Fractional operators with exponential kernels and a Lyapunov type inequality

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1 Abdeljwd Advnces in Difference Equions (2017) 2017:313 DOI /s RESEARCH Open Access Frcionl operors wih exponenil kernels nd Lypunov ype inequliy Thbe Abdeljwd* * Correspondence: bdeljwd@psu.edu.s Deprmen of Mhemics nd Physicl Sciences, Prince Suln Universiy, P.O. Box 66833, Riydh, 11586, Sudi Arbi Absrc In his ricle, we exend frcionl clculus wih nonsingulr exponenil kernels, iniied recenly by Cpuo nd Fbrizio, o higher order. The exension is given o boh lef nd righ frcionl derivives nd inegrls. We prove exisence nd uniqueness heorems for he Cpuo (CFC) nd Riemnn (CFR) ype iniil vlue problems by using Bnch conrcion heorem. Then we prove Lypunov ype inequliy for he Riemnn ype frcionl boundry vlue problems wihin he exponenil kernels. Illusrive exmples re nlyzed nd n pplicion bou Surm-Liouville eigenvlue problem in he sense of his frcionl clculus is given s well. Keywords: CFC frcionl derivive; CFR frcionl derivive; Lypunov inequliy; boundry vlue problem; higher order; exponenil kernel 1 Inroducion nd preliminries Frcionl clculus [ ] hs been rcive o mny reserchers in he ls hree decdes or so. Some reserchers hve found i necessry o define new frcionl derivives wih differen singulr or nonsingulr kernels in order o provide more sufficien re o model more rel-world problems in differen fields of science nd engineering [, ]. In [ ] he uhors sudied new ype of frcionl derivives where he kernel is of exponenil ype nd in [, ] he uhors sudied new frcionl derivives wih Mig-Leffler kernels. For he discree couner prs we refer o he work in [ ]. In his work we exend he frcionl clculus wih exponenil kernels proposed nd sudied in [, ] o higher order, prove some exisence nd uniqueness heorems nd prove Lypnouv ype inequliies for boundry vlue problems in he frme of his clculus. The exension is chieved for boh lef nd righ frcionl derivives nd inegrls so h we prepre for inegrion by prs in higher order o serve frcionl vriionl clculus in he frme of his clculus [, ]. Definiion ([ ]) For α >, R nd f rel-vlued funcion defined on [, ), he lef Riemnn Liouville frcionl inegrl is defined by α I f () = (α) ( s)α f (s) ds. The Auhor(s) This ricle is disribued under he erms of he Creive Commons Aribuion 4.0 Inernionl License (hp://creivecommons.org/licenses/by/4.0/), which permis unresriced use, disribuion, nd reproducion in ny medium, provided you give pproprie credi o he originl uhor(s) nd he source, provide link o he Creive Commons license, nd indice if chnges were mde.

2 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 2 of 11 This is frcionlizing of he n-iered inegrl ( I n f )()= 1 (n 1)! ( s)n 1 f (s) ds.therigh frcionl inegrl ending b is defined by ( I α b f ) ()= 1 Ɣ(α) (s ) α 1 f (s) ds. Definiion 2 ([8, 15]) Le f H 1 (, b), < b, α [0, 1], hen he definiion of he new (lef Cpuo) frcionl derivive in he sense of Cpuo nd Fbrizio becomes D α f ) ()= B(α) 1 α f ( x)α [ α (x)e 1 α ] dx (1) nd in he lef Riemnn-Liouville sense hs he following form: D α f ) ()= B(α) d 1 α d The ssocied frcionl inegrl is Iα f ) ()= 1 α B(α) f ()+ α B(α) ( x)α [ α f (x)e 1 α ] dx. (2) f (s) ds, (3) where B(α) > 0 is normlizion funcion sisfying B(0) = B(1) = 1. In he righ cse we hve D α b f ) ()= B(α) 1 α f (x )α [ α (x)e 1 α ] dx (4) nd in he righ Riemnn-Liouville sense hs he following form: D α b f ) ()= B(α) d 1 α d The ssocied frcionl inegrl is Ib α f ) ()= 1 α B(α) f ()+ α B(α) (x )α [ α f (x)e 1 α ] dx. (5) f (s) ds. (6) In [8, 15], i ws verified h IαCFR D α f )()=f() nd D αcf Iα f )()=f(). Also, in he righ cse Ib αcfr D α b f )()=f() nd(cfr D α b CF Ib α f )()=f(). From [8, 15]werecllhe relion beween he Riemnn-Liouville nd Cpuo new derivives s D α f ) ()= D α f ) () B(α) 1 α f ()e α 1 α ( )α. (7) In nex secion, we exend Definiion 2 o rbirry α >0. Lemm 1 ([15]) For 0<α <1,we hve IαCFC D α f ) (x)=f (x) f ()

