Abstrct nd Applied Anlysis Volume 2012, Article ID 406757, 13 pges doi:10.1155/2012/406757 Reserch Article On the Definitions of Nbl Frctionl Opertors Thbet Abdeljwd 1 nd Ferhn M. Atici 2 1 Deprtment of Mthemtics, Çnky University, 06530 Ankr, Turkey 2 Deprtment of Mthemtics, Western Kentucky University, Bowling Green, KY 42101, USA Correspondence should be ddressed to Ferhn M. Atici, ferhn.tici@wku.edu Received 12 April 2012; Accepted 4 September 2012 Acdemic Editor: Dumitru Bǎlenu Copyright q 2012 T. Abdeljwd nd F. M. Atici. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. We show tht two recent definitions of discrete nbl frctionl sum opertors re relted. Obtining such reltion between two opertors llows one to prove bsic properties of the one opertor by using the known properties of the other. We illustrte this ide with proving power rule nd commuttive property of discrete frctionl sum opertors. We lso introduce nd prove summtion by prts formuls for the right nd left frctionl sum nd difference opertors, wherewe employ the Riemnn-Liouvilledefinitionof the frctionldifference. We formlize initil vlue problems for nonliner frctionl difference equtions s n ppliction of our findings. An lterntive definition for the nbl right frctionl difference opertor is lso introduced. 1. Introduction The following definitions of the bckwrd nbl discrete frctionl sum opertors were given in 1, 2, respectively. For ny given positive rel number α, we hve f t 1 t ( ) α 1f s, t ρ s 1.1 where t { 1, 2,...}, nd α f t 1 t ( ) α 1f s, t ρ s 1.2 s where t {, 1, 2,...}.
2 Abstrct nd Applied Anlysis One significnt difference between these two opertors is tht the sum in 1.1 strts t 1ndthesumin 1.2 strts t. In this pper we im to nswer the following question: Do these two definitions led the development of the theory of the nbl frctionl difference equtions in two directions? In order to nswer this question, we first obtin reltion between the opertors in 1.1 nd 1.2. Then we illustrte how such reltion helps one to prove bsic properties of the one opertor if similr properties of the other re lredy known. In recent yers, discrete frctionl clculus gins gret del of interest by severl mthemticins. First Miller nd Ross 3 nd then Gry nd Zhng 1 introduced discrete versions of the Riemnn-Liouville left frctionl integrls nd derivtives, clled the frctionl sums nd differences with the delt nd nbl opertors, respectively. For recent developments of the theory, we refer the reder to the ppers 2, 4 19. For further reding in this re, we refer the reder to the books on frctionl differentil equtions 20 23. The pper is orgnized s follows. In Section 2, we summrize some of bsic nottions nd definitions in discrete nbl clculus. We employ the Riemnn-Liouville definition of the frctionl difference. In Section 3, we obtin two reltions between the opertors nd α. So by the use of these reltions we prove some properties for -opertor. Section 4 is devoted to summtion by prts formuls. In Section 5, we formlize initil vlue problems nd obtin corresponding summtion eqution with -opertor. This section cn be considered s n ppliction of the results in Section 3. Finlly, in Section 6, definition of the nbl right frctionl difference resembling the nbl right frctionl sum is formulted. This definition cn be used to prove continuity of the nbl right frctionl differences with respect to the order α. 2. Nottions nd Bsic Definitions Definition 2.1. i For nturl number m,them rising scending fctoril of t is defined by m 1 t m t k, t 0 1. 2.1 k 0 ii For ny rel number α, the rising function is defined by t α Γ t α, t R {..., 2, 1, 0}, 0 α 0. 2.2 Γ t Throughout this pper, we will use the following nottions. i For rel numbers nd b, we denote N {, 1,...} nd b N {...,b 1, b}. ii For n N, we define Δ n f t : 1 n Δ n f t. 2.3
Abstrct nd Applied Anlysis 3 Definition 2.2. Let ρ t t 1 be the bckwrd jump opertor. Then i the nbl left frctionl sum of order α>0 strting from is defined by f t 1 t ( t ρ s ) α 1f s, t N 1 2.4 ii the nbl right frctionl sum of order α>0 ending t b is defined by b f t 1 ( ) α 1f s 1 s ρ t σ s t α 1 f s, t b 1 N. 2.5 We wnt to point out tht the nbl left frctionl sum opertor hs the following chrcteristics. i mps functions defined on N to functions defined on N. ii n f t stisfies the nth order discrete initil vlue problem n y t f t, i y 0, i 0, 1,...,n 1. 2.6 iii The Cuchy function t ρ s n 1 /Γ n stisfies n y t 0. In the sme mnner, it is worth noting tht the nbl right frctionl sum opertor hs the following chrcteristics. i b mps functions defined on b N to functions defined on b N. ii b n f t stisfies the nth order discrete initil vlue problem Δ n y t f t, Δ i y b 0, i 0, 1,...,n 1. 2.7 iii The Cuchy function s ρ t n 1 /Γ n stisfies Δ n y t 0. Definition 2.3. i The nbl left frctionl difference of order α>0 is defined by α f t n n α n f t Γ n α t ( t ρ s ) n α 1f s, t N 1. 2.8 ii The nbl right frctionl difference of order α>0 is defined by b α f t Δ n b n α Δ n ( ) n α 1f s, f t s ρ t t b 1 N. 2.9 Γ n α Here nd throughout the pper n α 1, where α is the gretest integer less thn or equl α.
