Existence of Solutions for Volterra Integro-Differential Equations with Implicit Derivative

Transcription

1 Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Volume 6 Iue 4 8 PP -5 ISSN X Pri & ISSN Olie DOI: hp://xoiorg/43/ wwwarcjouralorg Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive Giueppe Aichii Giueppe Coi Diparimeo i Maemaica e Iormaica U Dii Uiverià i Fireze Fireze Ial *Correpoig Auhor: Giueppe Aichii Diparimeo i Maemaica e Iormaica U Dii Uiverià i Fireze Fireze Ial Abrac: The purpoe o hi paper i o u he exiece o coiuou oluio o a iegro-iereial euaio wih implici erivaive o boue ierval I our iveigaio we aime o exee he ue o ixe-poi heorem or upperemicoiuou mappig wih acclic value i a Baach pace o ge he reul Kewor: Meric pace mulivalue map abolue rerac cohomolog acclic e iegro-iereial euaio ucio euicoiuou a uiorml boue Acoli-Arzelà heorem INTRODUCTION I hi paper we u i a abrac eig he olvabili o a oliear iegro-iereial euaio o Volerra pe wih implici erivaive lie: x'= x x' x = We will loo or oluio o hee euaio i he Baach pace o all real C ucio which are eie i he real ierval [ ] Euaio i a pecial cae o iegro-iereial euaio I i well ow ha he heor o iegro-iereial euaio ha bee emergig a a impora area o iveigaio i rece ear a ha bee evelope ver rapil ue o he ac ha uch euaio i a wie rage o applicaio moellig aeuael ma real procee oberve i phic chemir biolog a egieerig ee eg [] [] [3] [4] [5] [6] a he receio herei Paricularl he iegro-iereial euaio are ivolve hrough moellig i he ramewor o hea low i maerial ieic heor elecrical egieerig vehicular raic heor biolog ueuig heor populaio amic corol heor mahemaical ecoomic mechaic For example Balachara a Somauaram ee [] prove a exiece heorem or he opimal corol o oliear em havig a implici erivaive wih uaraic perormace ivolvig a iegroiereial erm alo b uig he ame meho o ucioal aali i e a ixe poi heorem The iegro-iereial euaio have bee uie i variou paper wih he help o everal ool o ucioal aali opolog a ixe poi heor For iace we ca reer o [] [] [4] [5] [6] [7] So he crucial e o our approach i orer o i oluio o euaio coi i he ue o a ver ueul ixe poi heorem or mulivalue compac upperemicoiuou map wih acclic value i a Baach pace PRELIMINARIES AND NOTATIONS Le B = C J R be a Baach pace o all coiuoul iereiable ucio eie o J = [a b] wih he orm x max x x' where x max x a b x ' max x' a b a Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page

2 Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive A ube A C J R i a relaivel compac e i a ol i he ucio o A are euicoiuou a uiorml boue ogeher wih heir erivaive o J Le M be a ube o he Baach pace B a le T : M B be a map Le be a iiieimal euece o poiive real umber A euece T o map T : M B i ai o be a - T approximaio o T i x T x or ever x M Le u eoe b CB he amil o all oemp a compac ube o B a eoe b B r he ball o B eie b B r xb: x r a b B r i cloure Le X be a ube o B; a mulivalue map S : X CB i ai o be upperemicoiuou u c i he graph o S i cloe ie or a euece x X x x a we have S x A mappig rom B o CB i ai o be compac i i e boue e io relaivel compac e We a ha A B i a R e i he pace B i A i he ierecio o couable ecreaig euece o abolue rerac coaie i B I i ow ha R e i a acclic e ie i i acclic wih repec o a coiuou heor o cohomolog ee or iace [8] Le M be a cloe a oemp ube o B a le T : M B be a compac mappig Le T : M B be a - approximaio o T where T are compac mappig I i ow ha i he euaio x - T x = ha a mo oe oluio belogig o B or ever aural umber he he e o ixe poi o T i a compac R e ee [9] The well-ow Growall Lemma rom he aar heor o Oriar Diereial Euaio will be ue Propoiio : Le u v g : J : R be coiuou a oegaive ucio; moreover aume ha g i a oecreaig ucio o J I he ollowig ieuali hol: u g+ vu J he we have: a u g exp v J a The ollowig propoiio ca be euce rom Theorem o [] a i will be ueul i he euel Propoiio : Le B be a Baach pace le M be a cloe a covex ube o B Aume ha S : M CB be a upperemicoiuou compac mulivalue map wih acclic value The i SM M S ha a ixe poi 3 MAIN RESULT We wa o eal wih he exiece o oluio o iegro-iereial euaio The ollowig heorem hol Theorem: Aume ha he ollowig coiio hol: i : [ ] R R R i a coiuou ucio uch ha or ever x [ ] R R we have x a x b a b ii : [ ] [ ] R i a C ucio uch ha here exi a coiuou oegaive ucio h : J : R aiig he ollowig coiio: Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page

