Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives

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1 Abdll Advnces in Difference Equtions (2018) 2018:107 R E S E A R C H Open Access Oscilltion of differentil equtions in the frme of nonlocl frctionl derivtives generted by conformble derivtives Bheldin Abdll * * Correspondence: bbdllh@psu.edu.s Deprtment of Mthemtics nd Generl Sciences, Prince Sultn University, Riydh, Sudi Arbi Abstrct Recently, Jrd et l. in (Adv. Differ. Equ. 2017:247, 2017) defined new clss of nonlocl generlized frctionl derivtives, clled conformble frctionl derivtives (CFDs), bsed on conformble derivtives. In this pper, sufficient conditions re estblished for the oscilltion of solutions of generlized frctionl differentil equtions of the form { D α, x(t)f 1 (t, x)=r(t)f 2 (t, x), t >, lim t I j α, x(t)=b j (j =1,2,..., m), where m = α, D α, is the left-frctionl conformble derivtive of order α C, Re(α) 0 in the Riemnn Liouville setting nd I α, is the left-frctionl conformble integrl opertor. he results re lso obtined for CFDs in the Cputo setting. Exmples re provided to demonstrte the effectiveness of the min result. MSC: 34A08; 34C10; 26A33 Keywords: Frctionl conformble integrls; Frctionl conformble derivtives; Frctionl differentil equtions; Oscilltion theory 1 Introduction Frctionl clculus is still being developed continuously nd its opertors re used to model complex systems where the kernel of the frctionl opertors reflects the nonloclity [2, 3]. he singulrity of the kernel of the frctionl opertors hs recently motivted reserchers to present new types of frctionl opertors with nonsingulr kernels nd their discrete versions [4 12]. his new trend dded nother pproch in defining frctionl derivtives nd integrls. In clssicl frctionl opertors, the frctionlizing process depends on iterting the weighted usul integrls or some locl type derivtives in the wy to get the fctoril function nd then replce it by the gmm function. he theory of frctionl clculus with opertors hving nonsingulr kernels depends on limiting pproch vi dirc delt functions. Indeed, the frctionl derivtive with the nonsingulr kernel is first defined so tht in the limiting cse α 0wegetthefunctionitselfnd when α 1 we get the usul derivtive of the function. hen the corresponding integrl opertors re evluted by the help of Lplce trnsforms for functions whose convolu- he Author(s) his rticle is distributed under the terms of the Cretive Commons Attribution 4.0 Interntionl License ( which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 2 of 15 tion with the nonsingulr kernel vnishes t the strting point. he oscilltion theory for frctionl differentil nd difference equtions ws studied by some uthors (see [13 21]), thus severl definitions of frctionl derivtives nd frctionl integrl opertors exist in the literture. In this rticle, we study the oscilltion of frctionl opertors defined by the first pproch mentioned bove. Nmely, we investigte the oscilltion of clss of generlized frctionl derivtives defined in [1] by iterting the locl conformble derivtive developed in [22]. We shll nme this derivtive the conformble frctionl derivtive (CFD). Although both the CFD with its Cputo setting nd the Ktugmpol-type derivtive studied in [23 25] coincide when = 0, they re very different from ech other. In fct, the kernel of CFDsdepends on the end points nd b which cuses mny differences from the Ktugmpol-type one. We shll study the oscilltion of conformble frctionl differentil eqution of the form D α, x(t)f 1 (t, x)=r(t)f 2 (t, x), t >, (1) lim t I j α, x(t)=b j (j =1,2,...,m), where m = α, D α, is the left-frctionl conformble derivtive of order α C, Re(α) 0 in the Riemnn Liouville setting nd I α, is the left-frctionl conformble integrl opertor. he objective of this pper is to study the oscilltion of conformble frctionl differentil equtions of the form (1). his will generlize the results obtined in [13, 14]whenwe tke =0. his pper is orgnized s follows. Section 2 introduces some nottions nd provides the definitions of conformble frctionl integrl nd differentil opertors together with some bsic properties nd lemms tht re needed in the proofs of the min theorems. In Sect. 3, the min theorems re presented. Section 4 is devoted to the results obtined for the conformble frctionl opertors in the Cputo setting where we lso remrk the oscilltion of Ktugmpol-type frctionl opertors. Exmples re provided in Sect. 5 to demonstrte the effectiveness of the min theorems. 2 Nottions nd preliminry ssertions he left conformble derivtive strting from of function f :[, ) R of order 0 < 1isdefinedby ( f ) f (t ɛ(t ) ) f(t) (t)=lim. ɛ 0 ɛ If ( f )(t)existson(, b), then ( f )()=lim t ( f )(t). If f is differentible, then ( f ) (t)=(t ) f (t). (2) he corresponding left conformble integrl is defined s I f (x)= x dt f (t), 0< 1. (t ) For the extension to the higher order >1,see[22].

