Existence and Uniqueness Theorems for Generalized Set Differential Equations

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1 Ieraoal Joural of Corol Scece ad Egeerg, (): -6 DOI: 593/jCorol Exsece ad Uqueess Teorems for Geeralzed Se Dffereal Equaos Adrej Ploov,,*, Naala Srp Deparme of Opmal Corol & Ecoomc Cyberecs, Odessa Naoal Uversy amed afer II Mecov, Odessa, 656, Urae Deparme of Appled Maemacs, Odessa Sae Academy of Cvl Egeerg ad Arcecure, Odessa, 659, Urae Absrac I s paper e cocep of geeralzed dffereably for se-valued mappgs proposed by AV Ploov, NV Srp s used Te geeralzed se-valued dffereal equaos w geeralzed dervave are cosdered ad e exsece ad uqueess eorems are proved Keywords se-valued mappg, geeralzed dervave, exsece ad uqueess eorems Iroduco Te cocep of dervave for se-valued mappg was frs eered by M Huuara[] Te e problems of dffereably of fuzzy mappgs were cosdered by T F Brdglad[], JN Tyur[3], HT Bas ad MQ Jacobs[4], AV Ploov[5, 6], AN Vyu[7], B Bede ad SG Gal[8], AV Ploov ad NV Srp[9] Te properes of e ese dervaves were cosdered [-8] FS de Blas ad F Iervolo begu sudyg of se-valued dffereal equaos (SDEs) semlear merc spaces[,9-] Now developed e eory of SDEs as a depede dscple Te properes of soluos, e mpulsve SDEs, corol sysems ad asympoc meods for SDEs were cosdered[5,6,9-,6-4] O e oer ad, SDEs are useful oer areas of maemacs For example, SDEs are used as a auxlary ool o prove e exsece resuls for dffereal clusos Also, oe ca employ SDEs e vesgao of fuzzy dffereal equaos Moreover, SDEs are a aural geeralzao of usual ordary dffereal equaos fe (or fe) dmesoal Baac spaces[9] I[9] a ew cocep of a dervave of a se-valued mappg a geeralzes e cocep of Huuara dervave was eered ad a ew ype of a se-valued dffereal equao suc a e dameer of s soluo ca weer crease or decrease (for example, o be perodc) was cosdered I e deologcal sese s defo of e dervave s close o e defos proposed [5,6,8] * Correspodg auor: a-ploov@ure (Adrej Ploov) Publsed ole a p://jouralsapuborg/corol Copyrg Scefc & Academc Publsg All Rgs Reserved I s paper e geeralzed se-valued dffereal equaos w geeralzed dervave are cosdered ad e exsece ad uqueess eorems are proved Te Geeralzed Dervave Le cov ( R ) be a space of all oempy covex closed ses of R w Hausdorff merc ( AB, ) = m{ r : A B+ Sr(), B A+ Sr() }, were A, B cov( R ), Sr () = { s R : s r} Defo [] Le, Y cov( R ) A se Z cov( R ) suc a = Y + Z s called a Huuara dfferece of e ses ad Y ad s deoed by Y From Rådsröm's Embeddg Lemma[5] follows a f s dfferece exss, e s uque Le I = [ ', "] R ; : I cov( R ) be a se-valued mappg; (, + ) I be a -egbourood of a po I ; > For ay (, + ) cosder e followg Huuara dffereces f ese dffereces exs () ( ), () ( ) (), ( ) (), () ( ), () (3) (4) Te dffereces () ad ()[(3) ad (4)] are called e rg[lef] dffereces From e defo of e Huuara dfferece follows a bo oe-sded dffereces exs oly e case we ( ) F + { f( )} for [, + ) or (, ] If all dffereces ()-(4) exs e