3 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 3 of 11 nd I α b CFC D α b f ) (x)=f (x) f (b). One of our min purposes in his ricle is o obin he corresponding resul of he following populr Lypunov inequliy resul for CFR boundry vlue problems. Theorem 1 ([21]) If he boundry vlue problem y ()+q()y()=0, (, b), y()=y(b)=0, hs nonrivil soluion, where q is rel coninuous funcion, hen q(s) ds > 4 b. (8) The generlizion of he bove Lypunov inequliy o frcionl boundry vlue problems hs been he ineres of some reserchers in he ls few yers. For exmples, we refer he reder o [22 26]. For discree frcionl counerprs of Lypunov inequliies we refero[27] ndforheq-frcionl ypes we refer o [28]. For recen exensions o higher order nd Lypunov ype inequliies for frcionl operors wih Mig-Leffler kernels nd frcionl difference operors wih discree exponenil kernels we refer o [29]nd [30], respecively. For he Lypnunov inequliies of frcionl difference operors wih discree Mig-Leffer kernels we refer o [31]. 2 The higher order frcionl derivives nd inegrls Definiion 3 Le n < α n +1ndf be such h f (n) H 1 (, b). Se β = α n. Then β (0, 1] nd we define D α f ) ()= D β f (n)) (). (9) In he lef Riemnn-Liouville sense hs his he following form: D α f ) ()= D β f (n)) (). (10) The ssocied frcionl inegrl is Iα f ) ()= ( I ncf Iβ f ) (). (11) Noe h if we use he convenion h ( I 0 f )()=f()henforhecse0<α 1wehve β = α nd hence ( I α f )()=( I α f )() s in Definiion 2. Also, he convenion f (0) ()=f() leds o D α f )()= D α f )()nd D α f )()= D α f )()for0<α 1. Remrk 1 In Definiion 3, ifweleα = n +1henβ =1ndhence D α f )() = D 1 f (n) )()=f (n+1) (). Also, by noing h I1 f )()=( I 1 f )(), we see h for α = n +1 we hve Iα f )() =( I n+1 f )(). Also, for 0 < α 1 we reobin he conceps defined in Definiion 2. Therefore, our generlizion o he higher order cse is vlid. Anlogously, in he righ cse we hve he following exension.

4 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 4 of 11 Definiion 4 Le n < α n +1ndf be such h f (n) H 1 (, b). Se β = α n. Then β (0, 1] nd we define D α b f ) ()= D β b ( 1)n f (n)) (). (12) In he righ Riemnn-Liouville sense i hs he following form: D α b f ) ()= D β b ( 1)n f (n)) (). (13) The ssocied frcionl inegrl is I α b f ) ()= ( I n b CF I β b f ) (). (14) The nex proposiion explins he cion of he higher order inegrl operor CF Iα on he higher order CFR nd CFC derivives nd vice vers, nd he cion of he CFR derivive on he CF inegrl. Proposiion 1 For u() defined on [, b] nd α (n, n +1],for some n N 0, we hve: D αcf Iα u)()=u(). IαCFR D α u)()=u() n 1 u (k) () ( ) k. IαCFC D α u)()=u() n u (k) () ( ) k. Proof ByDefiniion3 nd he semen fer Definiion 2 we hve D αcf Iα u ) ( ()= CFR D β dn d n I ncf ) Iβ u () = D β CF Iβ u ) ()=u(), (15) where β = α n. ByDefiniion3 nd he semen fer Definiion 2 we hve IαCFR D α u ) ()= ( I ncf CFR Iβ D β u (n)) () n 1 = I n u (n) ()=u() u (k) () ( ) k. (16) By Lemm 1 pplied o f ()=u (n) () we hve IαCFC D α u ) ()= I n I β CFC D β u (n) ()= I n[ u (n) () u (n) () ] n 1 = u() = u() n Similrly,forherighcsewehvehefollowing. u (k) () ( ) k u (n) ( )n () n! u (k) () ( ) k. (17)

