EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar

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1 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded PIN-4363 (MS) INDIA E-ail:-palikar@rediffailco Absrac I his paper, he exisece resul of oliear firs order ordiary rado differeial equaios is proved usig rado fixed poi heore Idex Ters: Rado differeial equaio, uli-valued fucio, iiial value proble 2MaheaicsSubjecClassificaios: 6H25,47H4, 47N2 I STATEMENT OF THE PROBLEM Le R deoe he real lie ad R, he se of oegaive real ubers Le C( R, R ) deoe he class of realvalued fucios defied ad coiuous o R Give a easurable space (Ω, A) ad a easurable fucio x : Ω C ( R, R), cosider he iiial value proble of oliear firs order ordiary rado differeial equaios ( RDE) x' ) k ) x ) = f x ), ) a e R x(, ) = q( ) () forall Ω, where k : R Ω R, q : Ω R ad f : R R Ω R By a rado soluio of he RDE () I ea a easurable fucio x : Ω AC ( R, R) ha saisfies he equaios i (), where AC( R, R) is he space of absoluely coiuous real-valued fucios defied o R The iiial value probles of ordiary differeial equaios have bee sudied i he lieraure o bouded as well as ubouded ierals of he real lie for differe aspecs of he soluio See for exaple, Buro ad Furuochi [2], Dhage [3], Siilarly, he iiial value proble of rado differeial equaios have also bee discussed i he lieraure for exisece heores o bouded iervals, however, he sudy of such rado equaios has o bee ade o ubouded iervals of he real lie for ay aspecs of he rado soluios Soe resuls alog his lies appear i Ioh [5], Bharucha-Reid [] ad Dhage [4] Therefore, oliear rado differeial equaios o ubouded iervals eed o pay aeio o he exisece of he rado soluios The prese paper proposes o discuss he exisece resuls for rado differeial equaios () o he righ half R of he real lie R The classical fixed poi heory, i paricular, rado versio of Schauder s fixed poi heore will be eployed o prove he ai resul of his paper I clai ha he resuls of his paper is ew o he lieraure o rado differeial equaios II AUXILIARY RESULTS The followig heore is ofe used i he sudy of oliear discoiuous rado differeial equaios I also eed his resul i he subseque par of his paper Theore 2 (Ioh[5]) Le X be a o-epy, closed covex bouded subse of he separable Baach space E ad le Q : Ω X X be a copac ad coiuous rado operaor The he rado equaio Q( )x = x has a rado soluio, ha is here is a easurable fucio ξ : Ω X such ha Q( ) ξ ( ) = ξ ( ) for all Ω Theore22(Carahéodory) Le Q : Ω E E be a appig such ha Q(, x) is easurable for all x E ad Q ) is coiuous for all Ω The he ap x) Q x) is joily easurable

2 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 2 ISSN III EXISTENCE RESULTS I eed he followig defiiio i he sequel Defiiio 4 A fucio f : R R Ω R is called rado Carahéodoryif i) he ap f x, ) is easurable for all R ad x R ad (ii) he ap x) f x, ) is joily coiuous for all Ω Furherore, a rado Carahéodory fucio f : R R Ω R is called rado L - Carahéodory, if here exiss a fucio f x, ) h( ) a e R h L ( R, R) such ha for all Ω ad x R The fucio h is he growh fucio of f o R R Ω I cosider he followig se of hypoheses follows: ( H) The fucio k : R R R is easurable ad bouded Moreover, for each Ω, li K ) = Where K ) k( s, ) = ( H) The fucio q : Ω R is easurable ad bouded Moreover, ess sup q( ) = c Ω for soe real uber c > ( H The fucio f is rado L -Carahéodory wih growh fucio h o R Moreover, K ) k ( s, ) li e e h( s) = forall Ω Reark 42 If he hypohesis ( H hol, he he fucio : K (, ) k ( s, ) ) ( ) w = e e h s is coiuous ad he uber w R Ω R defied by K (, ) k ( s, ) sup ) sup ( ) w= w = e e h s exiss for all Ω Hypohesis ( H has bee cosidered i a uber of papers i he lieraure See for exaple, Dhage [3], Buro ad Furuochi [2] ad he refereces herei IV MAIN RESULT Theore 43 Assue ha he hypoheses ( H ) hrough ( H hold The he RDE() adis a rado soluio Proof Now RDE () is equivale o he rado equaio for all R ad Ω K (, ) K (, ) K ( s, ) ) ( ) ), ) x = q e e e f s x s (4),

