Boundary Value Problems of Conformable. Fractional Differential Equation with Impulses

Transcription

1 Applied Meicl Scieces Vol 2 28 o HIKARI Ld www-irico ps://doiorg/2988/s28823 Boudry Vlue Probles of Coforble Frciol Differeil Equio wi Ipulses Arisr Tgvree Ci Tipryoo d Apisi Ppogpu Depre of Meics Fculy of Sciece Burp Uiversiy Coburi 23 Tild Copyrig 28 Arisr Tgvree Ci Tipryoo d Apisi Ppogpu Tis ricle is disribued uder e Creive Coos Aribuio Licese wic peris uresriced use disribuio d reproducio i y ediu provided e origil wor is properly cied Absrc I is pper we cosider e exisece of boudry vlue probles for coforble frciol differeil equios wi ipulses By usig e oooe ierive eod coupled wi lower d upper soluios we obi e exrel soluios of e boudry vlue proble Keywords: Coforble frciol derivive Boudry vlue probles Moooe ierive ecique Ipulsive differeil equio Iroducio Frciol clculus s bee sudied for ree ceuries s e coveiol clculus cully e resercers ve sudied i sice 695 Te ides of frciol clculus re used by eicis pysiciss d egieers Recely frciol differeil equios re vlued ools i e odelig for y peoe i vrious fields of sciece d egieerig Frciol clculus eciques re widely used i e ipulsive frciol differeil equios see [ ] Te oooe ierive ecique coupled wi e eod of lower d upper soluios provide effecive o prove cosrucive exisece soluio for iiil d boudry vlue probles for ipulsive differeil equios Tis ecique provides sufficie codiios for exisece of soluio; we refer e reder o series of ppers [ 2 6] Correspodig uor

2 378 Arisr Tgvree e l Tis reserc focuses o e coceps of ipulsive coforble frciol differeil equios by usig e oooe ierive eod coupled wi lower d upper soluios We ve cosidered e followig boudry vlue probles for coforble frciol differeil equios wi ipulses 2 x 2 x x D D f x x w J T x I D D x D x I x xt D x D xt () 2 2 were f C J wc J w I I C x x x Te res of e pper is orgized s follows: I Secio 2 we recll soe defiiios d resuls fro coforble frciol clculus I secio 3 we defie e lower d upper soluios we prove e exisece d uiqueess for e lier proble d e copriso resul Te i resuls re coied i Secio 4 d coclusios re coied i Secio 5 2 Preliiries I is secio we prese preliiry resuls eeded i our proofs ler Defiiio 2 [4]Te coforble frciol derivive srig fro poi of fucio f :[ ) of order is defied by: D f f f() li We we wrie D If ( D f )( ) ( D f )( ) li( D f )() exiss o b e If f is differeible e D f ( ) f coforble frciol derivive of f of order exiss o clled differeible o I ddiio if e e f is

3 Boudry vlue probles of coforble frciol 379 Le 22 [4] If (] 2 p d e fucio f g be differeible o e : i D f g D f D g ( ) 2 2 p p- D ii ( ) p( ) for ll cos fucio iii s f ( ) D iv D ( fg) f D g g D f f g D f f D g v g ( ) g g D for ll fucios 2 Defiiio 23 [4] Le (] Te coforble frciol iegrl srig fro poi of fucio f :[ ) of order is defied s: f I f s s ds Teore 24 [4] (Me vlue eore) Le iervl cd f :[ ) be give fucio sisfies: (i) f is coiuous o cd (ii) f is differeible for soe o cd d le e ere exiss cos ec d suc D f ( e) f c f d d c Le 25 [4] If f :[ ) be differeible for ll d e: I D f f f Propery 26 Le bcdbe rel uber Te followig relios old:

4 38 Arisr Tgvree e l b s b i s z Q z dz ds b z z Q z dz d b b ii s z Q z dz ds d c z Q z dz c 3 Meods I is secio we will defie e lower d upper soluios d we prove e exisece d uiqueess d copriso priciple for e lier proble Firs we cover e lier proble io ipulsive iegrl equio Nex we prove exisece d uiqueess of e ipulsive iegrl equio by usig e Bc fixed poi eore Filly we ge ew proof of copriso priciples Le J T J J 2 T J J 2 be sub-iervls of J d e se PC J x : J : x x \ wi x exis d is coiuous everywere excep for soe wi x d x x x PC J d PC J e ors x sup x : J d x x x x is coiuous everywere excep for soe d x x d PC J x PC( J ): x PC exis d re Bc spce wi Le PC PC PC 2 E PC( J ) C ( J ) A fucio x E is clled soluio of e proble () if i sisfies () Defiiio 3 A fucio E is clled lower soluio of e boudry vlue proble () if i sisfies: I D D D f w J D D I T D D T (2) Siilrly fucio E is clled upper soluio of e boudry vlue proble () if i sisfies:

5 Boudry vlue probles of coforble frciol 38 I D D D f w J D D I T D D T (3) Le f fi ( ) for Ji i 2 Te ipulsive iegrl oio is defied s b p p b f s dˆ s f () s ds f () s ds f () s ds bj p p q p q were b p q l l We deoe C z z Z l ) l ˆ Qs () Mp s Np ws s R ( s Q s ds Now we cosider e followig boudry vlue probles of coforble frciol differeil equio wi ipulses subjec o boudry codiio s: D D p Mp Np w p L Dp D p D p L p p p T D p D p T (4) were M N L L 2 re coss d E Le 32 Le p E be soluio of (4) if d oly if ppc J is soluio of e ipulsive iegrl equio:

6 382 Arisr Tgvree e l T ˆ p s Qsds () Lp Z ( ) Z l T j ˆ CQsds () Lp j j j j () ˆ CQsds Lp j j L p j L Dp 2 L Dp wi s x ; d l (5) Proof Suppose p is soluio of (4) We use e coforble frciol iegrl of order of e firs equio for proble (4) fro o J o ge D p D p I Q Le we ve D p D p I Q Iegrig (4) fro o J gi we ge D p D p I Q Fro (4) D p D p L p (6) (7) (8) we ve D p D p L p (9) Subsiuig (9) io (8) i follows D p D p I Q I Q L p Repeig e bove process for J we ve Dp Dp ˆ s Qsds L p () l Puig T i () d fro (4) D p D pt we obi

7 Boudry vlue probles of coforble frciol 383 T D p s ˆ l Q s ds L p () Subsiuig () io () we ge ( ) () Dp RT L p R L p (2) Usig e coforble frciol iegrl of order o bo sides of proble (2) fro o J we ge p p I R( T) I Lp I Rs () I L p d le we ve p p I RT I L p ( ) I R() s I L p (3) Iegrig (2) fro o J us p p I RT I L p ( ) I R() s I L p Fro (4) p L D p we ge (4) p p L D p (5) Fro (3) (4) d (5) we obi

8 384 Arisr Tgvree e l p p I RT I RT I Rs ( ) ( ) ( ) I R s I L p () I L p I L p I L p L D p Repeig e bove process for J we ve p p I RT I RT I R s i i ( ) ( ) ( ) i i i i I R s I L p i i () I L p I L p i i i I L p L D p (6) Puig T i (6) we obi p T p I RT I RT I RT ( ) ( ) ( ) I RT ( ) T I R() s I R() s I R() s I R s 3 I R s T I L p () 4 () I L p I L p I L p T I L p I L p I L 2 p I L p T L D p (7)

9 Boudry vlue probles of coforble frciol 385 Fro (4) p pt d usig propery 26 we ge T p ( ) ˆ sl Qs ds L p Z ( ) Z T j Cd z L p j j j j j Cdz L p j j L p L D p L D p (8) Terefore we obi e iegrl equio (5) s required Coversely i is esily sow by direc copuio e iegrl equio (5) sisfies e ipulsive boudry vlue proble (4) Te proof is copleed We deoe M N L M N i i ii j L j j j M N i i i i L j j L L j MN MN L L

10 386 Arisr Tgvree e l Le 33 For M N L L 2 If e (4) s uique soluio p i E (9) x Proof For y p E we defie operor F by Fy s My s Ny w s s ds L y ˆ ( ) T l T zl zl MyzNywz zdz l j Ly ( ) j j j zl z MyzNywz z dz j Ly j j Ly j L D y L D y (2) By direc copuio we ve FxFy PC M N L MN ii i i j M N Lj j i i j i i L j j L L j xy PC