3 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 3 of 15 Definition 2.1 ([1]) he left-frctionl conformble integrl opertor is defined by I α, f (x)= 1 x ( (x ) (t ) f (t) dt, (3) Ɣ(α) (t ) where α C, Re(α) 0. Definition 2.2 ([1]) he left-frctionl conformble derivtive of order α C, Re(α) 0 in the Riemnn Liouville setting is defined by D α, f (x)= m ( I m α,) f (x) m = x ( (x ) (t ) Ɣ(m α) ) m α 1 f (t) dt (t ), where m = Re(α), m = }{{ nd } α is the left conformble differentil opertor presented in (2). m times Definition 2.3 ([1]) he left-cputo frctionl conformble derivtive of order α C, Re(α) 0isdefinedby C Dα, f (x)= I m α,( m f (x) ) = 1 Ɣ(m α) x ( (x ) (t ) ) m α 1 m f (t) dt. (4) (t ) he following identity (see [1]) is essentil to solving liner conformble frctionl differentil equtions: (I α, (t ) ν ) (x)= 1 Ɣ(ν) α Ɣ(α v) (x )(αν 1), Re(ν)>0. (5) Lemm 2.1 (Young s inequlity [26]) (i) Let X, Y 0, u >1, nd 1 u 1 v =1, then XY 1 u Xu 1 v Y v. (ii) Let X 0, Y >0, 0<u <1, nd 1 u 1 v =1, then XY 1 u Xu 1 v Y v, where equlities hold if nd only if Y = X u 1. 3 Oscilltion of conformble frctionl differentil equtions in the frme of Riemnn In this section we study the oscilltion theory for eqution (1). Lemm 3.1 ([1]) Let Re(α)>0,m = [ Re(α)], f L(, b), nd I α, f C, m [, b]. hen I α,( D α, f (x) ) = f (x) m D α j, f () α j Ɣ(α j 1) (x )α j. Using Lemm 3.1, we cn write the solution representtion of (1)s x(t)= m D α j, x() α j Ɣ(α j 1) (t )α j I α, F(t, x), (6) where F(t, x)=r(t)f 2 (t, x) f 1 (t, x)nd >0.

4 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 4 of 15 Asolutionof(1) is sid to be oscilltory if it hs rbitrrily lrge zeros on (0, ); otherwise, it is clled nonoscilltory. An eqution is sid to be oscilltory if ll of its solutions re oscilltory. We prove our results under the following ssumptions: xf i (t, x)>0 (i =1,2),x 0,t 0, (7) f 1 (t, x) p 1 (t) x β nd f 2 (t, x) p 2 (t) x γ, x 0,t 0, (8) f1 (t, x) p1 (t) x β nd f2 (t, x) p2 (t) x γ, x 0,t 0, (9) where p 1, p 2 C([0, ), (0, )) nd β, γ re positive constnts. Define nd (t)=ɣ(α) (t, 1 )= m D α j, f () α j Ɣ(α j 1) (t )α j (10) ( (t ) (s ) F(s, x(s)) ds. (11) (s ) heorem 3.2 Let f 2 =0in (1) nd condition (7) hold. If ( (t ) (s ) =, (12) (s ) nd ( (t ) (s ) for every sufficiently lrge, then every solution of (1) is oscilltory. =, (13) (s ) Proof Let x(t) be nonoscilltory solution of eqution (1)withf 2 =0.Supposetht 1 > is lrge enough so tht x(t) >0fort 1.Hence,(7) impliesthtf 1 (t, x) >0fort 1. Using (3), weget from (6) Ɣ(α)x(t) =Ɣ(α) m D α j, f () (t α j )α j Ɣ(α j 1) ( (t ) (s ) F(s, x(s)) ds (s ) ( (t ) (s ) r(s) f 1 (s, x(s)) ds 1 (s ) ( (t ) (s ) (t) (t, 1 ), (14) (s ) where nd re defined in (10)nd(11), respectively. 1