3 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 3 ISSN Se E = BC ( R, R ) For a give fucio x : Ω E, defie a appig Q o K ) K ) K ( s, ) ( ) ) ( ) ), ) Ω E by Q x = q e e e f s x s (4 for all R ad Ω For he sake of coveiece, I wrie Q ( ) x ) = Q ( ) x( ) oiig he double appearace of erge i io Q( ) Clearly, Q defies a appig Q : Ω E E To see his, le x E be arbirary The for each Ω, he coiuiy of ap Q( ) x( ) follows fro he fac ha expoeial e f ( s, x ( s, ), ) d s K ) ad he idefiie iegral are coiuous fucios of o R Nex, I show ha he fucio Q( ) x : R R is bouded for each Ω Now by hypoheses ( H ) ad ( H, K (, ) K (, ) K ( s, ) ( ) ( ) ( ) ), ) Q x q e e e f s x s (43) K (, ) K (, ) K ( s, ) c e e e h( s) c W for all Ω As a resul, Q : Ω E E Defie a closed ball B r () i he Baach space E ceered a origi of radius r = c W fro (43), Q( ) x c W for all Ω ad x E Hece Q : Ω E B r (), ad i paricular, Q defies a ap Q : Ω Br () Br () Now I show ha Q saisfies all he codiios of Theore 2 wih X = r () Firsly, I show ha Q is a rado operaor o Ω B r ( ) i o B r ( ) By hypohesis ( H, he ap f x, ) is easurable by he Carahéodory heore Sice a coiuous fucio is easurable, he K ) K ) ap e is easurable ad so he produc e f x ), ) is easurable i for all R ad x R Sice he iegral is a lii of he fiie su of easurable fucios, I have ha he fucio K ( s, ) e f ( s, x( s, ), ) is easurable Siilarly, he ap K (, ) K (, ) K ( s, ) ( ) ), ) q e e e f s x s is easurable for all R Cosequely, he ap Q( ) x is easurable for all x E ad ha Q is a rado operaor o Ω r () Secodly, B I show ha he rado operaor Q( ) K (, ) K ( s, ) li w ) = li e e h( s) = is coiuous o B r () By hypohesis ( H ε, here is a real uber T > such ha w( ) < for all 4 T I show ha he coiuiy of he rado operaor Q( ) i he followig wo cases: Case I Le [, T] ad le { x } be a sequece of pois i B r () such ha x by he doiaed covergece heore, B x as The,

4 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 4 ISSN Q x q e e e f s x s K (, ) K (, ) K ( s, ) = q( ) e li e e f ( s, x ( s, ), ) K ) K ) K ( s, ) li ( ) ( ) = li ( ) ), ) for all [, T] ad Ω K ) ( ) K ) li K ( s, ) ), ) ( ) = q e e e f s x s K ) K ) K( s, ) ( ) ), ) ( ) = q e e e f s x s = Q( ) x( ) Case II Suppose ha T The, K ) K ( s, ) K ) K ( s, ) ( ) ( ) ( ) ( ) = ), ) ), ) Q x Q x e e f s x s e e f s x s K (, ) K ( s, ) K (, ) K ( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) K (, ) K ( s, ) K (, ) K ( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) 2 ( ) <ε for all T ad Ω Sice ε is arbirary, oe has li Q( ) x ( ) = Q( ) x ( ) all T ad Ω Now cobiig he Case I wih Case II, I coclude ha Q( ) is a poiwise coiuous rado operaor o B r () io iself Furher, i is show below ha he faily of fucios { Q( ) x } is a equicoiuous se i E for a fixed Ω Hece, he above covergece is uifor o R ad cosequely, Q( ) is a coiuous rado operaor o B r () io iself Nex, I show ha Q( ) is a copac rado operaor o r () Le Ω be fixed ad cosider a sequece { Q x } ( ) of pois i r () B B To fiish, i is eough o show ha he sequece { Q x } subsequece for each Ω Clearly{ Q( ) x } is a uiforly bouded subse of r () I show ha i is a equi-coiuous sequece of fucios o R B ( ) has a Cauchy ε Le ε > be give Sice li w ) =, here exiss a real uber T > such for < for T I 4 cosider he followig hree cases:,, T The, Case I Le [ ] Q( ) x ( ) Q ( ) x( ) K (, ) K (, ) K ( s, ) K ( 2, ) K ( 2, ) K ( s, ) q( ) e e e f ( s, x ( s, ), ) q( ) e e e f ( s, x( s, ), ) K(, ) K( 2, ) q( ) e e K (, ) K ( s, ) K ( 2, ) K ( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), )

