Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels

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1 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 DOI /s RSARCH Open Access Discrete frctionl differences with nonsingulr discrete Mittg-Leffler kernels Thbet Abdeljwd1 nd Dumitru Blenu2,3* * Correspondence: dumitru@cnky.edu.tr 2 Deprtment of Mthemtics, Çnky University, Ankr, 06530, Turkey 3 Institute of Spce Sciences, Mgurele-Buchrest, Romni Full list of uthor informtion is vilble t the end of the rticle Abstrct In this mnuscript we propose the discrete versions for the recently introduced frctionl derivtives with nonsingulr Mittg-Leffler function. The properties of such frctionl differences re studied nd the discrete integrtion by prts formuls re proved. Then discrete vritionl problem is considered with n illustrtive exmple. Finlly, some more tools for these derivtives nd their discrete versions hve been obtined. Keywords: discrete frctionl derivtive; modified Mittg-Leffler function; discrete Mittg-Leffler function; discrete nbl Lplce trnsform; convolution; discrete ABR frctionl derivtive 1 Introduction nd preliminries Frctionl clculus hs become n importnt mthemticl tool used in severl brnches of science nd engineering in order to describe better the properties of non-locl complex systems [ ]. Recently some uthors hve introduced new non-locl derivtives with nonsingulr kernels nd they pplied them successfully to some rel world problems [ ]. However, severl res where the frctionl clculus cn be pplied successfully remin still not deeply investigted, e.g. the thermoelsticity of bodies with microstructure see [ ] for exmple nd the references therein). Finding the discrete counterprts of these new frctionl opertors is n importnt step to pply them to model the dynmics of complex systems. In the following we recll nd prove some results in discrete frctionl clculus tht will be necessry in proceeding to obtin our discrete results see [ ]). Definition [ ] i) Let m be nturl number, then the m rising fctoril of t is written s tm = m t + k), t =. ) k= ii) For ny rel number the α rising function becomes tα = t + α), t) such tht t R \ {...,,, }, α =. ) 2016 Abdeljwd nd Blenu. This 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 uthors) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 2 of 18 In ddition, we hve t α) = αt α 1, 3) hence t α is incresing on N 0,whereρt)=t 1. Definition 2 See [31, 32]) Let ρt) =t 1 be the bckwrd jump opertor. Thus, for function f : N = {, +1, +2,...} R, the nbl left frctionl sum of order α >0becomes α f t)= 1 Ɣα) t ) α 1f t ρs) s), t N+1. The nbl right frctionl sum of order α >0forf : b N = {b, b 1,b 2,...} Ris written s b α f t)= 1 Ɣα) s=t s ρt) ) α 1f s)= 1 Ɣα) The nbl left frctionl difference of order α >0 hs the form α f t)= n n α) n f t)= Ɣn α) ) α 1f σ s) t s), t b 1 N. s=t t ) n α 1f t ρs) s), t N+1, nd the nbl right frctionl difference of order α >0 is defined s b α f t)= n b n α) f t)= 1)n n ) n α 1f s ρt) s), t b 1 N. Ɣn α) s=t The left Cputo frctionl difference of order α >0strtedbyα)= + n 1,n =[α]+1 is written s C α α) f ) t)= n α α) n f t), t N +n, nd the right Cputo frctionl difference of order α > 0 ending t bα)=b n + 1 hs the following form: Cbα) α f ) t)= bα) n α n f t), t b n N. The Q-opertor ction, Qf )t) =f + b t), ws used in [31, 32] to connect left nd right frctionl sums nd differences. We recll the following results: αqf )t)=q b α f t). αqf )t)=q b α f t). C αqf )t)=qc b α f t). The mixing of nbl nd delt opertors in defining right frctionl differences plys crucil role in obtining the bove dul identities.