2 Adrej Ploov e al: Exsece ad Uqueess Teorems for Geeralzed Se Dffereal Equaos ( ) F + { f( )} -egbourood of e po If for all (, + ) ere exss oly oe of e oe-sded dffereces, e usg e properes of e Huuara dfferece, we ge a e mappg dam : I R + e -egbourood of e po ca be: a) o-decreasg o (, + ); b) o-creasg o (, + ); c) o-decreasg o (, ) ad o-creasg o (, + ); d) o-creasg o (, ) ad o-decreasg o (, + ) Hece, for eac of e above meoed cases oly oe of combaos of dffereces s possble: a) () ad (3); b) () ad (4); c) () ad (3); d) () ad (4) Cosder four ypes of lms correspodg o oe of e dfferece ypes: + lm ( ) ( ) ; (5) lm ( ) ( ) ; (6) + lm ( ) ( ) ; (7) lm ( ) ( ) (8) So s possble o say a e po o more a wo lms ca exs (as we assumed a ere exs oly wo of four Huuara dffereces) Cosderg all above we ave a ere ca exs oly e followg combaos of lms: a) (5) ad (7); b) (6) ad (8); c) (6) ad (7); d) (5) ad (8) Defo [9] If e correspodg wo lms exs ad are equal we wll say a e mappg () s dffereable e geeralzed sese e po ad deoe e geeralzed dervave by D ( ) Le us say a e se-valued mappg : I cov( R ) s dffereable e geeralzed sese o e erval I f s dffereable e geeralzed sese a every po of s erval Remar Properes of e geeralzed dervave ave bee cosdered [9] Defo 3[9] Te se-valued mappg :[, T] cov( R ) s called absoluely couous o e erval [, T ] f ere exs a measurable se-valued mappg G () ad a sysem of ervals [, + ], =,, m, m+ = T suc a for all [, + ], =,, m () = ( ) Gsds () or () = ( ) + Gsds () Teorem [9] Le a se-valued mappg :[, T] cov( R ) s absoluely couous o e erval [, T ] Te e se-valued mappg () s dffereable e geeralzed sese almos everywere o e erval [, T ] ad D () = G() almos everywere o [, T ] 3 Geeralzed Dffereal Equaos w e Geeralzed Dervave Frs cosder a dffereal equao w e geeralzed dervave a s smlar o a dffereal equao w e Huuara dervave, e D = F(, ), ( ) =, (9) were D () s e geeralzed dervave of a se-valued mappg :[, T ] cov( R ), F :[, T ] cov( R ) cov( R ) s a se-valued mappg, cov( R ) Defo 4 A se-valued mappg :[, T] cov( R ) s sad o be soluo of dffereal equao (9) f s absoluely couous ad sasfes (9) almos everywere o [, T ] Remar Ule e case of dffereal equaos w Huuara dervave, f a dffereal equao w e geeralzed dervave (9) as a soluo e ere exss a fe umber of soluos rrespecve of e codos o e rg-ad sde of e equao Example Cosder e followg dffereal equao w e geeralzed dervave D = [,], () = [, ] () I s easy o cec a e followg se-valued mappgs are e soluos of equao (): ( ) = [, + ], [,], ( ) = [ +, ], [,], [, + ], [,5], 3() = [ 5 +,5 ], [5,], [, +, ], [,5], 4 ( ) = [ 5,5 + ], [5,5], [ 5 +, 5 ], [5,] Also s possble o cosruc oer soluos, us oly () wll be e soluo of e correspodg dffereal equao w e Huuara dervave DH = [,], () = [, ] ad () ad () are soluos of e dffereal equao w e geeralzed dervave ( e sese of[8]) Terefore we wll cosder e oer dffereal equao w e geeralzed dervave: D ( ) Φ ( φ( )) F (, ( )) =Φ( φ( )) F(, ( )), () ( ) =, were [, T] ; :[, T ] cov( R ) ; cov( R ) ; F, F :[, T ] cov( R ) cov( R ) are se-valued mappgs;,, φ :[, T] Rs a couous fuco; fuco φ > Φ ( φ) =, φ Defo 5 A se-valued mappg :[, T] cov( R ) s called e soluo of dffereal equao () f s couous ad o ay suberval [ τ, τ ] [, T], were fuco φ () of cosa sgs, sasfes e egral equao ( ) + Φ φ ( ( s)) F ( s, ( s)) ds = ( τ) + Φ( φ( s)) F( s, ( s)) ds τ If o e erval [ τ, τ ] e fuco φ () >, e () τ

