On Stability of a Class of Fractional Differential Equations
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- Clarence Dalton
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1 Proceedgs of the Pkst Acdey of Sceces 49 (: (0 Coyrght Pkst Acdey of Sceces SSN: Pkst Acdey of Sceces Orgl Artcle O Stblty of Clss of Frctol Dfferetl Equtos Rbh W. brh* sttute of Mthetcl Sceces, Uversty Mly, Kul Luur 50603, Mlys Abstrct: ths er, e cosder the Hyers-Ul stblty for frctol dfferetl equtos of the for: D f ( = G( f (, f ( ;, < colex Bch sce. Furtherore, lctos re llustrted. Keyords: Alytc fucto; Ut dsk; Hyers-Ul stblty; Adssble fuctos; Frctol clculus; Frctol dfferetl equto. NTRODUCTON A clsscl roble the theory of fuctol equtos s tht: f fucto f roxtely stsfes fuctol equto E he does there exsts exct soluto of E hch f roxtes. 940, Ul [] osed the questo of the stblty of Cuchy equto d 94, D. H. Hyers solved t []. 978, Rsss [3] rovded geerlto of Hyers, theore rovg the exstece of uque ler gs er roxte ddtve gs. The roble hs bee cosdered for y dfferet tyes of sces [4-6]. Recetly, L d Hu [7] dscussed d roved the Hyers-Ul stblty of scl tye of fte olyol equto, d Bdkh, Meerj d Gordj [8] troduced the Hyers-Ul stblty of geerled fte olyol equto. Flly, Rsss [9] osed Cuchy tye ddtve fuctol equto d vestgted the geerlsed Hyers-Ul roduct-su stblty of ths equto. The clss of frctol dfferetl equtos of vrous tyes lys ortt roles d tools ot oly thetcs but lso hyscs, cotrol systes, dycl systes d egeerg to crete the thetcl odelg of y hyscl heoe. Nturlly, such equtos requred to be solved. My studes o frctol clculus d frctol dfferetl equtos, volvg dfferet oertors such s Re-Louvlle oertors [0], Erdély-Kober oertors [], Weyl-Res oertors [], Cuto oertors [3] d Grüld-Letkov oertors [4], hve ered durg the st three decdes. The exstece of ostve soluto d ult-ostve solutos for oler frctol dfferetl equto re estblshed d studed [5]. Moreover, usg the cocets of the subordto d suerordto of lytc fuctos, the exstece of lytc solutos for frctol dfferetl equtos colex do re suggested d osed [6-8]. Srvstv d O [9] gve deftos for frctol oertors (dervtve d tegrl the colex -le C s follos:.. Defto: The frctol dervtve of order s defed, for fucto f ( D d f ( := ( d 0 f ( ( d, here the fucto f ( s lytc slycoected rego of the colex -le C cotg the org d the ultlcty of Receved, Deceber 0; Acceted, Mrch 0 *El: rbhbrh@yhoo.co
2 40 Rbh W. brh ( s reoved requrg ( be rel he ( > 0. log to.. Defto: The frctol tegrl of order > 0 s defed, for fucto f (, f ( := f ( ( ( 0 d ; > 0, here the fucto f ( s lytc slycoected rego of the colex -le (C cotg the org d the ultlcty of ( s reoved requrg log ( to be rel he ( > 0... Rerk: D d ( =, > ( ( =, >. ( [7], t s sho the relto D f ( = D f ( = f (. Let U := { C: < } be the oe ut dsk the colex le C d H deote the sce of ll lytc fuctos o U. Here e suose tht H s toologcl vector sce edoed th the toology of ufor covergece over coct subsets of U. Also for C d N, let H [, ] be the subsce of H cosstg of fuctos of the for f ( =, U. Defto.3. Let be rel uber. We sy tht = f ( ( =0 hs the geerled Hyers-Ul stblty f there exsts costt K > 0 th the follog roerty: for every > 0, U = U U, f (, ( =0 =0 (0, the