3 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 3 of 18 In our mnuscript, we use the properties of the discrete version of Q-opertor to define nd confirm our definitions of frctionl differences with discrete Mittg-Leffler function kernels. Definition 3 [33] Letfunctionf be defined on N 0. Then the nbl discrete Lplce trnsform hs the form N f z)= 1 z) t 1 f t). 4) t=1 More generlly for function f it is defined on N by N f z)= 1 z) t 1 f t). 5) t=+1 If f t, s) denotes function of two vribles, we hve explicitly to show to which prmeter we use the trnsform. Lemm 1 [30] For ny α R \{..., 2, 1,0}, i) N t α 1 )z)= Ɣα) z α, 1 z <1, ii) N t α 1 b t )z)= bα 1 Ɣα) z+b 1) α, 1 z < b. Remrk 1 We cn generlize i) of Lemm 1 to N t ) α 1 )s) =1 s) Ɣα) s α.herewe ccept N 0 = N. Definition 4 [2] The Mittg-Leffler function of one prmeter hs the following form: α z)= z k Ɣαk +1) z C; Reα)>0 ), 6) nd the one with two prmeters α nd β becomes α,β z)= z k Ɣαk + β) z, β C; Reα)>0 ), 7) where α,1 z)= α z). Since in generl it is not true tht t α ) β = t αβ nd b) α = α b α in generl, we define for the ske of discretiztion the following modified) versions of Mittg-Leffler functions which lso gree with the time scle clculus nottions. Definition 5 Modified clssicl Mittg-Leffler functions) The Mittg-Leffler function of one prmeter is defined by α λ, z)= α λz α ) = λ k z αk Ɣαk +1) 0 λ R, z C; Reα)>0 ), 8)

4 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 4 of 18 nd the one with two prmeters α nd β by α,β λ, z)=z β 1 α,β λz α ) = λ k z αk+β 1 Ɣαk + β) 0 λ R, z, β C; Reα)>0 ), 9) where α,1 λ, z)= α λ, z). Agreeing with Definition 5, theuthorin[31, 32] defined the following discrete versions of Mittg-Leffler functions. Definition 6 Nbl discrete Mittg-Leffler) see [31 33]) For λ R, λ <1,ndα, β, z C with Reα) > 0, the nbl discrete Mittg-Leffler functions is α,β λ, z)= For β =1,wehve λ k α λ, z) α,1 λ, z)= z kα+β 1 Ɣαk + β). 10) λ k z kα, λ <1. 11) Ɣαk +1) The generlized ML of three prmeters ws defined in the literture by ρ α,β z)= z k ρ) k k!ɣαk + β), 12) where ρ) k = ρρ +1) ρ + k 1).Noticetht1) k = k!sothtα,β 1 z)= α,βz). TopsstothediscreteprocesswedefinethefollowingversionofML function of three prmeters: ρ α,β λ, z)= λ k z αk+β 1 ρ) k k!ɣαk + β). 13) Definition 7 The nbl) discrete generl ML function ofthreeprmeters α, β, ndρ is defined by ρ λ, z)= λ k z kα+β 1 ρ) α,β k k!ɣαk + β). 14) Notice tht 1 λ, z)= α,β α,β λ, z). Proposition 1 Summtion nd difference of discrete ML functions) t α λ, z)=λ α,α λ, z). t 1,β λ, z)=λ 1,β+1 λ, z). t γ λ, α,β z)=γ λ, z). α,β 1 z t=+1 α,β λ, t )= α,β+1 λ, z ).