11 Boudry vlue probles of coforble frciol 387 Fx Fy x y (2) PC PC Ad e se eod d le x { } we ve Fx Fy PC MN L M N L x y PC M N M N L L xy PC x y (22) PC x PC PC by e Bc fixed-poi eore F s uique fixed poi p E d by Le 32 p is lso e uique soluio of (4) Te proof is copleed I view of (2) d (22) Fx Fy x x y s Le 34 Assue p E sisfies: D D p Mp Np w J p L D p D p D p L p p p T D p D p T (23) were coss M N L L 2 x { } d e sisfy: M N 2 2 Lv Lv v v Te p o J Proof Fro Le 34 we prove by cordicio (24)

12 388 Arisr Tgvree e l Firs we sow if p If p J J if b fro (23) we ve D D p Mp Np w b M N (25) Ad fro (23) D p D p Le S By e vlue eore we ge (26) D D p S D p D p (27) Fro (25) we ve D p D (28) Subsiuig (28) io (27) we obi D p D p Fro (26) d (29) we ve (29) D p D p D p D p D p D p 2 D p D p Suig up bove iequliies we obi Puig T ; D p D pt D p Fro (23) we ve D p (3) (3) D p D pt (32) Subsiuig (3) io (32) we ge D p D p us D p Hece fro (3) D p d e secod iequliy of (23)

13 Boudry vlue probles of coforble frciol 389 p is odecresig fucio erefore p p pt if p Te p p pt pt p L D p L D p e Fro J Tis cordics o p pt Tus we le J ere exis J i p ( ) cosider p ( ) sow p If p i 2 d if p b Te or p ( ) We oly Te cse we p ( ) e proof is siilr Nex we for soe d proof cocludes by cordicio for ll J Fro (23) e secod iequliy e p p Terefore wic cordics p pt e p J j j { 2 } Fro (23) we ge suc p i p L D p L ( ) p( ) p is icresig fucio e i p p(t) for soe Tus ere exiss D D p M N (33) d Le S j j D p D p L (34) By e vlue eore we ve D j D p S j j j D p j j D p j (35) Fro (33) we ve ( ) D D p M N ( M N) (36) j Te D pj D p j j Fro (34) we ve D p D p D p L D p Fro (36) d (37) we ve (37) j j j j j j j

14 39 Arisr Tgvree e l j j j j j j2 j j j j j j ( M N) D p D p L ( M N) D p D p L (38) ( M N) D p D p ( M N) D p D p 2 2 Suig up e bove iequliies we obi M N M N L j j D p j D p j v v Lv v v Fro p us Le D p we ge j j j M N M N D p Lv (39) v v J 2 If e by e bove we ve M N D p Lv v v (4) Le J 2 If e e bove eod ogeer by usig (23) d (39) gives M N D p L Terefore fro (4) d (4) we ve 2 v (4) v v M N 2 D p Lv (42) v Filly o sows p () for ll We prove by cordicio Suppose ere exiss J r { 2 } ssue e i r r suc p Cse we

15 Boudry vlue probles of coforble frciol 39 Le Ur r d by e vlue eore we ve p pr r D p U r r (43) Fro (23) we ve p p p p L D p r r r r r (44) Subsiuig (43) io (44) i follows p p r r Lrr r DpU r r Fro (42) siilrly we ve M N 2 p p r r Lrr r Lv v ( M N)( 2) p r p r r r Lr r r2 Lv v ( M N)( 2) p i p i Suig up bove iequliies we obi v Lv p p L v i 2 v i r r v v v v v r i Terefore ( M N)( 2) Lv v p p L v v v v v v v i ( M N)( 2) Fro p d p Lv v we ve