5 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 5 ISSN K(, ) K( 2, ) q( ) e e K(, ) K( s, ) K( 2, ) K( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) K( 2, ) K( s, ) K( 2, ) K( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) K (, ) K ( 2, ) q ( ) e e K (, ) K ( 2, ) K ( s, ) e e e f ( s, x ( s, ), ) K ( 2, ) K ( s, ) K ( s, ) e e f ( s, x ( s, ), ) e f ( s, x( s, ), ) c e h e e p( ) p( ) k K (, ) K ( 2, ) L for all N, where k is he boud for k (, ) ad p( ) are coiuous o [ ] Q( ) x ( ) Q ( ) x( ) as 2 Case II If, [ T, ], he Q( ) x ( ) Q ( ) x( ) K ( s, ) ad ( ) p = e h( s) Sice he fucios e K ),T,hey are uiforly coiuous here Hece, uiforly for all, [, T] K (, ) K ( s, ) K ( 2, ) K ( s, ) ad for all N e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) K (, ) K ( s, ) K ( 2, ) K ( s, ) e e f ( s, x ( s, ), ) e e f ( s, x( s, ), ) w( ) w( ) ε ε < ε 4 4 Sice ε is arbirary, oe has Q( ) x ( ) Q ( ) x( ) as uiforly for all N Case III If < T < 2, he Q ( ) x ( ) Q ( ) x ( ) Q ( ) x ( ) Q ( ) x ( T ) 2 As, T ad T, ad so ad Q ( ) x ( ) Q ( ) x ( T) Q ( ) x ( T) Q ( ) x ( T ) Hece as uiforly for all N Q ( ) x ( T ) Q ( ) x ( ) Q ( ) x ( ) Q ( ) x( ) as uiforly for all < T ad 2 > T ad for all N Thus, i all hree cases Q( ) x ( ) Q ( ) x ( ) as 2 uiforly for all, 2 R ad for all N 2

6 Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 6 ISSN This shows ha { Q( ) x } heore yiel ha { Q x } Wihou loss of geeraliy, call he subsequece o be he sequece iself I show ha { Q x } is a equicoiuous sequece i X Now a applicaio of Arzelá-Ascoli ( ) has a uiforly coverge subsequece o he copac subse [, T ] of R ( ) is Cauchy i X Now shows ha Q( ) x ( ) Q ( ) x( ) as for all [, T ] The for give ε > here exis N such ha p k p K ( s, ) sup ), ) ), ) pt ε e e f s x s f s x s < 2 for all, Therefore, if,,he we have Q( ) x Q ( ) x = K (, ) K ( s, ) sup < ), ) ), ) e e f s x s f s x s p K ( p, ) K ( s, ) sup pt ), ) ), ) e e f s x s f s x s p K ( p, ) K ( s, ) sup p T ), ) ), ) <ε This shows ha { } e e f s x s f s x s Q ( ) x Q ( )( B r ()) ( B r ()) coverges o a poi i X As Q( )( Br ()) Q( )( Br ()) Hece Q ( )( B r ()) is Cauchy Sice X is coplee, { Q x } is closed, he sequece { Q x } ( ) is relaively copac for each Ω ( ) coverges o a poi i ad cosequely Q is a coiuous ad copac rado operaor o Ω ( r ()) Now a applicaio of Theore 2 o he operaor B Q( ) o ( B r ()) yiel ha Q has a fixed poi i ( B r ()) which furher iplies ha he RDE () rado soluio o R ACKNOWLEDGMENT This research aricle is soe oupu resul of Mior Research Projec fuded by UGC, New Delhi REFERENCES [] AT Bharucha-Reid, Rado Iegral Equaios, Acadeic Press, New York, 972 [2] TA Buro, T Furuochi, A oe o sabiliy by Schauder s heore, Fukcial Ekvac44 (2), [3] BC Dhage, O global exisece ad araciviy resuls for oliear rado iegral equaios, Paaer Mah J 9 (29), 97 [4] BCDhage, Soe algebraic ad opological rado fixed poi heores wih applicaios o oliear rado iegral equaios, Takag J Mah 35(24), [5] SIoh, Rado fixed poi heores wih applicaios o rado differeial equaios i Baach spaces, J Mah Aal Appl 67 (979),26-273