5 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 5 of 18 Definition 8 [31, 32] Letfunctionf be defined on N 0.Thus,for0<α 1itsα-order Cputo frctionl derivtive is C α 0 f t)= ) 0 f t) = 1 Ɣ1 α) t ) α f t ρs) s), s=1 where ρs)=s 1nd α 0 f t)= 1 Ɣα) t s=1 t ρs))α 1 f s) is the nbl left frctionl sum of order α. We recll tht if f is defined on N 0,then C 0 αf t)isdefinedonn 1 = {1,2,3,...}. For the Cputo frctionl difference of order n 1<α n strting from α)= + n 1 we refer to Section 5 in [31]. xmple 1 [31, 32] Let0<α 1, R, nd consider the nbl left Cputo nonhomogeneous frctionl difference eqution C α 0 yt)=λyt)+f t), y0) = 0, t N 0. 15) Thus, the solution of 15)iswrittens yt)= 0 α λ, t)+ t ) α,α λ, t ρs) f s). 16) s=1 Remrk 2 [31, 32] The solution of 15)withα =1nd 0 =1is yt)= λ k tk k! + t s=1 k t ρs))k λ f s). k! The nbl discrete exponentil function ê λ t,0)=1 λ) t representsthefirstprtofthe bove solution, with λ < 1. For more detils see [34], p.118. For the rest of this section, we summrize some fcts s regrds the discrete Lplce trnsform of Mittg-Leffler type nd convolution type functions see [33] for some detils). Definition 9 See [33]) Let s R, 0<α <1,ndf, g : N R be functions. The nbl discrete convolution of f with g is defined by f g)t)= t g t ρs) ) f s). 17) In the bove, ρs) = s 1 isthe bckwrdjumpingopertor usedin -nlysisfor the time scle Z. This opertor is necessry to prove for exmple discrete convolution theorem s shown below. Also it is necessry to obtin dul reltions between the left nd right frctionl sums nd differences vi the Q-opertor.

6 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 6 of 18 Proposition 2 For ny α R \{..., 2, 1,0}, s R, nd f, gdefinedonn we hve N f g) ) z)=n f )z)n g)z). 18) Proof N f g) ) t z)= 1 z) t 1 f s)g t ρs) ) t=+1 +1 t = 1 z) t 1 f s)g t ρs) ) t=s +1 = 1 z) r 1 1 z) s 1 f s)gr) r=1 =N f )z)n g)z), where the chnge of vrible r = t ρs)wsused. For the cse =0ndgt)=t α wereferto[33]. Lemm 2 [33] Let f be function defined on N 0. Thus, N f t) )) z)=zn f )z) f 0). 19) We cn generlize Lemm 2 s follows. Lemm 3 Let f be function defined on N. The following result holds: N f t) )) z)=zn f )z) 1 z) f ). 20) More generlly, Nα) n f ) n 1 z)=z n N α) f )z) 1 z) α) z n 1 i i f + 1). 21) Lemm 4 [30] For ny positive rel number ν, i=0 N 1 ν 1) f s)=s ν N 1 f )s). For the following lemm we will present n lterntive proof without using convolutions s ws done in [33]. Lemm 5 [33] Let f be defined on N 0 nd 0<α 1. Then N C α 0 f ) z)=z α N f )z) z α 1 f 0). 22)

7 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 7 of 18 Proof From the definition nd Lemm 3.2 in [35]wehve C 0 α f ) t)= ) 0 f ) t)= ) 0 f ) t α t) Ɣ1 α) f 0). Apply the nbl discrete Lplce trnsform nd mke use of Lemm 2 nd Lemm 1 to get N C 0 α f ) z)=z N ) 0 f ) z) ) 0 f ) Ɣ1 α) 0) f 0). Ɣ1 α)z Then the result follows by Lemm 4 with =1nd ) 0 f )0) = 0. Remrk 3 Lemm 5 cn be generlized s follows. For f defined on N nd 0 < α 1, we hve N C α f ) z)=z α N f )z) 1 z) z α 1 f ). 23) This cn be proved by mking use of Remrk 1. Lemm 6 [33] Let 0<α 1 nd f be defined on N 0. Then: i) N α λ, t))z)= zα 1 z α λ. ii) N α,α λ, t))z)= 1 z α λ. Proof We just repet the proof of ii) due to the clcultion error in Lemm 4ii) in [33]. ii) First it is esy to see tht α λ, t)=λ α,α λ, t). Indeed, α λ, z)= λ k kαzkα 1 Ɣαk +1). Since dividing by blls of Gmm function leds to zero, we then hve α λ, t) = zkα 1 k=1 λk Ɣαk) = λ zkα+α 1 λk Ɣαk+α) = λ α,αλ, t). If we use N together with i) nd Lemm 2, we conclude tht N α,α λ, t) ) z)=λ 1[ z N α λ, t) ) z) α λ,0) ] [ z = λ 1 α ] z α λ 1 = 1 z α λ. 2 Discrete frctionl differences with discrete Mittg-Leffler kernels Definition 10 Let f be defined on N b N, < b, α [0, 1], then the nbl discrete new left Cputo) frctionl difference in the sense of Atngn nd Blenu is defined by ABC α f ) t)= Bα) t ) α s f s) α, t ρs) = Bα) [ )] α f t) α, t 24)