4 Abstrct nd Applied Anlysis Regrding the domins of the frctionl difference opertors we observe the following. i The nbl left frctionl difference α mps functions defined on N to functions defined on N n on N if we think f 0before. ii The nbl right frctionl difference b α mps functions defined on b N to functions defined on b n N on b N if we think f 0fterb. 3. A Reltion between the Opertors In this section we illustrte how two opertors, Lemm 3.1. The following holds: i α 1f t α f t, nd α nd α ii α f t 1/ t 1 α 1 f f t. re relted. Proof. The proof of i follows immeditely from the bove definitions 1.1 nd 1.2. For the proof of ii, we hve α f t 1 t ( ) α 1f s t ρ s s 1 t 1 α 1 f 1 t ( t ρ s ) α 1f s 3.1 1 t 1 α 1 f f t. Next three lemms show tht the bove reltions on the opertors 1.1 nd 1.2 help us to prove some identities nd properties for the opertor by the use of known identities for the opertor α. Lemm 3.2. The following holds: f t f t t α 1 f. 3.2 Proof. It follows from Lemm 3.1 nd Theorem 2.1 in 13 f t α 1 f t α f t t 1 α 1 f { } 1 t 1 α 1 f f t t 1 α 1 f
Abstrct nd Applied Anlysis 5 α 1 t 1 α 2 f f t t 1 α 1 f f t t α 1 f. 3.3 Lemm 3.3. Let α>0 nd β> 1. Then for t N, the following equlity holds t μ Γ ( μ 1 ) Γ ( μ α 1 ) t α μ. 3.4 Proof. It follows from Theorem 2.1 in 13 t μ α 1 t μ Γ ( μ 1 ) Γ ( μ α 1 ) t α μ. 3.5 Lemm 3.4. Let f be rel-vlued function defined on N, nd let α, β > 0. Then β f t α β f t β f t. 3.6 Proof. It follows from Lemm 3.1 nd Theorem 2.1 in 2 β f t α 1 β 1f t α β 1 f t α β f t. 3.7 Remrk 3.5. Let α>0ndn α 1. Then, by Lemm 3.2 we hve α f t n( ) n α f t n( ) n α f t 3.8 or α f t n [ n α f t ] t n α 1 Γ n α f. 3.9 Then, using the identity n t n α 1 t α 1 Γ n α Γ α 3.10 we verified tht 3.2 is vlid for ny rel α. By using Lemm 3.1, Remrk 3.5, nd the identity t α 1 α 1 t α 2,we rrive inductively t the following generliztion.
6 Abstrct nd Applied Anlysis Theorem 3.6. For ny rel number α nd ny positive integer p, the following equlity holds: p 1 p 1 p f t p p 1 f t k 0 ( t ( p 1 )) α p k Γ ( α k p 1 ) k f ( p 1 ), 3.11 where f is defined on N. Lemm 3.7. For ny α>0, the following equlity holds: b Δf t Δ b f t b t α 1 f b. 3.12 Proof. By using of the following summtion by prts formul Δ s [ (ρ s ρ t ) α 1f s ] α 1 ( s ρ t ) α 2 f s ( s ρ t ) α 1Δf s 3.13 we hve b Δf t 1 ( ) α 1Δf s s ρ t 1 [ ( ) ] (ρ s ) α 1f s ( ) α 2f s Δ s ρ t α 1 s ρ t 3.14 1 Γ α 1 ( ) α 2f s b t α 1 s ρ t f b. On the other hnd, Δ b f t 1 ( ) α 1f s 1 ( ) 3.15 α 2f s, Δ t s ρ t s ρ t Γ α 1 where the identity Δ t ( s ρ t ) α 1 α 1 ( s ρ t ) α 2 3.16 nd the convention tht 0 α 1 0reused.