5 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 5 of 11 Proposiion 2 For u() defined on [, b] nd α (n, n +1],for some n N 0, we hve: D α b CF I α b u)()=u(). I α b CFR D α b u)()=u() n 1 ( 1) k u (k) (b) (b ) k. I α b CFC D α b u)()=u() n ( 1) k u (k) (b) (b ) k. Exmple 1 Consider he iniil vlue problem: D α y ) ()=K(), [, b], (18) where K() is coninuouson [, b]. We consider wo cses depending on he order α. Assume 0<α 1, y()=c nd K()=0. By pplying CF Iα nd mking use of Proposiion 1, we ge he soluion y()=c + 1 α B(α) K()+ α B(α) K(s) ds. Noice h he condiion K()=0verifies h he iniil condiion y()=c.also noice h when α 1 we reobin he soluion of he ordinry iniil vlue problem y ()=K(), y()=c. Assume 1<α 2, K()=0, y()=c 1, y ()=c 2. By pplying CF Iα nd mking use of Proposiion 1 nd Definiion 3 wih β = α 1,wegehesoluion y()=c 1 + c 2 ( )+ 2 α K(s) ds + α 1 ( s)k(s) ds. Noice h he soluion y() verifies y()=c 1 wihou he use of K()=0. However, i verifies y ()=c 2 under he ssumpion K()=0. Also, noe h when α 2 we reobin he soluion of he second order ordinry iniil vlue problem y ()=K(). In he nex secion, we prove exisence nd uniqueness heorems for some ypes of CFC nd CFR iniil vlue problems. Exmple 2 Consider he CFC boundry vlue problem D α y ) ()+q()y()=0, 1<α 2, < < b, y()=y(b)=0. (19) Then β = α 1 nd by Proposiion 1 pplying he operor CF Iα will resul in he soluion y()=c 1 + c 2 ( ) Iα q( )y( ) ) (). Bu Iα q( )y( ))()= 1 β B(β) q(s)y(s) ds + β y()=c 1 + c 2 ( ) 2 α B(β) I 2 q()y(). Hence, he soluion hs he form q(s)y(s) ds α 1 The boundry condiions imply h c 1 =0nd c 2 = 2 α (b ) q(s)y(s) ds + α 1 (b ) ( s)q(s)y(s) ds. (b s)q(s)y(s) ds.

6 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 6 of 11 Hence, (2 α)( ) y()= (b ) 2 α (α 1)( ) q(s)y(s) ds (b ) q(s)y(s) ds α 1 (b s)q(s)y(s) ds ( s)q(s)y(s) ds. (20) 3 Exisence nd uniqueness heorems for he iniil vlue problem ypes In his secion we prove exisence uniqueness heorems for ABC nd ABR ype iniil vlue problems. Theorem 2 Consider he sysem D α y ) ()=f (, y() ), [, b], 0 < α 1, y()=c, (21) such h f (, y()) = 0, A( 1 α B(α) + α(b ) B(α) )<1,nd f (, y 1 ) f (, y 2 ) A y 1 y 2, A >0.Here f :[, b] R R nd y :[, b] R. Then he sysem (21) hs unique soluion of he form y()=c + CF Iα f (, y() ). (22) Proof Firs, wih he help of Proposiion 1, (7) nd king ino ccoun h f (, y()) = 0, i is srighforwrd o prove h y() sisfies he sysem (21) if nd only if i sisfies (22). Le X = {x : mx [,b] x() < } be he Bnch spce endowed wih he norm x = mx [,b] x().onx define he liner operor (Tx)()=c + CF Iα f (, x() ). Then, for rbirry x 1, x 2 X nd [, b], we hve by ssumpion (Tx 1 )() (Tx 2 )() = CF Iα[ f (, x 1 () ) f (, x 2 () )] ( ) 1 α α(b ) A + x 1 x 2, (23) B(α) B(α) nd hence T is conrcion. By he Bnch conrcion principle, here exiss unique x X such h Tx = x nd hence he proof is complee. Remrk 2 Similr exisence nd uniqueness heorems cn be proved for he sysem (21) wih higher order by mking use of Proposiion 1. The condiion f (, y()) = 0 lwys cnnobevoidedswehveseeninexmple1wih f (, y()) = K(). As resul of Theorem 2 we conclude h he frcionl liner iniil vlue problem D α y ) ()=μy(), μ R, [, b], 0 < α 1, y()=c, only cn hve he rivil soluion unless α = 1. Indeed, he soluion sisfies y() =c + μ 1 α αμ y()+ B(α) B(α) y(s) ds.thissoluionisonlyverified if (1 α)y()=0.