3 Ieraoal Joural of Corol Scece ad Egeerg, (): -6 3 sasfes e egral equao ( ) = ( τ ) + F ( s, ( s)) ds τ for [ τ, τ] ad dam () creases If o e erval [ τ, τ ] e fuco φ () <, e we ave e τ, τ ( ) + F ( s, ( s)) ds = ( ) ( ) = ( τ ) F ( s, ( s)) ds ad dam () decreases τ If o e erval [ τ, τ ] e fuco φ(), e we ave () τ ( ) So we ca eer e oer equvale defo of a soluo of equao () Defo 6 A se-valued mappg :[, T] cov( R ) s called e soluo of dffereal equao () f s absoluely couous, sasfes () almos everywere o [, T ] ad creases f φ( ) >, dam ( ) = s cosa f φ( ) =, decreases f φ( ) < Example Cosder e followg dffereal equao w geeralzed dervave D Φ ( s )[, 4] =Φ (s )[,3], () = [, 4] () As s > for (, ) we ave ( ) = [,4] + [,3] ds = [,4] + [,3 ] = [ +,4 + 3 ] for [, ] So for = we ge ( ) = [ +,4 + 3 ] Furer as s < for (, ) we ave ( ) = [ +,4+ 3 ] [,4] ds = = [ +,4+ 3 ] ds [,4] = = [ +,4 + 3 ] ( )[,4] = = [ +, ] So for = < we ge = Fgure Te grap of a soluo of sysem () + 4 I meas a e soluo exss oly for, 3 (see fg ) Example 3 Cosder e same dffereal equao w geeralzed dervave bu w ϕ ( ) = s() : D Φ(s )[, 4] =Φ ( s )[,3], () = [, 4] (3) As s < for, e we ave ( ) = [,4] [,4] = [+,4 4 ] for, Furer as s > for, e we ge 5 7 ( ) = +,4 + [,3] ds = +, Furer as s < for, e we ge 5 7 ( ) = +,4 [,4] ds,4 4 5 = So for = + < we ave = 3 3 Fgure Te grap of a soluo of sysem (3) 3 I meas a e soluo exss oly for, + 3 (see fg ) Remar 3 I s obvous a e mappgs F (, ), F (, ) defe oly o ow muc e mappg () cages case of s "decrease"( F(, ) ) or "crease"( F (, )) ad fuco φ() defes wa wll occur o () ["decrease" or "crease"] If φ() rrespecve of F (, ( )) ad F (, ( )) e mappg () wll be cosa Example 4 Cosder e dffereal equao from Example w φ() for [, ] Te ( ) [, 4] for [, ] Remar 4 If we ae φ() e we wll ave ( ) = [,4] [,4] ds = [+,4 4 ] Te for = we ge 3 8 = So e soluo exss 3 3

4 4 Adrej Ploov e al: Exsece ad Uqueess Teorems for Geeralzed Se Dffereal Equaos for, 3 (see fg 3) Fgure 3 Te grap of a soluo of sysem () for ( ) So for all φ () we ca guaraee e exsece of soluo of e dffereal equao o e erval ϕ, 3 D ( ) Φ ( φ( ))[, 4] =Φ ( φ( ))[,3], () = [, 4] Le CC( R ) ( ) be a space of all oempy srcly covex closed ses of R ad all eleme of R [7] Te followg eorem of exsece of e soluo of equao () for case CC( R ) olds: Teorem Le e se-valued mappgs F(, ), (, ): ( ) ( F ) R CC R CC R e doma Q = {(, ) R CC( R ): [,, (, ) b} sasfy e followg codos: ) for ay fxed e se-valued mappgs F (, ), F (, ) are measurable; ) for almos every fxed e se-valued mappgs F(,), F (,) are couous; ) F(, ) m(), F(, ) m(), were m (), m () are summable o [, ; v) φ () s couous ad as e fe umber of ervals were sg( φ ( )) = ±, v) Te ere exss a soluo of equao () defed o e erval [,, were d > sasfes e codos a) d a; b) ϕ ( + d) b, were c) µ [, + d] 33 θ m () s ds were θ m C(, ψ) C(, ψ) =, ψ = ( ψ) ( ψ ψ ) C = x + + x,, max x ϕ () = m ( s) ds, =, ; µ [, [, : φ( ) < for µ [, Proof Le us cosder some cases ) φ () > for [, Te equao () s e ordary dffereal equao w Huuara dervave DH ( ) = F(, ), ( ) = (4) Terefore, usg[7] we ge a e equao () as a soluo () defed o [,, were d sasfes e codo d = m{ a, γ}, () = +γ m s ds b ) φ() for [, Te equao () s e ordary dffereal equao w Huuara dervave D ( ) = {}, ( ) = ad erefore, ( ) s e soluo of () o [, 3) φ () < for [, Te equao () s e equao w e geeralzed dervave D () F (, ()) = {}, ( ) = (5) Accordg o Defo 5 cosder e followg egral equao ( ) = F ( s, ( s)) ds (6) for [, ad prove e exsece of soluo o e some erval [, 3a) As F(, ) m() for (, ) Q, e F (, ) S m () (), were S ( a) = { x R : x a r} r So F ( s, ) ds Sm ( s) () ds = S () m ( s) ds Defe by S ( ) = S () m ( s) ds