there exsts soe equto ( such tht K, (, U, N. U tht stsfes the reset er, e study the geerled Hyers-Ul stblty for holoorhc solutos of the frctol dfferetl equto colex Bch sces X d Y D f ( = G( f (, f ( ;, <, ( here U Y d f : U X re holoorhc fuctos such tht f (0 = ( s the ero vector X. Recetly, the uthors studed the ul stblty for dfferet tyes of frctol dfferetl equtos [0-].. RESULTS ths secto e reset extesos of the geerled Hyers-Ul stblty to holoorhc vector-vlued fuctos. Let X, Y rereset colex Bch sce. The clss of dssble fuctos G ( X, Y, cossts of those fuctos g : X U Y tht stsfy the dssblty codtos: g( r, ks;, he r =, s =, ( U, k. We eed the follog results: (3.. Le: [3] Let g G( X, Y. f f : U X s the holoorhc vector-vlued fuctos defed the ut dsk U th f (0 =, the
3 O Stblty of Clss of Frctol Dfferetl Equtos 4 g( f (, f ( ; < f( <. (4.. Theore: Eq. (, f G G( X, Y s the holoorhc vector-vlued fucto defed the ut dsk U the G( f (, f ( ; < G( f (, f ( ;. (5 Proof. By cotuty of G, the frctol dfferetl equto ( hs t lest oe holoorhc soluto f. Accordg to Rerk., the soluto f ( of the roble ( tkes the for f ( = G( f (, f ( ;. Therefore, vrtue of Le., e obt the sserto (5... Theore: Let G G( X, Y be holoorhc vector-vlued fuctos defed the ut dsk U the the equto ( hs the geerled Hyers-Ul stblty for U. Proof. Assue tht G := (, =0 U therefore, Rerk., e hve G( = =0 = f (. Also, U d thus. Accordg to Theore., e hve f ( <=. Let > 0 d U be such tht (. ( = = We ll sho tht there exsts costt K deedet of such tht u K, U, u U d stsfes (. We ut the fucto f ( = =, 0,0 < <, thus, for U, e obt u (, =0 = ( ( ( =0 ( ( ( 6( := K. (6 Wthout loss of geerlty, e cosder = x ( yeldg u = < = = f ( f ( u f ( f ( u f ( u u =, = Ths coletes the roof. 3. APPLCATONS u ths secto, e troduce soe lctos of fuctos to cheve the geerled Hyers-Ul stblty. 3.. Exle: Cosder the fucto U R R G r s r s b (, ; = (, th 0.5, b 0 d G (,,0 = 0. Our s to ly Theore.. ths follos sce.
4 4 Rbh W. brh G( r, ks; = ( r k s b k b = (, he r = s =, U. Hece Theore., e hve : f 0.5, b 0 d f : U X s holoorhc vector-vlued fucto defed U, th f (0 =, the ( f ( f ( b < f ( <. Cosequetly, G( f (, f ( ; <, thus ve of Theore., geerled Hyers-Ul stblty. f hs the 3.. Exle: Assue the fucto X s (, ; = (, =, G r s G r s re th G (, =. By lyg Theore., e eed to sho tht G G( X, X. Sce G( r, ks = k = e, re ks he r = s =, k. Hece, Theore., e hve : For f : U X s holoorhc vector-vlued fucto defed U, th f (0 =, the, f ( ( < f e f( <. Cosequetly, G( f (, f ( ; <; thus ve of Theore., geerled Hyers-Ul stblty Exle: Let, b : U C stsfy ( b(, f hs the for every, > d U. Cosder the fucto Y G( r, s; = ( r b( s, th G (, =. No for r = s =, e hve G( r, s; = ( b( d thus G G( X, Y. f f : U X s holoorhc vector-vlued fucto defed U, th f (0 =, the ( f ( b( f ( f ( <. < Hece ccordg to Theore., f hs the geerled Hyers-Ul stblty Exle: Let :U C be fucto such tht ( > 0, ( for every U. Cosder the fucto Y s G( r, s; = r, ( th G (, =. No for r = s =, e hve k G( r, ks; =, ( k d thus G G( X, Y. f f : U X s holoorhc vector-vlued fucto defed U, th f (0 =, the f ( f( < ( f( <. Hece, ccordg to Theore., f hs the geerled Hyers-Ul stblty.
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