Ulam stability for fractional differential equations in the sense of Caputo operator

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1 Sogklaakar J. S. Tehol. 4 (6) Nov. - De Orgal Artle Ulam stablty for fratoal dfferetal equatos the sese of Caputo operator Rabha W. Ibrahm* Isttute of Mathematal Sees Uversty Malaya 56 Kuala Lumpur Malaysa. Reeved 19 Aprl 212; Aepted 15 August 212 Abstrat I ths paper we osder the Hyers-Ulam stablty for the followg fratoal dfferetal equatos the sese of omplex Caputo fratoal dervatve defed the ut dsk: D ß f()=g(f() D á f()f ();) <á<1<ß<2. Furthermore a geeralato of the admssble futos omplex Baah spaes s mposed applatos are llustrated. Keywords: aalyt futo ut dsk Hyers-Ulam stablty admssble futos fratoal alulus omplex fratoal dfferetal equato Caputo fratoal dervatve 1. Itroduto A lassal problem the theory of futoal equatos s that: If a futo f approxmately satsfes futoal Equato E whe does there exst a exat soluto of E whh f approxmates. Ulam (1964) mposed the questo of the stablty of Cauhy equato 1941 solved t (Hyers 1957). Rassas (1978) provded a geeralato of Hyers theorem provg the exstee of uque lear mappgs ear approxmate addtve mappgs. The problem has bee osdered for may dfferet types of spaes (Hyers 198; Hyers Rassas1992; Hyers et al.1998). Reetly L Hua (29) dsussed proved the Hyers- Ulam stablty of spaal type of fte polyomal equato Bdkham et al. (21) trodued the Hyers-Ulam stablty of geeraled fte polyomal equato. Fally Rassas (211) mposed a Cauhy type addtve futoal equato vestgated the geeraled Hyers-Ulam produt-sum stablty of ths equato. The lass of fratoal dfferetal equatos of varous types plays mportat roles tools ot oly * Correspodg author. Emal address: rabhabrahm@yahoo.om mathemats but also physs otrol systems dyamal systems egeerg to reate the mathematal modelg of may physal pheomea. Naturally suh equatos requred to be solved. There are dfferet fratoal operators appeared durg the past three deades suh as Rema- Louvlle operators Erdély-Kober operators Weyl-Res operators Grüwald-Letkov operators (Podluby 1999). The ma advatage of Caputo fratoal dervatve s that the fratoal dfferetal equatos wth Caputo fratoal dervatve use the tal odtos (ludg the mxed boudary odtos) o the same harater as for the teger-order dfferetal equatos (Podluby 1999). I the preset work we wll show aother advatage of Caputo fratoal dervatve based o admssble futos omplex Baah spaes. 2. Prelmares Let U := { C : < 1} be the ope ut dsk the omplex plae C H deote the spae of all aalyt futos o U. Here we suppose that H as a topologal vetor spae edowed wth the topology of uform overgee over ompat subsets of U. Also for a C m N let H [ a m] be the subspae of H osstg of