I s obvously, a f < < < + a, e {} = S ( ) S ( ) S ( ) S ( + a) As CC( R ) ad, e ere exss d > suc a e se S () ca be embedded e se for all [, (e ere exss ζ () suc a S () + ζ () ) ad s o embedded for > + d Ad, s obvously, a d ca be foud ou from e equao θ m s ds = + d () Terefore, for all (, ) Q = {(, ) R CC( R ): [,, (, ) b} e se F (, ) s ds s embedded e se 3b) As (, ) ( F ) CC R for all (, ) Q, e (, ) ( F ) ds CC R for all (, ) Q [7] Terefore, as CC( R ) ad a e se S () ca be embedded e se for all [,, e e Huuara dfferece (, ) F s ds exss for all (, ) Q [7] 3c) Le us fd d > suc a + d m () s ds = b ad cosder d= m{ ad,, d} 3d) Coose ay aural Sequeally o e ervals

5 Ieraoal Joural of Corol Scece ad Egeerg, (): -6 5 d + + ( + ), =, =,, le us buld e successve approxmaos of e soluo () = for, ( ) = F ( s, ( s )) ds for [, (7) By 3b) () s exs ad () CC( R ) for all N ad [, Also by codos ) ad ) of e eorem () s couous o [, d] Besdes + for all N ( ( ), ) = F( s, ( s )) ds, {}, F( s, ( s )) ds ( F( s, ( s ),{} ) ds m () s ds ϕ ( + d) b Hece, follows a e sequece of e se-valued () uformly bouded: = ( ),{} (,{}) + b mappgs { } ( ) Le us sow a e se-valued mappgs () are equcouous For ay α < α, [, ad ay aural e equaly olds ( ( α), ( )) = α = F( s, ( s )) ds, F( s, ( s )) ds = (, = F s ( s )) ds,{} ( F( s, ( s )),{}) ds α α m () s ds = ϕ ( ) ϕ ( α) α Te fuco ϕ () s absoluely couous o [, as e egral of e summable fuco w a varable op lm Hece, for ay ε > ere exss δε ( ) > suc a for all α, suc a α < δ e equaly ( ( α), ( )) ε < s far, e sequece { () } = s equcouous Accordg o Asol eorem[8] we ca coose a uformly covergg subsequece of e sequece { () } = Is lm s a couous se-valued mappg a we wll deoe by () As ad e frs summad s less a ε for d = < δ vew of e equcouy of e se-valued mappgs { () } = { ( s )} =, e alog e cose subsequece coverges o () Owg o e eorem codos (5) s possble o pass o e lm uder e sg of e egral We receve a e se-valued mappg () sasfes equao (6) ad ( ) =, e () s e soluo of (5) o e erval [, ( ( s ), s ( )) ( ( s ), ( s)) + ( ( s), ( s)), 4) I case we e fuco φ () cages sg o e erval [,, e exsece of e soluo s proved combg cases )-3) Te eorem s proved Teorem 3 Le e se-valued mappgs F(, ), (, ): ( ) ( F ) R CC R CC R e doma Q = {(, ) R CC( R ): [,, (, ) b} sasfy e codos of Teorem ad sasfy e codos ( F(, '), F(, ")) L( ', "), ( F (, '), F (, ")) L ( ', ") for all (, '),(, ") Q Te ere exss e uque soluo of equao () defed o e erval [, Te proof s smlar o[7,4] Fally we cosder example for case CC( R ) Example 6 Cosder e followg dffereal equao w geeralzed dervave D Φ( ) =Φ( ), () = S(), (8) were : R CC( R ) S (),, I s obvous a () = e s e soluo of S,5 (), > e dffereal equao (8) (see fg 4) Fgure 4 Te grap of a soluo of sysem (8) Remar 5 Also s possble o prove e smlar resuls f CC( R ) be a space of all oempy M- srogly covex closed ses of R ad all eleme of R [9] Remar 6 Le's oce a we cosdered some couous fuco φ() bu s also possble o ae φ (, ) = dam () c(), for example, were c ()- s e dameer of some ealo se-valued mappg 4 Coclusos I s paper e cocep of geeralzed dffereably (proposed [9]) for se-valued mappgs s used Te ew ype of e se-valued dffereal equao geeralzed se dffereal equaos s cosdered Te exsece ad uqueess eorems for se-valued dffereal equaos

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