8 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 8 of 18 nd in the left Riemnn sense by ABR α f ) t)= Bα) t t ) α f s) α, t ρs) = Bα) [ )] α t f t) α, t. 25) It is be noted tht since for 0 < α < 1 α we hve 1 < λ = 2 <0,then αλ, t) isconvergent for ny t N. For exmple, α λ,1)=) provided tht 0 < α < 1 2.Hence,ll the AB-type frctionl differences will converge under the restriction 0 < α < 1 2.Alsonote tht since t α is incresing on N 0, α λ, t) is monotone decresing for 0 < α < 1 2, t >0,nd λ = α <0see[36] for the continuous cse α t α 1 )). We cn show tht lim σ 0 σ α 1 σ, t ρs)) = δ s t)= { 1, t = s, s in [6] we cn show tht, for α 0, we hve ABR 0, t s, α =1,whichisthedeltDircfunctiononthetimescleZ, nd hence α f )t) f t)ndforα 1, we hve σ, t ρs)) = 1 α)t ρs), σ =, nd hence for ex- α ABR α f )t) f t). Notice tht 1 1 mple lim ABR α 1 α f ) t)=lim Bα) t α 1 t f s)1 α) t s = f t). Above we hve mde used of the fct tht the nbl discrete exponentil function hs the form e λ t, ρs)) = 1 1 λ )t ρs) nd 1 λ, t ρs)) = e λ t, ρs)). To derive the proper frctionl difference for the bove proposed frctionl difference we consider the eqution ABR α f ) t)=ut). 26) Apply N to 26) bove nd mke use of Lemm 3,Proposition2 with gt)= α λ, t)with λ = α, nd Lemm 6, Bα) N f t) α λ, t) ) z)= Bα) z N f t) α λ, t) ) z) 0 = Bα) [ z z α 1 ] N f )z) z α λ Tht is, = N ut) ) z). 27) N f )z)= N ut) ) z) λ N ut) ) z). 28) Bα) Bα) z α Apply the inverse of N nd use of Proposition 2 nd Lemm 1 to conclude tht f t)= Bα) ut)+ α α Bα) u) t). 29) This suggests the following definition for the frctionl sum corresponding to the frctionl difference with discrete Mittg-Leffler function kernel.

9 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 9 of 18 Definition 11 The frctionl sum ssocited to ABR α f )t)withorder0<α <1isdefined by AB α f ) t)= Bα) f t)+ α α Bα) f ) t). 30) It is cler tht α = 0 gives the function f nd α =1gives t f s). From the definition of the discrete frctionl integrl we hve ABR αabr α f ) t)=f t). On the other hnd we hve the following. Theorem 1 For ny 0<α 1 nd f defined on N, ABR α f )t) stisfies the eqution ABR α g ) t)=ft). 31) Proof From the definition of frctionl sum the eqution in the sttement of the theorem is equivlent to Bα) gt)+ α α Bα) g) t)=ft). 32) Apply the discrete Lplce trnsform N nd mke use of Lemm 4 to obtin Bα) Gs)+ α Bα) s α Gs)=Fs), 33) where Gs)=N g)s)ndfs)=n f )s). From this it follows tht Gs)= s α Bα) Bα) s α Fs)= Fs), 34) 1 α)s α + α s α λ where λ = α. Finlly, pply the inverse of N nd use the discrete convolution theorem, Proposition 2,or27)toconcludethtgt)= ABR α f )t). Theorem 2 The reltion between the Cputo nd Riemnn frctionl differences with ML kernels) We hve ABC α f ) t)= ABR α f ) t) f) Bα) αλ, t ). 35) Proof From 27)wehve ABR N α f ) z)= Bα) [ z α ] N f )z), 36) z α λ where λ = α. On the other hnd, we hve N ABC α f ) z)= Bα) N f t) α λ, t) ) z) = Bα) N f )z) N α λ, t) ) z)