5 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 5 of 15 Multiplying (14)by )1 α,weget ) 1 α 0< Ɣ(α)x(t) ) 1 α t ) 1 α (t)( (t, 1 ) ( (t ) (s ). (15) (s ) 1 ke 2 > 1. We consider two cses s follows. Cse (1): Let 0 < α 1. hen m =1nd )1 α (t)=b 1 t α (t ) α. Since the function h 1 (t)=t α (t ) α is decresing for >0ndα <1,wegetfor t 2 ) 1 α (t) = b 1 t α (t ) α b 1 α 2 ( 2 ) α := c 1 ( 2 ). (16) he function h 2 (t)=t α [(t ) (s ) ] α 1 is decresing for >0ndα <1.hus,we get ) 1 α (t, 1 ) ( = t ) 1 α ( (t ) (s ) [ ( ) ( )] ds r(s)f2 s, x(s) f1 s, x(s) (s ) ) 1 α ( (t ) (s ) ( ) ( ) r(s)f2 s, x(s) f1 s, x(s) ds (s ) ( ) 1 α ( 2 (2 ) (s ) r(s)f2 ( s, x(s) ) f1 ( s, x(s) ) ds (s ) := c 2 ( 1, 2 ) fort 2. (17) hen, from eqution (15)ndfort 2,weget hence ( (t ) (s ) 1 ( (t ) (s ) which contrdicts condition (12). (s ) [ c 1 ( 2 )c 2 ( 1, 2 ) ], (s ) [ c 1 ( 2 )c 2 ( 1, 2 ) ] >,

6 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 6 of 15 Cse (2): Let α >1.henm 2. Also, ) α <1forα >1nd > 0. he function t h 3 (t)=(t ) j is decresing for j >1nd >0.hus,fort 2,wehve ) 1 α ( (t) = t ) 1 α m b j (t ) α j Ɣ(α) α j Ɣ(α j 1) ( ) t α m b j (t ) j = Ɣ(α) t 1 j Ɣ(α j 1) Ɣ(α) Ɣ(α) m m b j (t ) j 1 j Ɣ(α j 1) b j ( 2 ) j 1 j Ɣ(α j 1) := c 3( 2 ). (18) Also, since )1 α <1nd( (t ) (s ) t <1forα >1nd >0,weget ) 1 α (t, 1 ) ( = t ) 1 α ( (t ) (s ) [ ( ) ( )] ds r(s)f2 s, x(s) f1 s, x(s) (s ) ( (t ) (s ) ( ) ( ) r(s)f t 2 s, x(s) f1 s, x(s) ds (s ) ( ) ( ) r(s)f2 s, x(s) f1 s, x(s) ds (s ) := c 4( 1 ). (19) From (15), (18), nd (19), we conclude tht ( (t ) (s ) (s ) [ c 3 ( 2 )c 4 ( 1 ) ] 1 for t 2.Hence ( (t ) (s ) (s ) [ c 3 ( 2 )c 4 ( 1 ) ] >, which contrdicts condition (12). herefore, we conclude tht x(t) is oscilltory. In cse x(t) is eventully negtive, similr rguments led to contrdiction with condition (13). heorem 3.3 Let conditions (7) nd (8) hold with β > γ. If ( (t ) (s ) [r(s)h(s)] ds =, (20) (s ) nd ( (t ) (s ) [r(s) H(s)] ds (s ) =, (21)