3 Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive h a h or ever [ ] [] iii Puig h up h he ollowig ieuali hol: h a hb The i he coiio i ii iii hol euaio ha a lea oe oluio Proo Le be a ucio belogig o C J R a coier he ollowig iegral euaio: = τ τ Le : C J R C J R be he map which aociae o ever C J R he e o oluio o euaio 3 Clearl puig 3 ha he ixe poi o he map are he oluio o euaio To ha aim he ollowig ep i he proo have o be eablihe: a here exi a cloe a covex e C uch ha C C ; b he map i a upperemicoiuou a compac map; c he e i a acclic e or ever C a Le C = ' M Le C; we have: M x we have x = a x = ; o B he cloe ball o C J R ; hece C i a ol i M a So ha we have: The i ollow: h a b h a hb h am hb h am h a ; hece M i h b h b Sice h a hb he he la ieuali hol Moreover we have: ' Hece we obai: ' Thu we have h The coiio C C abh M am am b ham b ham b h am h ' M i h a h b hb ; ice h a hb he he la ieuali hol hb i aiie b We wa o prove ha he map : C C i compac Le C a ix For a [ ] we have: Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page

4 Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page 3 ' ' = + + b a + h + h B he coiui o he ucio h a i ollow ha here exi uch ha ' ' or a [ ] Sice M a M ' i ollow ha he e C i relaivel compac Le u ow how ha he mulivalue map i upperemicoiuou Le be a euece covergig o i he C - orm ie le Aume ha i he C - orm; we ee o how ha From he Domiae Lebegue Covergece Theorem i ollow: a Hece we have: ie c Now we wa o how ha or ever ixe C he e i a acclic e Coier he ollowig iegral euaio: 4

5 Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive Puig g he he euaio 4 ca be wrie i he ollowig wa: g Coier he operaor H : C C J R eie a ollow: H g Oberve ha he operaor H i compac Clearl he ixe poi o he operaor 6 are he oluio o euaio 5 For ever aural umber here exi a Lipchiz ucio g : [ ] R R uch ha we have or ever z [ ] R : g g z L z a g Le H : C C J R be he operaor eie a ollow: g h ee [9] H g 7 The operaor H eie i 7 i a compac operaor or ever aural umber Moreover we have: g g h H H h ' H ' H h g g g g g g h g g Le ow C mo oe oluio Le z be aoher oluio; we have: Coier he euaio z Uig ormula we obai: a h h H ; we wa o prove ha hi euaio ha a g g z L z h h z Hece = z or ever The we ca coclue ha he e o he oluio o euaio 4 i a acclic e 4 EXAMPLE Le u coier he ollowig iegro-iereial euaio wih implici erivaive: co i ' l ' x x x 4 x x B recallig our reul we have or ever [ ] [ ]: co a o 4 h 4 4 I aalogou wa we have: h 4 So we ca pu: a = b = h 4 3 Fiall he we obai: h a hb 4 The we ca a ha he coiio o our heorem are aiie a ha he euaio 8 ami a lea oe oluio Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page 4

6 Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive REFERENCES [] Aichii G Coi G Exiece o oluio or Volerra iegral euaio epeig o erivaive Pioeer Joural o Mahemaic a Mahemaical Sciece 8 pp [] Balachara K Somauaram D Exiece o opimal corol or oliear em wih uaraic perormace J Auralia Mah Soc Ser B 9 pp [3] Hoai N T K Loi N V Exiece o oluio or ome Hammerei pe iegro-iereial icluio Elecroic Joural o Diereial Euaio 78 pp-8 7 [4] Mallia Arjua M Selvi S Exiece reul or impulive mixe Volerra-Freholm iegro-iereial icluio wih olocal coiio I Joural o Mahemaical Sciece a Applicaio Paper 6 9 [5] Pachpae B G Applicaio o Lera-Schauer aleraive o ome Volerra iegral a iegroiereial euaio Iia Joural Pure a Applie Mahemaic 6 pp [6] Pachpae B G Implici pe Volerra iegro-iereial euaio Tamag Joural o Mahemaic 4 pp 97-7 [7] Siora A Exiece heor or oliear Volerra iegral a iereial euaio Joural Ieualiie a Applicaio 6 pp [8] Gabor G O he acclici o ixe poi e o mulivalue map Topological Meho i Noliear Aali 4 pp [9] Lar J M Rober RAale o liéaire mulivoue Cahier mahémaiue e la eciio No 76 Pari 976 [] Fizparic P M Perhi W W Fixe poi heorem or mulivalue ocompac acclic mappig Paciic Joural o Mahemaic 54 pp Ciaio: G Aichii G Coi " Exiece o Soluio or Volerra Iegro-Diereial Euaio wih Implici Derivaive " Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch vol 6 o 4 p -5 8 hp://xoiorg/43/ Coprigh: 8 Auhor Thi i a ope-acce aricle iribue uer he erm o he Creaive Commo Aribuio Licee which permi urerice ue iribuio a reproucio i a meium provie he origial auhor a ource are creie Ieraioal Joural o Scieiic a Iovaive Mahemaical Reearch IJSIMR Page 5