16 392 Arisr Tgvree e l p L v v Hece v v v v v i ( M N)( 2) Lv v v v v v Lvv v i ( M N)( 2) Lv v (45) Tis is cordicio o (24) Cse2 If e r i e proof is siilr (45) d we ve v v v v Lvv v i M N 2 Lv v (46) Fro (45) d (46) is is cordicio o (24) Te proof is copleed 4 Resuls d Discussio Teore 4 Assue e followig codiios old: (H ) Te fucios d re lower d upper soluios of e boudry vlue proble () respecively suc (H 2 ) Te fucio f sisfies o J ; f x 2 y2 fx y M x2x N y2 y were 2 2 (H 3 ) Te fucios I d xx w y y w J I sisfy I D x I D y L D x y I x I y L x y were y x 2

17 Boudry vlue probles of coforble frciol 393 (H 4 ) Coss M N L L 2 sisfy (9) d (24) Te ere exis oooe sequeces li uiforly coverge o J d xil soluios of proble () d x E suc li x x x re iil d Proof Firs we cosider e ierio forul Mu Nu w L u u 2 D D u Mu Nu w f u u w u I D u L D u D u Du Du I u u u T D u D u T (47) Here u or u 2 By Le 33 e ierio forul (47) s uique soluio Nex i wo seps we sow e sequeces re oooe sequece d Sep We cli d Le p we ve d f w M Nw Mp Np w D D p f w M N w p I D I D L D D L D p

18 394 Arisr Tgvree e l d D p D I D I L D p L p d p T p T D T D D pt d D p D D By Le 34 p wic iplies for ll J Siilrly we c prove Sep 2 We prove we Le p e for J d by H 2 we ve d by H ; M N w M Nw f w M N w Mp Np D D p M N w f w 3 p I D L D D I D L D D L D p d D I L D p L p D p D I L

19 Boudry vlue probles of coforble frciol 395 d p T T p T D T D T D pt d D p D D By Le 34 p wic iplies Siilrly we c prove d Fro sep d sep 2 we prove d e followig iequliies: old Terefore ere exis fucios li x uiforly coverge o J sisfy proble () for ll J for ll J re oooe sequece d x x suc li x d Clerly x d x Filly we sow x x re iil d xil soluios of e proble Le x x be y soluio of proble () wic sisfies J Also suppose ere exiss posiive ieger suc for J x Le p x e for J we ve d d M Nw f x xw Mp Np w D x D D p M N w f w p I D L D D I L D p D x I x D p D I L D p L p

20 396 Arisr Tgvree e l D p D D x D T D x T D pt d p x T x T p T d for ll J By Le 34 p wic iplies p By siilr eod we c sow p J x for ll J wic iplies x p x is copleed for ll Terefore Te proof 5 Coclusios We ve cosidered boudry vlue probles for coforble frciol differeil equios wi ipulses By usig e oooe ierive eod coupled wi lower d upper soluios we obi e exrel soluios of e boudry vlue probles for coforble frciol differeil equios wi ipulses Refereces [] Jili Li Periodic boudry vlue proble for e secod-order ipulsive iegro- differeil equios Applied Meics d Copuio 98 (28) ps://doiorg/6/jc27879 [2] Ju J Nieo Ros Rodriguez-Lpoez New copriso resuls for ipulsive iegro differeil equios d pplicios J M Al Appl 328 (27) ps://doiorg/6/jj26629 [3] Supw Aswsri Soiris K Nouys Pollri Tirus d Jessd Triboo Periodic boudry vlue probles for coforble frciol iegro-differeil equios Boudry Vlue Probles 26 (26) 22 ps://doiorg/86/s [4] Tber Abdeljwd O coforble frciol clculus Jourl of Copuiol d Applied Meics 279 (25) ps://doiorg/6/jc246 [5] Weer Yuuo Suep Sui Soiris K Nouys d Jessd Triboo Boudry vlue probles for ipulsive uli-order Hdrd frciol differeil equios Boudry Vlue Probles 25 (25) 48

21 Boudry vlue probles of coforble frciol 397 ps://doiorg/86/s [6] Wei Dig Mo H Jurog Mi Periodic boudry vlue proble for e secod order ipulsive fuciol differeil equios Copu M Appl 5 (25) ps://doiorg/6/jcw253 [7] Xi Fu d Xiio Bo Soe exisece resuls for olier frciol differeil equios wi ipulsive d frciol iegrl boudry codiios Advce i Differece Equio 24 (24) 29 ps://doiorg/86/ Received: Februry 4 28; Publised: Mrc 9 28