2 72 futos of the form f a a a U m m 1 ( ) = m m1. Srvastava Owa (1989) posed deftos for fratoal operators (dervatve tegral) the omplex -plae C as follows: Defto 2.1 The fratoal dervatve of order < < 1 s defed for a futo f () 1 d f ( ) f ( ) := d (1 ) d ( ) where the futo f () s aalyt smplyoeted rego of the omplex -plae C otag the org the multplty of ( ) s removed requrg log ( ) to be real whe ( ) >. Defto 2.2 The fratoal tegral of order > s defed for a futo f () 1 I f f d ( ) ( ) := ( )( 1 ) ; > where the futo f () s aalyt smply-oeted rego of the omplex -plae (C) otag the org 1 the multplty of ( ) s removed requrg log ( ) to be real whe ( ) >. Note that Defto orrespod to the Rema-Louvlle dervatve tegral respetvely the real form. Remark 2.1 D ( 1) = > 1 ( 1) I ( 1) = > 1. ( 1) It was show that (Ibrahm Darus 28) I f ( ) = I f ( ) = f ( ) f () =. Defto 2. The Caputo fratoal dervatve of order > s defed for a futo f () ( ) 1 f ( ) 1 ( ) ( ) f ( ) := d where = [ ] 1 (the otato [] sts for the largest teger ot greater tha ) the futo f () s aalyt smply-oeted rego of the omplex -plae C otag the org the multplty of ( ) s removed 1 requrg log ( ) to be real whe ( ) >. Remark 2.2 The followg relatos hold: () Represetato D f ( ) = I D f ( ) 1 < < ; () The Caputo fratoal dervatve of the power futo ( 1) = = ; ( 1) () (v) Learty I D f ( ) = f ( ) U f () = (1); D ( f ( ) g( )) = D f ( ) D g( ); (v) No-ommutato D D f ( ) D D f ( ). More detals o fratoal dervatves ther propertes applatos a be foud Klbas et al. (26); Sabater et al. (27); L et al. (29) L et al. (211). We ext trodue the geeraled Hyers-Ulam stablty depedg o the propertes of the fratoal operators. Reetly the author studed the geeraled Hyers-Ulam stablty for varous types of fratoal dfferetal equatos (Ibrahm 211; Ibrahm 212abd). Defto 2.4 Let p be a real umber. We say that a = f ( ) (1) = has the geeraled Hyers-Ulam stablty f there exsts a ostat K > wth the followg property: for every > wu = U U f p a a w ( ) = = 2 the there exsts some U that satsfes equato (1) suh that w K ( wu N). I the preset paper we study the geeraled Hyers- Ulam stablty for holomorph solutos of the fratoal dfferetal equato omplex Baah spaes X Y where D f ( ) = G( f ( ) D f ( ) f ( ); ) (2) ( < < 1 < < 2) G : X U Y f : U X are holomorph futos suh that f () = ( s the ero vetor X).. Geeraled Hyers-Ulam stablty I ths seto we preset extesos of the geeraled Hyers-Ulam stablty to holomorph vetor-valued futos. Let X Y represet omplex Baah spaes. The lass of admssble futos G ( X Y ) ossts of those futos g : X U Y that satsfy the admssblty odtos: g( r ks lt; 1 whe r s t 1 ()

3 7 We eed the followg results: Lemma.1 (Hll 1957) If f : D X s holomorph the f s a subharmo of D C. It follows that f a have o maxmum D uless f s of ostat value throughout D. Lemma.2 (Mller Moau 2) Let f : U X be the holomorph vetor-valued futo defed the ut dsk U wth f () = (the ero elemet of X). If there exsts a U suh that the f ( ) max f = f ( ) f ( 1. Theorem.1 Let G G( X Y ). If f : U X s a holomorph vetor-valued futo defed the ut dsk U wth f () = the G( f ( ) D f ( ) f ( ); ) 1 f ( ) 1. (4) Proof From Defto 2. we observe that D f ( ) = f ( ) (1 ) ( ) f ( ) (2 ) 1 d f ( ) U. (2 ) Assume that f ( ) 1 for U. Thus there exsts a pot U for whh f ( ) 1. Aordg to Lemma.1 we have f ( ) 1 ( U = { : < = r }) max r f ( ) f ( ) 1. I vew of Lemma.2 at the pot there s a ostat 1 suh that Therefore f ( ) f ( ). f ( ) f ( ) D f (2 ) (2 ) (2 ) ( ) = = osequetly we obta (2 ) f ( ) = f ( ) 1 = f ( ) = 1 1. We put k := 1 (2 ) we dedue l := ; hee from Equato () G( f ( ) D f ( ) f ( ); ) G( f ( ) k[ D f ( )/ k] l[ f ( )/ l]; ) 1 whh otradts the hypothess (4) we must have f < 1. Corollary.1 Assume the problem (2). If G G( X Y ) s the holomorph vetor-valued futo defed the ut dsk U the G( f ( ) f ( ) f ( ); ) < 1 I G( f ( ) D f ( ) f ( ); ) < 1. (5) Proof By otuty of the fratoal dfferetal equato (2) has at least oe holomorph soluto f satsfyg ( f () = f () = ). Aordg to Remark 2.2 the soluto f() of the problem (2) takes the form f ( ) = I G( f ( ) f ( ) f ( ); ). Therefore vrtue of Theorem.1 we obta the Asserto (5). Theorem.2 Let G G( X Y ) be holomorph vetor-valued futos defed the ut dsk U the the Equato (2) has the geeraled Hyers-Ulam stablty for U. Proof Assume that G( ) := U = therefore Remark 2.1 we have I G( ) = a = f ( ). = Also U. thus 1. Aordg to Theorem.1 we have f ( ) < 1 =. Let > w U be suh that p a ( ). 2 a w =1 =1 We wll show that there exsts a ostat K depedet of suh that w u K w U u U satsfes (1). We put the futo 1 f ( w) = aw (6) a =1 ( a < < 1)