7 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 7 of 11 Theorem 3 Consider he sysem D α y ) ()=f (, y() ), [, b], 1 < α 2, y()=c, (24) A (α 1)(b )2 such h ((2 α)(b )+ )<1nd f (, y B(α 1) 2 1 ) f(, y 2 ) A y 1 y 2, A >0. Also, f :[, b] R R nd y :[, b] R. Then he sysem (21) hs unique soluion of he form y()=c + CF Iα f (, y() ) = c + 2 α f ( s, y(s) ) ds + α 1 ( I 2 f (, y( ) )) (). (25) Proof If we pply CF Iα o sysem (24) nd mke use of Proposiion 1 wih β = α 1,henwe obin he represenion (25). Conversely, if we pply CFR D α, mke use of Proposiion 1 nd noe h CFR D α = CFR D β d d c =0, we obin he sysem (24). Hence, y() sisfies he sysem (24) if nd only if i sisfies (25). Le X = {x : mx [,b] x() < } be he Bnch spce endowed wih he norm x = mx [,b] x().onx define he liner operor (Tx)()=c + CF Iα f (, x() ). Then, for rbirry x 1, x 2 X nd [, b], we hve by ssumpion (Tx1 )() (Tx 2 )() = CF Iα[ f (, x 1 () ) f (, x 2 () )] ) A (α 1)(b )2 ((2 α)(b )+ x 1 x 2, (26) 2 nd hence T is conrcion. By he Bnch conrcion principle, here exiss unique x X such h Tx = x nd hence he proof is complee. 4 The Lypunov inequliy for he CFR boundry vlue problem In his secion, we prove Lypunov inequliy for n CFR boundry vlue problem of order 2 < α 3. Consider he boundry vlue problem D α y ) ()+q()y()=0, 2<α 3, (, b), y()=y(b)=0. (27) Lemm 2 y() is soluion of he boundry vlue problem (27) ifndonlyifisisfieshe inegrl equion y()= G(, s)t ( s, y(s) ) ds, (28)

8 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 8 of 11 where G(, s)= }, s b, b ( ( )(b s) ( s)), s b, b { ( )(b s) nd T (, y() ) = Iβ q( )y( ) ) ()= 1 β B(β) q()y()+ β ( I 1 q( )y( ) ) (), β = α 2. B(β) Proof Apply he inegrl CF Iα o (27) nd mke use of Definiion 3 nd Proposiion 1 wih n =2ndβ = α 2oobin 1 b y()=c 1 + c 2 ( ) ( I 2 T (, y( ) )) () = c 1 + c 2 ( ) ( s)t ( s, y(s) ) ds. (29) The condiion y()=0implieshc 1 = 0 nd he condiion y(b)=0implieshc 2 = (b s)t(s, y(s)) ds nd hence y()= b (b s)t ( s, y(s) ) ds Then he resul follows by spliing he inegrl (b s)t ( s, y(s) ) ds = (b s)t ( s, y(s) ) ds + ( s)q(s)t ( s, y(s) ) ds. (b s)t ( s, y(s) ) ds. Lemm 3 The Green s funcion G(, s) defined in Lemm 2 hs he following properies: G(, s) 0 for ll, s b. mx [,b] G(, s)=g(s, s) for s [, b]. H(s, s) hs unique mximum, given by ( + b mx G(s, s)=g s [,b] 2, + b ) = 2 (b ). 4 Proof Iisclerhg 1 (, s)= ( )(b s) 0. Regrding he pr g b 2 (, s)=( ( )(b s) ( s)) we b see h ( s)= b (b ( + (s )(b ) ( ) )) nd h + (s )(b ) ( ) s if nd only if s. Hence, we conclude h g 2 (, s) 0 s well. Hence, he proof of he firs pr is complee. Clerly, g 1 (, s) is n incresing funcion in. Differeniing g 2 wih respec o for every fixed s we see h g 2 is decresing funcion in. Leg(s)=G(s, s)= (s )(b s). Then one cn show h g (s)=0if s = +b b proof is concluded by verifying h g( +b 2 nd hence he 2 b )= 4. Inhenexlemm,weesimeT(, y()) for funcion y C[, b]. Lemm 4 For y C[, b] nd 2<α 3, β = α 2,we hve for ny [, b] T (, y() ) R() y,

