), s) (2.4) we obtin sup P (t) P (α) < δ, t [α,β] becuse of (2.4). This mens x(t) < ε for t [α, β], becuse x 0 is regulrly stble with respect to perturbtions. Therefore the trivil solution of (2.1) is regulrly stble. 2.2 Lypunov-type theorems for generlized ODEs Before presenting direct method of Lypunov functionls for the stbility concepts described in the previous section, we need some uxiliry results. Lemm 2.8 (Proposition from [36]). Assume tht < < b < nd f, g : [, b] R re left-continuous functions on (, b]. If for every σ [, b], there is δ(σ) > 0 such tht for every η (0, δ(σ)) the inequlity f(σ + η) f(σ) g(σ + η) g(σ)

45 2.2 Lypunov-type theorems for generlized ODEs 43 holds, then f(t) f() g(t) g() for ll t [, b]. Lemm 2.8 is simple rel nlysis result tht relies on the properties of supremum nd continuity. The next lemm, though very technicl, is going to be proved in detil, becuse mny fetures of the theory of generlized ODEs discussed in the first chpter re employed here nd there re some technicl dpttions we mke from the proof of Lemm 6.1 from [15]. We remind the reder G = [0, ) B c [0, ), where B c = {x X; x < c}. Lemm 2.9. Let F F(G, h). Suppose V : [0, ) B ρ R, where ρ < c, is continuous from the left on (0, ) for x B ρ nd stisfies V (t, z) V (t, y) K z y z, y B ρ, t [0, ) (2.5) where K is positive constnt. Furthermore, suppose there exists function Φ : B ρ R such tht for every solution x : [α, β] B ρ, [α, β] [0, ), of the generlized ODE (2.1), we hve V (t, x(t)) = lim sup η 0+ V (t + η, x(t + η)) V (t, x(t)) η Φ(x(t)), t [α, β]. (2.6) If x : [γ, υ] B ρ, 0 γ < υ <, is left-continuous on (γ, υ] nd regulted on [γ, υ], then V (υ, x(υ)) V (γ, x(γ)) K sup where M = sup t [γ,υ] Φ(x(t)). t [γ,υ] x(t) x(γ) D s F (t, x(τ), s) + M(υ γ) (2.7) γ Proof. Given function x : [γ, υ] B ρ which is left-continuous on (γ, υ] nd regulted on [γ, υ] [0, ), we cn pply Theorem 1.18 to prove the integrl υ γ D sf (t, x(τ), s) exists.

46 44 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs Tke σ [γ, υ]. From our locl existence nd uniqueness theorem (Theorem 1.19), we know (2.1) hs locl unique solution x : [σ, σ + η 1 (σ)] B ρ with initil condition x(σ) = x(σ). From the integrbility on subintervls, there is η 2 > 0 such tht η 2 η 1 (σ), [σ, σ + η 2 ] [σ, υ] nd the integrl σ+η 2 σ D s [F (t, x(τ), s) F (t, x(τ), s)] exists. Thus, given ε > 0, there is guge δ on [σ, σ + η 2 ] corresponding to ε in the definition of the lst integrl. We cn tke smller η 2 stisfying η 2 < δ(σ). By (2.6), there is 0 < η η 2 such tht V (σ + η, x(σ + η)) V (σ, x(σ)) ηφ(x(σ)) ηm. (2.8) From the Sks-Henstock Lemm (Lemm 1.6), we cn get inequlities similr to (1.4): σ+η F (t, x(σ), σ + η) F (t, x(σ), σ) D s F (t, x(τ), s) < ηε 2K σ (2.9) nd F (t, x(σ), σ + η) F (t, x(σ), σ) σ+η Since x(σ) = x(σ), from (2.9) nd (2.10), we obtin Indeed, we hve σ+η σ σ D s F (t, x(τ), s) < ηε 2K. (2.10) D s [F (t, x(τ), s) F (t, x(τ), s)] ηε K. (2.11) σ+η σ+η D s [F (t, x(τ), s) F (t, x(τ), s)] = D s [F (t, x(τ), s) F (t, x(τ), s)]+ σ σ +F (t, x(σ), σ + η) F (t, x(τ), σ + η) + F (t, x(σ), σ) F (t, x(τ), σ) Moreover, (2.5) implies ηε 2K + ηε 2K. V (σ + η, x(σ + η)) V (σ + η, x(σ + η))