4 74 thus for w U we obta w u = w f ( w) f ( w) u w f ( w) f ( w) u < w f ( w) w u 1 = w aw a =1 w u 1 a w w u. a =1 Wthout loss of geeralty we osder a = max 1( a ) yeldg 1 w u a w a (1 ) =1 p a ( ) a (1 ) 2 = ( ) (1 ) 2 p1 a 1 = 2 a (1 ) := K. Ths ompletes the proof. 4. Applatos p1 I ths seto we trodue some applatos of futos to aheve the geeraled Hyers-Ulam stablty. Example 4.1 Cosder the futo G : X U R G( r s t; ) = a( r s t ) b wth a.5 b G( ) =. Our am s to apply Corollary.1 ths follows se G( r ks t; ) a( r k s l t ) b = a(1 k l) b 1 whe r s t 1 U. Hee Corollary.1 we have : If a.5 b f : U X s a holomorph vetor-valued futo defed U wth f () = the a( f ( ) D f ( ) f ( ) ) b < 1 f ( ) < 1. Cosequetly I G( f ( ) D f ( ) f ( ); ) < 1 thus vew of Theorem.2 f has the geeraled Hyers- Ulam stablty. Example 4.2 Assume the futo G r s t G r s t re s t 1 ( ; ) = ( ) = G : X X wth G( ) =. By applyg Corollary.1 we eed to show that G G( X X ). Se G r ks lt re e ks lt 1 kl 1 ( ) 1 whe r s t 1 k 1 l 1. Hee Corollary.1 we have : For f : U X s a holomorph vetor-valued futo defed U wth f () = the Cosequetly D f ( ) f ( ) 1 f ( ) e < 1 f ( ) < 1. I G( f ( ) D f ( ) f ( ); ) < 1 thus vew of Theorem.2 f has the geeraled Hyers- Ulam stablty. Example 4. Let a b : U C satsfy a( ) b( ) ( ) 1 for every 1 > 1 U. Cosder the futo G : : X Y G( r s t; ) = a( ) r b( ) s ( ) t wth G( ) =. Now for r s t 1 we have G( r s t; ) = a( ) b( ) ( ) 1 thus G G( X Y ). If f : U X s a holomorph vetor-valued futo defed U wth f () = the a( ) f ( ) b( ) D f ( ) ( ) f ( ) < 1 f ( ) < 1. Hee aordg to Theorem.2 f has the geeraled Hyers- Ulam stablty. Referees Bdkham M. Meerj H. A. Gordj M. E. 21. Hyers- Ulam stablty of polyomal equatos. Abstrat Appled Aalyss Hll E. Phllps R. S Futoal Aalyss Semgroup Amera Mathematal Soety U.S.A. pp Hyers D. H O the stablty of lear futoal equato. Proeedgs of the Natoal Aademy of Sees Hyers D. H The stablty of homomorphsms related tops. Global Aalyss-Aalyss o Mafolds Hyers D. H. Rassas Th. M Approxmate homomorphsms Aequatoes mathematae

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