9 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 9 of 11 where [ 3 α R()= q() α 2 + B(α 2) B(α 2) q(s) ds ]. Theorem 4 If he boundry vlue problem (27) hs nonrivil soluion, where q() is rel-vlued coninuous funcion on [, b], hen R(s) ds > 4 b. (30) Proof Assume y Y = C[, b] is nonrivil soluion of he boundry vlue problem (27), where y = sup [,b] y(). By Lemm 2, y mus sisfy y()= G(, s)t ( s, y(s) ) ds. Then, by using he properies of he Green s funcion G(, s) proved in Lemm 3 nd Lemm 4, we come o he conclusion h y < b 4 R(s) ds y. From his (30) follows. Remrk 3 Noe h if α 2 +,henr()endso q() nd hence one obins he clssicl Lypunov inequliy (8). Exmple 3 Consider he following CFR Surm-Liouville eigenvlue problem (SLEP) of order 2 < α 3: 0 D α y ) ()+λy()=0, 0< <1,y(0) = y(1) = 0. (31) If λ is n eigenvlue of (31), hen by Theorem 4 wih q()=λ,wehve [ 3 α α 2 ( T()= λ + 0I 1 λ ) ] () B(α 2) B(α 2) [ 3 α = λ B(α 2) + α 2 B(α 2) Hence, we mus hve 1 [ ] 3 α T(s) ds = λ B(α 2) + α 2 >4. 2B(α 2) 0 Hence, [ ] 3 α λ >4 B(α 2) + α B(α 2) ]. (32)

10 Abdeljwd Advnces in Difference Equions (2017) 2017:313 Pge 10 of 11 Noice h he limiing cse α 2 + implies h λ > 4. This is he lower bound for he eigenvlues of he ordinry eigenvlue problem: y ()+λy()=0, 0< <1,y(0) = y(1) = 0. 5 Conclusions Frcionl derivives nd heir corresponding inegrl operors re of impornce in modeling vrious problems in engineering, science nd medicine. To provide he reserchers wih he possibiliy of modeling by mens of higher order rbirry dynmicl sysems we exended frcionl clculus whose derivives depend on nonsingulr exponenil funcion kernels o higher order. The corresponding higher order inegrl operors hve been defined s well nd confirmed. The righ frcionl exension is lso considered. To se up he bsic conceps we proved exisence nd uniqueness heorems by mens of he Bnch fixed poin heorem for iniil vlue problems in he frme of CFC nd CFR derivives. We relized h he condiion f (, y()) = 0 is necessry o gurnee unique soluion nd hence he frcionl liner iniil vlue problem wih consn coefficiens resuls in he rivil soluion unless he order is posiive ineger. We used our exension o higher order o prove Lypunov ype inequliy for CFR boundry vlue problem wih order 2 < α 3 nd hen obined he clssicl ordinry cse when α ends o 2 from he righ. This proves differen behvior from he clssicl frcionl cse, where he Lypunov inequliy ws proved for frcionl boundry problem of order 1 < α 2 nd he clssicl ordinry cse ws verified when α ends o 2 from lef. In connecion o his behvior, we propose he following open problem: Isipossibleoformule sequenil CFR boundry vlue problem whose Green s funcion is so nice s o prove Lypunov ype inequliy? Acknowledgemens The uhor would like o hnk Prince Suln Universiy for funding his work hrough reserch group Nonliner Anlysis Mehods in Applied Mhemics (NAMAM) group number RG-DES Compeing ineress The uhors declre h hey hve no compeing ineress. Auhor s conribuions The uhor red nd pproved he finl mnuscrip. Publisher s Noe Springer Nure remins neurl wih regrd o jurisdicionl clims in published mps nd insiuionl ffiliions. Received: 9 My 2017 Acceped: 18 July 2017 References 1. Smko, G, Kilbs, AA, Mrichev, OI: Frcionl Inegrls nd Derivives: Theory nd Applicions. Gordon & Brech, Yverdon (1993) 2. Podlubny, I: Frcionl Differenil Equions. Acdemic Press, Sn Diego (1999) 3. Kilbs, A, Srivsv, MH, Trujillo, JJ: Theory nd Applicion of Frcionl Differenil Equions. Mhemics Sudies, vol Norh-Hollnd, Amserdm (2006) 4. Tenreiro Mchdo, JA, Kirykov, V, Minrdi, F: A poser bou he recen hisory of frcionl clculus. Frc. Clc. Appl. Anl. 13(3), (2010) 5. Tenreiro Mchdo, JA: Frcionl dynmics of sysem wih pricles subjeced o impcs. Commun. Nonliner Sci. Numer. Simul. 16(12), (2011) 6. Blenu, D, Diehelm, K, Scls, E, Trujillo, JJ: Frcionl Clculus: Models nd Numericl Mehods, 2nd edn. (2016) 7. Bozkur, F, Abdeljwd, T, Hjji, MA: Sbiliy nlysis of frcionl order differenil equion model of brin umor growh depending on he densiy. Appl. Compu. Mh. 14(1),50-62 (2015) 8. Cpuo, M, Fbrizio, M: A new definiion of frcionl derivive wihou singulr kernl. Prog. Frc. Differ. Appl. 1(2), (2015)

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