7 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 7 of 15 for every sufficiently lrge, where H(s)= β γ γ γ [ γ β γ p2 (s) p1 (s) β then every solution of (1) is oscilltory. ] β β γ, (22) Proof Let x(t) be nonoscilltory solution of eqution (1), sy x(t)>0fort 1 >. Let s 1. Using conditions (7)nd(8), we get f 2 (s, x) f 1 (s, x) p 2 (s)x γ (s) p 1 (s)x β (s). Let X = x γ (s), Y = γ p 2(s) βp 1 (s), u = β γ,ndv = β, then from prt (i) of Lemm 2.1 we get β γ p 2 (s)x γ (s) p 1 (s)x β (s)= βp [ 1(s) x γ (s) γ p 2(s) γ βp 1 (s) γ β [XY 1u ] Xu = βp 1(s) γ ( x γ (s) ) β γ ] where H is defined by (22). hen from eqution (6) we obtin Ɣ(α)x(t) = (t) (t, 1 ) 1 ( (t ) (s ) [r(s)f 2(s, x(s)) f 1 (s, x(s))] ds (s ) ( (t ) (s ) (t) (t, 1 ) 1 [r(s)p 2(s)x γ (s) p 1 (s)x β (s)] ds (s ) ( (t ) (s ) (t) (t, 1 ) 1 βp 1(s) 1 γ v Y v = H(s), (23) [r(s)h(s)] ds (s ). (24) he rest of the proof is the sme s tht of heorem 3.2 nd hence is omitted. heorem 3.4 Let α 1 nd suppose tht (7) nd (9) hold with β < γ. If nd ( (t ) (s ) ( (t ) (s ) [r(s)h(s)] ds (s ) = (25) [r(s) H(s)] ds (s ) = (26) for every sufficiently lrge, where H is defined by (22), then every bounded solution of (1) is oscilltory.

8 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 8 of 15 Proof Let x(t) be boundednonoscilltorysolution of eqution (1). hen there exist constnts M 1 nd M 2 such tht M 1 x(t) M 2 for t. (27) Assume tht x is bounded eventully positive solution of (1). hen there exists 1 > such tht x(t) >0fort 1 >. Using conditions (7) nd(9), we get f 2 (s, x) f 1 (s, x) p 2 (s)x γ (s) p 1 (s)x β (s). Using (ii) of Lemm 2.1 nd similr to the proof of (23), we find p 2 (s)x γ (s) p 1 (s)x β (s) H(s) fors 1. From (6)ndsimilrto(24), we obtin ( (t ) (s ) Ɣ(α)x(t)= (t) (t, 1 ) Multiplying by )1 α,weget 1 [r(s)h(s)] ds (s ). ) 1 α ) 1 α t ) 1 α Ɣ(α)x(t) (t)( (t, 1 ) ( (t ) (s ) 1 [r(s)h(s)] ds (s ). (28) ke 2 > 1. We consider two cses s follows. Cse (1): Let α =1. hen(16) nd(17) re still correct. Hence, from (28) ndusing(27), we find tht ) 1 α M 2 Ɣ(α) Ɣ(α)x(t) c 1 ( 2 ) c 2 ( 1, 2 ) ( (t ) (s ) [r(s)h(s)] ds (s ) for t 2.hus,weget 1 ( (t ) (s ) [r(s)h(s)] ds 1 c 1 ( 2 )c 2 ( 1, 2 )M 2 Ɣ(α)<, (s ) which contrdicts condition (25). Cse (2): Let α >1. hen(18)nd(19) re still true. Hence, from (28) ndusing(27), we find tht ) 1 α M 2 Ɣ(α) c 3 ( 2 ) c 4 ( 1 ) ( (t ) (s ) [r(s)h(s)] ds (s ) 1

9 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 9 of 15 for t 2.Sincelim )1 α =0forα >1,weconcludetht ( (t ) (s ) 1 [r(s)h(s)] ds (s ) c 3 ( 2 )c 4 ( 1 )<, which contrdicts condition (25). herefore, we conclude tht x(t) isoscilltory. Incse x(t) is eventully bounded negtive, similr rguments led to contrdiction with condition (26). 4 Oscilltion of conformble frctionl differentil equtions in the frme of Cputo In this section, we study the oscilltion of conformble frctionl differentil equtions in the Cputo setting of the form C Dα, x(t)f 1 (t, x)=r(t)f 2 (t, x), t > k x()=b k (k =0,1,...,m 1), (29) where m = α nd C Dα, is defined by (4). Lemm 4.1 [1] Let f C, m [, b], α C. hen I α,( C D α, f (x) ) m 1 = f (x) k f ()(x ) k. k k! Using Lemm 4.1, the solution representtion of (29)cnbewrittens m 1 k x(t)= x()(t ) k k I α, F(t, x), (30) k! where F(t, x)=r(t)f 2 (t, x) f 1 (t, x)nd >0. Define m 1 χ(t)=ɣ(α) k x()(t ) k. (31) k k! heorem 4.2 Let f 2 =0in (29) nd condition (7) hold. If nd ) 1 m ( (t ) (s ) ) 1 m ( (t ) (s ) for every sufficiently lrge, then every solution of (29) is oscilltory. = (32) (s ) = (33) (s )