47 2.2 Lypunov-type theorems for generlized ODEs 45 K x(σ + η) x(σ + η) = K x(σ + η) x(σ) + x(σ) x(σ + η) = σ+η = K x(σ + η) x(σ) D s F (σ + η, x(τ), s) Then the inequlity bove, (2.11) nd (2.8) yield σ 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)) σ+η K x(σ + η) x(σ) D s F (σ + η, x(τ), s) + ηm +K σ σ+η K x(σ + η) x(σ) D s F (σ + η, x(τ), s) + σ+η σ σ D s [F (σ + η, x(τ), s) F (σ + η, x(τ), s)] + ηm σ+η K x(σ + η) x(σ) D s F (σ + η, x(τ), s) + η(ε + M). Given t, r [γ, υ], we define P (t, r) = x(r) σ r γ D s F (t, x(τ), s). The function P (t, ) is left-continuous on (γ, υ], since x is left-continuous on (γ, υ]. Thus P (t, σ + η) P (t, σ) = x(σ + η) x(σ) = x(σ + η) x(σ) σ+η γ σ+η σ D s F (t, x(τ), s) + D s F (t, x(τ), s). σ γ D s F (t, x(τ), s) = Now, we define f(t, r) = K P (t, r) P (t, σ) + (ε + M)t.

48 46 Chpter 2 Qulittive Properties of Solutions of Generlized ODEs Then V (σ + η, x(σ + η)) V (σ, x(σ)) K P (σ + η, σ + η) P (σ + η, σ) + (ε + M)η = = f(σ + η, σ + η) f(σ + η, σ) sup (f(t, σ + η) f(t, σ)). t [γ,υ] This mens we cn pply Lemm 2.8 in order to obtin V (υ, x(υ)) V (γ, x(γ)) sup {f(t, υ) f(t, γ)} = t [γ,υ] = K sup ( P (t, υ) P (t, σ) P (t, γ) P (t, σ) ) + (ε + M)(υ γ) t [γ,υ] K sup P (t, υ) P (t, γ) + (ε + M)(υ γ) = t [γ,υ] υ = K sup x(υ) x(γ) D s F (t, x(τ), s) + (ε + M)(υ γ) t [γ,υ] K sup t [γ,υ] Since ε > 0 is rbitrry, the result follows. γ x(t) x(γ) D s F (t, x(τ), s) + (ε + M)(υ γ). γ Now, we present definition of Lypunov functionl for generlized ODEs. It is borrowed from [15]. Definition We sy tht V : [0, ) B ρ R is Lypunov functionl with respect to the generlized ODE (2.1), if the following conditions re stisfied: (i) V (, x) : [0, ) R is continuous from the left on (0, ), for every x B ρ ; (ii) There is continuous nd incresing function b : R + R + stisfying b(0) = 0 (we sy tht such function is of Hhn clss) nd V (t, x) b( x ), for every t [0, ) nd x B ρ ;

49 2.2 Lypunov-type theorems for generlized ODEs 47 (iii) For every solution x : [γ, υ] B ρ of (2.1) with [γ, υ] [0, ), we hve V (t, x(t)) = lim sup η 0+ V (t + η, x(t + η)) V (t, x(t)) η 0, t [γ, υ], which mens the right-derivtive of V is non-positive long the solutions of (2.1). The next result presents sufficient conditions on Lypunov functionl for the regulr stbility of the trivil solution of (2.1). We borrowed the min ides from the proof of Theorem 6.3 in [15] nd mde simple dpttions. Theorem Let V : [0, ) B ρ R be Lypunov functionl where 0 < ρ < c. Suppose V stisfies the following conditions: (i) V (t, 0) = 0, t [0, ); (ii) There is constnt K > 0 such tht V (t, z) V (t, y) K z y, t [0, ) z, y B ρ. Then the trivil solution x 0 of (2.1) is regulrly stble. Proof. Let x : [γ, υ] X be regulted function on [γ, υ] [0, ). By condition (iii) from Definition 2.10 nd by Lemm 2.9, we my write V (t, x(t)) V (γ, x(γ)) + K sup r [γ,t] r x(r) x(γ) D s F (r, x(τ), s) for every t [γ, υ]. Since V is Lypunov functionl, there is function b : R + R + of Hhn clss stisfying γ V (t, x) b( x ), (2.12) for every (t, x) [γ, υ] B ρ.