10 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 10 of 18 = Bα) [ zn f )z) 1 z) f ) ] [ z α 1 ] z α λ [ N f )z) = Bα) From 36)nd37), we see tht N ABC z α ] 1 z) f ) Bα) z α λ [ z α 1 z α λ ]. 37) α f ) z)= ABR N α f ) z) 1 z) f ) Bα) [ z α 1 ]. 38) α z α λ Apply the inverse of N to 38) toconclude35). The fct tht N f t ))z) =1 z) N f t))z)wsusedbove. By mens of the ction of the Q-opertor on left nd right frctionl sums nd differences, we cn define the right frctionl sums AB b αf )t) nd differences AB b α f )t) s follows. Definition 12 The new right frctionl difference with ML kernel) For 0 < α <1,ndf defined on b N, the right frctionl difference of f is defined by ABR b α f ) t)= Bα) α t) f s) α, s ρt) ) ) s=t 39) nd the right Cputo one by ABC α b f ) t)= Bα) s=t s f )s) α α, s ρt) ) ). 40) Definition 13 The new right frctionl sum with ML kernel) For 0 < α <1,ndf defined on b N,therightfrctionlsumoff is defined by AB b α f ) t)= Bα) f t)+ α b α f ) t). 41) Bα) Theorem 3 For function f defined on b N nd 0<α <1,we hve ABR b αab b α f )t)=ft) nd AB b αabr b α f )t)=ft). If we pply the Q-opertor to both sides nd then replce f t) byqf )t)=f + b t), then we cn stte the following. Theorem 4 The reltion between the new right Cputo frctionl difference nd the new right Riemnn frctionl difference) We hve ABC α b f ) t)= ABR α b f ) t) f b) Bα) αλ, b t). 42) 3 Integrtion by prts for frctionl sums nd differences with discrete ML First we stte nd prove n integrtion by prts formul for frctionl sums.

11 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 11 of 18 Theorem 5 Integrtion by prts for the frctionl sums with ML kernels) For f nd g defined on N b N, b mod 1), nd 0<α <1,one hs gs) AB α f ) s)= Bα) = gs)f s)+ α Bα) f s) b α g ) s) f s) AB α b g) s). 43) Similrly, we hve gs) AB α b f ) s)= Bα) = gs)f s)+ α Bα) f s) α g) s) f s) AB α g ) s). 44) Proof The proof follows by the definition of the new left frctionl sum, the integrtion by prts formul for nbl clssicl frctionl sums see Proposition 37 in [31]), nd the definition of the new right frctionl sum. Theorem 6 Integrtion by prts for the frctionl differences with ML kernels) For f nd gdefinedonn b N, b mod 1), nd 0<α <1,one hs f s) ABR α g ) s)= gs) ABR α b f ) s). 45) Similrly, f s) ABR α b g) s)= gs) ABR α f ) s). 46) Proof The proof is chieved by Theorem 5 nd the previously proved fct tht the new frctionl sums nd differences re the inverses to ech other. Indeed, f s) AB α g ) s)= AB α = AB α = b AB b α b f ) s) AB f ) s) AB α g ) s) αab α g ) s) gs) AB b α f ) s). 47) Next, in order to present n integrtion by prts formul for Cputo type frctionl differences with ML kernels, we first define the discrete versions of the left) generlized