10 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 10 of 15 Proof Let x(t) be nonoscilltory solution of eqution (29)withf 2 =0.Supposetht 1 > is lrge enough so tht x(t)>0fort 1.Hence(7)impliesthtf 1 (t, x)>0fort 1.Using (3), we get from (30) m 1 k Ɣ(α)x(t) =Ɣ(α) x()(t ) k k k! ( (t ) (s ) F(s, x(s)) ds (s ) ( (t ) (s ) r(s) f 1 (s, x(s)) ds 1 (s ) ( (t ) (s ) χ(t) (t, 1 ), (34) (s ) 1 where χ nd re defined in (31)nd(11), respectively. Multiplying (34)by )1 m,weget ) 1 m 0< Ɣ(α)x(t) ) 1 m t ) 1 m χ(t)( (t, 1 ) ) 1 m ( (t ) (s ). (35) (s ) 1 ke 2 > 1. We consider two cses s follows. Cse (1): Let 0 < α 1. hen m =1nd )1 m χ(t)=ɣ(α)b 0. he function h 4 (t)=( (t ) (s ) is decresing for >0,t > 2 > s,ndα <1.hus, we get ) 1 m (t, 1 ) ( = (t ) (s ) [ ( ) ( )] ds r(s)f2 s, x(s) f1 s, x(s) (s ) ( (t ) (s ) ( ) ( ) r(s)f 2 s, x(s) f1 s, x(s) ds (s ) ( (2 ) (s ) ( ) ( ) r(s)f 2 s, x(s) f1 s, x(s) ds (s ) := c 5 ( 1, 2 ). hen, from eqution (35)ndfort 2,weget ) 1 m ( (t ) (s ) 1 (s ) [ Ɣ(α)b 0 c 5 ( 1, 2 ) ],

11 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 11 of 15 hence ) 1 m ( (t ) (s ) [ Ɣ(α)b 0 c 5 ( 1, 2 ) ] >, (s ) which contrdicts condition (32). Cse (2): Let α >1.henm 2. Also, ) m <1form 2nd > 0. he function t h 3 (t)=(t ) (k m1) is decresing for k < m 1nd >0.hus,fort 2,wehve ) 1 m ( χ(t) = t ) 1 m m 1 k Ɣ(α) x()(t ) k k k! ( ) t m m 1 k = Ɣ(α) x()(t ) (k m1) t k m1 k! m 1 Ɣ(α) m 1 Ɣ(α) k x() (t ) (k m1) k m1 k! k x() ( 2 ) (k m1) := c k m1 6 ( 2 ). (36) k! Also, since )1 m <1nd( (t ) (s ) t <1forα >1nd >0,ndsimilrto(19) we get ) 1 m (t, 1 ) c 4( 1 ). hen, from (35) nd(36), we get contrdiction with condition (32). herefore, we conclude tht x(t) isoscilltory.incsex(t) is eventully negtive, similr rguments led to contrdiction with condition (33). We stte the following two theorems without proof. heorem 4.3 Let conditions (7) nd (8) hold with β > γ. If ) 1 m ( (t ) (s ) [r(s)h(s)] ds = (s ) nd ) 1 m ( (t ) (s ) [r(s) H(s)] ds (s ) = for every sufficiently lrge, where H is defined by (22), then every solution of (29) is oscilltory. heorem 4.4 Let α 1 nd suppose tht (7) nd (9) hold with β < γ. If ) 1 m ( (t ) (s ) [r(s)h(s)] ds = (s )