12 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 12 of 18 frctionl integrl opertor introduced nd studied in [37], γ ρ,μ,ω, +ϕ) x)= x x t) μ 1 γ ρ,μ[ ωx t) ρ ] ϕt) dt, x >, 48) where ρ,μz)= γ γ ) k z k is the generlized Mittg-Leffler function which is defined Ɣρk+μ)k! for complex ρ, μ, γ Reρ) >0)[3, 37]. For our purposes we just introduce the discrete version for γ =1. Definition 14 The discrete left) generlized frctionl integrl opertor is defined by 1 ρ,μ,ω, +ϕ ) t)= t ) μ 1ρ,μ ) t ρs) ω, t ρs) ϕs), t N. The discrete right) generlized frctionl integrl opertor is defined by 1 ρ,μ,ω,b ϕ ) ) μ 1ρ,μ ) t)= s ρt) ω, s ρt) ϕs), t b N. s=t Theorem 7 Integrtion by prts for Cputo frctionl differences with ML kernels) For functions f nd g defined on N b N, we hve f s) ABC 1 α g ) s)= gs) ABR b 1 α f ) s)+g ρt) ) Bα) 1 α,1,λ,b f ) t) b. 49) s= s= Similrly, b f s) ABC b+1 α g) s)= where λ = α. b gs) ABR +1 α f ) s) g σ t) ) Bα) 1 α,1,λ, + f ) t) b, 50) Proof By 35)ndTheorem6,wehve f s) ABC 1 α g ) [ ABR s)= f s) 1 α g ) s) g 1) Bα) ) ] α λ, s ρ) s= s= = gs) ABR b 1 α f ) s) g 1) Bα) ) f s) α λ, s ρ) s= = gs) ABR b 1 α f ) s)+g ρt) ) Bα) 1 α,1,λ,b f ) t) b. s= s= The second prt follows by 42) nd the second prt of Theorem 6.

13 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 13 of 18 4 Discrete frctionl uler-lgrnge equtions We prove the uler-lgrnge equtions for Lgrngin contining the left new discrete Cputo derivtive. Theorem 8 Let 0<α 1 be non-integer,, b R, < b, b mod 1). Assume tht the functionl of the form Jf )= L t, f ρ t), ABC 1 α f t) ) t= hs locl extremum in S = {y :N 1 b 1 N) R : y 1)=A, yb 1)=B} t some f S, where L :N 1 b 1 N) R R R. Then [ L1 s)+ ABR α b 1 L 2s) ] =0, for ll s N 1 b 1 N), 51) where L 1 s)= L L f ρ s) nd L 2 s)= s). ABC 1 α f Proof Without loss of generlity, ssume tht J hs locl mximum in S t f.hence,there exists n ɛ >0suchthtJ f ) Jf) 0 for ll f S with f f = sup t N b N f t) ft) < ɛ. For ny f S there is n η H = {y :N 1 b 1 N) R : y 1)=yb 1)=0} such tht f = f + ɛη. Then the ɛ-tylor s theorem nd the ssumption implies tht the first vrition quntity δjη, y) = b 1 t= [ηρ t)l 1 t)+ ABC 1 α η)t)l 2 t)] dt = 0, for ll η H. Tomkethe prmeter η free,weusetheintegrtionbyprtseqution49)toobtin δjη, f )= η ρ s) [ L 1 s)+ ABR b 1 α L 2s) ] + η ρ t) Bα) ) 1 α,1, α L,b 2 t) b =0, s= for ll η H, nd hence the result follows by the discrete fundmentl lemm of the clculus of vrition. The term 1 α,1, α L 2 )t) b,b = 0 bove is clled the nturl boundry condition. Similrly, if we llow the Lgrngin to depend on the discrete right Cputo frctionl derivtive, we cn stte the following. Theorem 9 Let 0<α 1 be non-integer,, b R, < b, b mod 1). Assume tht the functionl J of the form Jf )= b L t, f σ t), ABC b+1 α f t)) +1 hs locl extremum in S = {y :N +1 b+1 N) R : y +1)=A, yb +1)=B} t some f S, where L :N +1 b+1 N) R R R. Then [ L1 s)+ ABR +1 α L 2 s) ] =0, for ll s N +1 b+1 N), 52) where L 1 s)= L f σ s) nd L 2 s)= L ABC α b+1 f s).