12 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 12 of 15 nd ) 1 m ( (t ) (s ) [r(s) H(s)] ds (s ) = for every sufficiently lrge, where H is defined by (22), then every bounded solution of (29) is oscilltory. 5 Exmples In this section, we construct numericl exmples to illustrte the effectiveness of our theoreticl results. Exmple 5.1 Consider the Riemnn conformble frctionl differentil eqution D α, x(t)x 5 (t) ln(t e) = 2α (t ) (2 α) [(t ) 10 (t ) 2 3 ] ln(t e)x 1 3 (t) ln(t e), Ɣ(3 α) lim t I 1 α, x(t)=0, 0<α <1, >0, (37) where m =1,f 1 (t, x)=x 5 (t) ln(t e), r(t)= 2α (t ) (2 α) [(t ) 10 (t ) 2 3 ] ln(t e), nd Ɣ(3 α) f 2 (t, x)=x 1 3 (t) ln(t e). It is esy to verify tht conditions (7)nd(8) re stisfied for β =5, γ = 1 3 nd p 1(t)=p 2 (t)=ln(t e). However, we show in the following tht condition (20) does not hold. For every sufficiently lrge 1ndllt,wehver(t)>0.Clculting H(s)sdefinedby(22), we find tht H(s) = 14(15) ln(s e) hen, using (5)with ν =1,weget ( (t ) (s ) [r(s)h(s)] ds (s ) ( (t ) (s ) H(s) ds (s ) ( (t ) 0.77 (s ) (s ) 0 ds (s ) ) 1 α = 0.77 Ɣ(α) ( I α, (s ) 0) (t) 0.77t ( ) t α = =. α t However, using (5)withν = 3, one cn esily verify tht x(t)=(t ) 2 is nonoscilltory solution of (37). he initil condition is lso stisfied becuse I 1 α, (t ) 2 = 2α 1 (t ) (3 α). Ɣ(4 α) Exmple 5.2 Consider the conformble frctionl differentil eqution D α, x(t)x 3 (t)=sin t, lim t I 1 α, x(t)=0, 0<α <1, (38)

13 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 13 of 15 where f 1 (t, x)=x 3 (t), r(t)=sin t, ndf 2 (t, x) = 0. hen condition (7) holds.furthermore, one cn esily check tht nd ( (t ) (s ) sin sds = (s ) ( (t ) (s ) sin sds =. (s ) his shows tht conditions (12) nd(13) of heorem 3.2 hold. Hence, every solution of (38)isoscilltory. Exmple 5.3 Consider the Cputo conformble frctionl differentil eqution C Dα, x(t)e t x 3 (t)= α (t ) (1 α) (t ) 3 e t, Ɣ(2 α) x()=0, 0<α <1, >0, (39) where m =1,f 1 (t, x)=e t x 3 (t), r(t)= α (t ) (1 α),ndf Ɣ(2 α) 2 (t, x) = 0. hen condition (7) isstisfied. However, condition (32)doesnotholdsince ) 1 m ( (t ) (s ) (s ) ( (t ) (s ) (s ) 0 ds (s ) = 1 α α (t )α =. Using (2), (5)withν =2ndthefcttht C Dα, (t ) (ν 1) = I 1 α, (t ) (ν 1), one cn esily check tht x(t)=(t ) is nonoscilltory solution of (39). Remrk 5.1 he oscilltion of frctionl differentil equtions in the frme of Ktugmpol-type frctionl derivtives studied in [23 25] cn be investigted in similr wyswehvedoneinthisrticleforcfdsndtheircputosettings.heredercn verify sufficient conditions nd the proofs by observing the kernel which is free from the strting point. 6 Conclusion In this rticle, the oscilltion theory for conformble frctionl differentil equtions ws studied. Sufficient conditions for the oscilltion of solutions of Riemnn conformble frctionl differentil equtions of the form (1) were given in three theorems in Sect. 3. As 1 in these theorems, we get the results obtined in [13]nd[14]when =0.hemin pproch is bsed on pplying Young s inequlity which will help us in obtining shrper

14 Abdll Advnces in Difference Equtions (2018) 2018:107 Pge 14 of 15 conditions. he oscilltion for the Cputo conformble frctionl differentil equtions hs been investigted s well. Numericl exmples hve been presented to demonstrte the effectiveness of the obtined results. We shll discuss the cse when 0inthe future work. Nmely, we shll discuss the oscilltion of Hdmrd-type frctionl differentil equtions with kernels both depending or not depending on the strting point. Acknowledgements he uthor would like to thnk Prince Sultn University for funding this work through reserch group Nonliner Anlysis Methods in Applied Mthemtics (NAMAM) group number RG-DES Competing interests he uthor declres tht he hs no competing interests. Authors contributions he uthor red nd pproved the finl mnuscript. Publisher s Note Springer Nture remins neutrl with regrd to jurisdictionl clims in published mps nd institutionl ffilitions. Received: 2 Jnury 2018 Accepted: 12 Mrch 2018 References 1. Jrd, F., Uğurlu, E., Abdeljwd,., Blenu, D.: On new clss of frctionl opertors. Adv. Differ. Equ. 2017, 247 (2017) Kilbs, A.A., Srivstv, M.H., rujillo, J.J.: heory nd Appliction of Frctionl Differentil Equtions. North Hollnd Mthemtics Studies, vol. 204 (2006) 3. Smko, S.G., Kilbs, A.A., Mrichev, O.I.: Frctionl Integrls nd Derivtives: heory nd Applictions. Gordon & Brech, Yverdon (1993) 4. Cputo, M., Fbrizio, M.: A new definition of frctionl derivtive without singulr kernel. Prog. Frct. Differ. Appl. 1(2), (2015) 5. Atngn, A., Blenu, D.: New frctionl derivtive with non-locl nd non-singulr kernel. herm. Sci. 20(2), (2016) 6. Abdeljwd,.: Frctionl opertors with exponentil kernels nd Lypunov type inequlity. Adv. Differ. Equ. 2017, 313 (2017) 7. Abdeljwd,., Blenu, D.: Integrtion by prts nd its pplictions of new nonlocl frctionl derivtive with Mittg-Leffler nonsingulr kernel. J. Nonliner Sci. Appl. 10, (2017) 8. Abdeljwd,., Blenu, D.: Discrete frctionl differences with nonsingulr discrete Mittg-Leffler kernels. Adv. Differ. Equ. 2016, 232 (2016) Abdeljwd,.: A Lypunov type inequlity for frctionl opertors with nonsingulr Mittg-Leffler kernel. J. Inequl. Appl. 2017, 130 (2017) Al-Rife, M., Abdeljwd,.: Anlysis for frctionl diffusion equtions with frctionl derivtive with non-singulr kernel. Adv. Differ. Equ. 2017,315 (2017) Abdeljwd,., Blenu, D.: On frctionl derivtives with exponentil kernel nd their discrete versions. Rep. Mth. Phys. 80(1), (2017) 12. Abdeljwd,., Blenu, D.: Monotonicity results for frctionl difference opertors with discrete exponentil kernels. Adv. Differ. Equ. 2017, 78 (2017) Grce, S.R., Agrwl, R.P., Wong, P.J.Y., Zfer, A.: On the oscilltion of frctionl differentil equtions. Frct. Clc. Appl. Anl. 15, (2012) Chen, D.-X., Qu, P.-X., Ln, Y.-H.: Forced oscilltion of certin frctionl differentil equtions. Adv. Differ. Equ. 2013, 125 (2013) Alzbut, J., Abdeljwd,.: Sufficient conditions for the oscilltion of nonliner frctionl difference equtions. J. Frct. Clc. Appl. 5(1), (2014) 16. Abdll, B., Abodyeh, K., Abdeljwd,., Alzbut, J.: New oscilltion criteri for forced nonliner frctionl difference equtions. Vietnm J. Mth. 45(4), (2017) Abdll, B.: Onthe oscilltion of q-frctionl difference equtions. Adv. Differ. Equ. 2017, 254 (2017) Abdll, B., Alzbut, J., Abdeljwd,.: On the oscilltion of higher order frctionl difference equtions with mixed nonlinerities. Hcet. J. Mth. Stt. (2018) Gref, J.R., Grce, S.R., unc, E.: Asymptotic behvior of solutions of nonliner frctionl differentil equtions with Cputo-type Hdmrd derivtives. Frct. Clc. Appl. Anl. 20(1),71 87 (2017) 20. Grce, S.R., Gref, J.R., unc, E.: Asymptotic behvior of solutions of forced frctionl differentil equtions. Electron. J. Qul. heory Differ. Equ. 2016, 71 (2016) unc, E., unc, O.: On the oscilltion of clss of dmped frctionl differentil equtions. Miskolc Mth. Notes 17(1), (2016) 22. Abdeljwd,.: On conformble frctionl clculus. J. Comput. Appl. Mth. 279, (2013) 23. Ktugmpol, U.N.: New pproch to generlized frctionl integrl. Appl. Mth. Comput. 218, (2011)

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