14 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 14 of 18 Proof The proof is similr to Theorem 8 by pplying the second integrtion by prts eqution 50) to get the nturl boundry condition of the form 1 α,1, α L 2 )t) b,+ =0. xmple 2 WeherestudyninterestedphysiclctiontosupportTheorem8. Nmely, let us consider the following frctionl discrete ction: Jy) = b 1 t= [ 1 2 ABC 1 α yt)) 2 Vy ρ t))], where 0 < α <1ndwithy 1),yb 1)re ssigned or with the nturl boundry condition 1 ABC α,1, α,b 1 α yt) ) t) b =0. Then the uler-lgrnge eqution by pplying Theorem 8 is ABR α b 1 ABC 1 α y ) s) dv dy s) = 0 for ll s N 1 b 1 N). Here, we remrk tht it is of interest to del with the bove uler-lgrnge equtions obtined in the bove exmple, where we hve composition of discrete right nd discrete left type frctionl derivtives. For the ske of comprisons with the clssicl discrete frctionl uler-lgrnge equtions within nbl we refer to [38]. For clssicl frctionl dynmicl systems composed by the left nd right frctionl opertors under the presence of dely we refer to [39]. 5 Some tools nd properties for frctionl derivtives with nonsingulr ML kernels nd their discrete versions Theorem 10 [37] Let ρ, μ, γ, ν, σ, λ C Reρ), Reμ), Reν)>0),then x 0 x t) μ 1 γ ρ,μ λ[x t] ρ ) t ν 1 σ ρ,ν In prticulr, if γ =1,μ =1nd ρ = α, we hve λt ρ ) dt = x μ+ν 1 γ +σ ρ,μ+ν λx ρ ). 53) x 0 α λ[x t] α ) t ν 1 α,ν σ λt α ) dt = x ν α,1+ν 1+σ λx α ). 54) Remrk 4 If we use the modified nottion of ML functions, then 53)tkes the form x 0 ρ,μ γ λ, x t)σ +σ ρ,ν λ, t) dt = γ ρ,μ+ν λ, x), 55) nd 54) tkes the form x 0 α λ, x t)α,ν σ λ, t) dt = 1+σ α,1+ν λ, x). 56) From [3] we recll lso the following differentition formul, expressed in modified wy, which will be helpful. For α, μ, γ, λ C Reα)>0)ndn N we hve ) d n [ γ α,μ dz λ, z)] = α,μ n γ λ, z). 57)

15 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 15 of 18 Now, with the help of 56)nd57), we hve ABR 0 D α[ α,ν σ λ, x)] = Bα) d [ 1+σ α,1+ν dx λ, x)] = Bα) Similrly, with the help of 57)nd56), we hve 1+σ α,ν λ, x). 58) ABC 0 D α[ α,ν σ λ, x)] = Bα) Remrk 5 Noting tht = Bα) x 0 1+σ α,ν α λ, x t) d dt [ σ α,ν λ, t) ] dt λ, x). 59) α,ν 1 λ, x)=xν 1 α,ν 1 λx α ) = xv 1 Ɣν) λxα+ν 1 Ɣα + ν) nd α,ν 0 xν 1 λ, x)= Ɣν) 0, ν 0+, one cn conclude from 58)nd59)with σ = 1 tht the function [ ] gx)= lim ν 0 + Bα) 1 α,ν λ, x) = α Bα) x α 1 Ɣα) is nonzero function whose frctionl ABR nd ABC derivtive is zero. Note tht the α function gx) tends to the constnt function s α tends to 1. Bα)Ɣα) The proof of the following lemm follows by Lemm 1i) nd the definition of discrete ML functions in Definition 7. Lemm 7 For γ, α, β, λ C Reβ)>0),nd s C with Res)>0, λs α <1,we hve N γ α,β λ, t)) s)=s β[ 1 λs α] γ. In prticulr N α,β λ, t) ) s)=s β[ 1 λs α] 1. The proof of the following lemm just follows by pplying the discrete Lplce trnsform N nd its inverse in finl step vi the help of Lemm 7 nd the discrete convolution theorem. Lemm 8 Let ρ, μ, γ, ν, σ, λ C Reρ), Reμ), Reν)>0),then t s=1 γ ) ρ,μ λ, t ρs) σ ρ,ν λ, s)= γ +σ ρ,μ+ν λ, t). 60)

16 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 16 of 18 Now, with the help of Proposition 1iii) nd Lemm 8,we hve ABR 0 α[ α,ν σ λ, t)] = Bα) [ t 1+σ α,1+ν λ, t)] = Bα) 1+σ α,ν Similrly, with the help of Proposition 1iii) nd Lemm 8, we hve ABC 0 α[ α,ν σ λ, t)] = Bα) Remrk 6 Noting tht = Bα) t ) [ α λ, t ρs) t σ α,ν λ, s) ] s=1 1+σ α,ν λ, x). 61) λ, t). 62) nd α,ν 1 xv 1 λ, t)= Ɣν) λtα+ν 1 Ɣα + ν) α,ν 0 tν 1 λ, t)= Ɣν) 0, ν 0+, one cn conclude from 61)nd62)with σ = 1 tht the function [ ] ht)= lim ν 0 + Bα) 1 α,ν λ, t) = α Bα) t α 1 Ɣα) is nonzero function whose discrete frctionl ABR nd ABC derivtive is zero. Note tht the function ht) tends to the constnt function 1 s α tends to 1.As result of this,if the potentil function V in xmple 2,with = 1, is 0, then the solution tkes the form t α 1 yt)= α Bα) Ɣα). Using the following -version of eqution 14) in [7]: ABC D α f ) t)= ABR D α f ) t) Bα) f ) α λt ) α ), λ = α, 63) nd the identity see [3], p.78, forexmple) I α t ) β 1 μ,β [ λt ) μ ]) x)=x ) α+β 1 μ,α+β [ λx ) μ ], 64) we cn stte the following result which is very useful tool to solve frctionl dynmicl systems with Cputo frctionl derivtive with ML kernels. Proposition 3 For 0<α <1,we hve AB IαABC D α f ) x)=fx) f) α λx ) α ) α f )xα α,α+1 λx ) α ) = f x) f).

17 Abdeljwd nd Blenu Advnces in Difference qutions 2016) 2016:232 Pge 17 of 18 Similrly, AB I α b ABC D α b f ) x)=f x) f b). 65) Similrly,in the discrete cse by recllingthtsee[32], Proposition 3.9 or [35]) α t Ɣμ +1) )μ = Ɣμ + α +1 t )μ+α, we cn stte the following importnt result in clssicl discrete frctionl clculus. Theorem 11 Let [0, ), nd let α, ρ, μ, γ, λ C Reα) >0,Reμ) >0,Reρ) >0). Then for t > the reltions hold: αγ ρ,μ λ, t )=γ ρ,μ+α λ, t ). αγ ρ,μ λ, t )=γ ρ,μ α λ, t ). Then we cn stte the following. Proposition 4 For 0<α <1,we hve AB αabc α f ) t)=ft) f) α λ, t ) α f ) α,α+1λ, t ) = f t) f). Similrly, by the first prt nd the ction of the Q-opertor AB α b ABC α b f ) t)=f t) f b). 66) 6 Conclusions The modified versions of Mittg-Leffler functions enble us to tret esily the frctionl type derivtives with ML kernels nd enble us to obtin successfully their discrete versions. The Q-opertor nd its discrete version lwys provide n effective tool to confirm dul definitions nd reltions when pssing from left to right or vice vers.thereexistnon- constnt functions whose usul or discrete ABC frctionl derivtives re zeros. Hence zero potentil function in usul or discrete vritionl problem does not imply only constnt solution. The results obtined tend to the ordinry cse when α tends to 1. The discrete versions for AB type frctionl derivtives hve been defined nd their discrete frctionl integrls given with the help of the discrete Lplce trnsform. Competing interests The uthors declre tht they hve no competing interests. Authors contributions All uthors contributed eqully in this rticle. They red nd pproved the finl mnuscript. Author detils 1 Deprtment of Mthemtics nd Physicl Sciences, Prince Sultn University, P.O. Box 66833, Riydh, 11586, Sudi Arbi. 2 Deprtment of Mthemtics, Çnky University, Ankr, 06530, Turkey. 3 Institute of Spce Sciences, Mgurele-Buchrest, Romni. Received: 28 June 2016 Accepted: 23 August 2016

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