Abstrct nd Applied Anlysis 7 Remrk 3.8. Let α>0ndn α 1. Then, by the help of Lemm 3.7 we hve Δ b α f t Δ Δ n( ) b n α f t Δ n( ) Δ b n α f t 3.17 or Δ b α f t Δ n [ b n α Δf t ] b t n α 1 Γ n α f b. 3.18 Then, using the identity n b t n α 1 b t α 1 Δ Γ n α Γ α 3.19 we verified tht 3.12 is vlid for ny rel α. By using Lemm 3.7, Remrk 3.8, nd the identity Δ b t α 1 α 1 b t α 2,we rrive inductively t the following generliztion. Theorem 3.9. For ny rel number α nd ny positive integer p, the following equlity holds: b p 1 p 1 ( ) α p k b p 1 t Δ p f t Δ p b p 1 f t k 0 Γ ( α k p 1 ) Δ k f ( b p 1 ), 3.20 where f is defined on b N. We finish this section by stting the commuttive property for the right frctionl sum opertors without giving its proof. Lemm 3.10. Let f be rel vlued function defined on b N, nd let α, β>0. Then b[ ] b β f t b α β f t b β[ b f t ]. 3.21 4. Summtion by Prts Formuls for Frctionl Sums nd Differences We first stte summtion by prts formul for nbl frctionl sum opertors. Theorem 4.1. For α>0,, b R, f defined on N nd g defined on b N, the following equlity holds g s f s f s b g s. 4.1
8 Abstrct nd Applied Anlysis Proof. By the definition of the nbl left frctionl sum we hve g s f s 1 g s s r 1 ( s ρ r ) α 1f r. 4.2 If we interchnge the order of summtion we rech 4.1. By using Theorem 3.6, Lemm 3.4,nd n α f 0, we prove the following result. Theorem 4.2. For α>0, nd f defined in suitble domin N, the following re vlid α f t f t, 4.3 α f t f t, when α/ N, 4.4 n 1 α t k f t f t k f, when α n N. 4.5 k! k 0 We recll tht D α D α f t f t, where D α is the Riemnn-Liouville frctionl integrl, is vlid for sufficiently smooth functions such s continuous functions. As result of this it is possible to obtin integrtion by prts formul for certin clss of functions see 23 pge 76, nd for more detils see 22. Since discrete functions re continuous we see tht the term 1 α f t t,for0<α<1disppers in 4.4, with the ppliction of the convention tht f s 0. By using Theorem 3.9, Lemm 3.10,nd b n α f b 0, we obtin the following. Theorem 4.3. For α>0, nd f defined in suitble domin b N, we hve b b f t f t, 4.6 b b α f t f t, when α/ N, 4.7 b b α n 1 b t k f t f t Δ k f b, when α n N. 4.8 k! k 0 Theorem 4.4. Let α>0 be noninteger. If f is defined on b N nd g is defined on N,then f s α g s g s b α f s. 4.9 Proof. Eqution 4.7 implies tht f s α g s b ( b α f s ) α g s. 4.10
Abstrct nd Applied Anlysis 9 And by Theorem 4.1 we hve f s α g s b α f s α g s. 4.11 Then the result follows by 4.4. 5. Initil Vlue Problems Let us consider the following initil vlue problem for nonliner frctionl difference eqution α 1 y t f( t, y t ) for t 1, 2,..., 5.1 1 α 1 y t t y c, 5.2 where 0 <α<1nd is ny rel number. Apply the opertor to ech side of 5.1 to obtin α 1y t α f ( t, y t ). 5.3 Then using the definition of the frctionl difference nd sum opertors we obtin { 1 α 1 } y t f ( t, y t ) { { }} 1 α t 1 α y t y { } t 1 α α y t y f ( t, y t ), f ( t, y t ). 5.4 5.5 It follows from Lemm 3.2 tht { } t 1 α y { } t 1 α y t α 1 y. 5.6 Note tht { } t 1 α y 1 t 1 α y t 1 α 1 y. 5.7
10 Abstrct nd Applied Anlysis Hence we obtin { } t 1 α α y { 1 t 1 α y } t 1 α 1 y t α 1 y y { t 1 0} t 1 α 2 α 1 y t α 1 y 5.8 t 1 α 1 y, which follows from the power rule in Lemm 3.3. Let us put this expression bck in 5.5 nd use 4.4, we hve y t t 1 α 1 y f ( t, y t ). 5.9 Thus, we hve proved the following lemm. Lemm 5.1. y is solution of the initil vlue problem, 5.1, 5.2, if, nd only if, y hs the representtion 5.9. Remrk 5.2. A similr result hs been obtined in the pper 13 with the opertor α. And the initil vlue problem hs been defined in the following form α y t f ( t, y t ) for t 1, 2,..., 5.10 1 α y t t y c, 5.11 where 0 <α 1nd is ny rel number. The subscript of the term α y t on the left hnd side of 5.10 indictes directly tht the solution hs domin strts t. The nture of this nottion helps us to use the nbl trnsform esily s one cn see in the ppers 2, 12.Onthe other hnd, the subscript 1in 5.1 indictes tht the solution hs domin strts t. 6. An Alterntive Definition of Nbl Frctionl Differences Recently, the uthors in 18, by the help of nbl Leibniz s Rule, hve rewritten the nbl left frctionl difference in form similr to the definition of the nbl left frctionl sum. In this section, we do this for the nbl right frctionl differences. The following delt Leibniz s Rule will be used: g s, t Δ t g s, t g t, t 1. Δ t b 1 6.1
Abstrct nd Applied Anlysis 11 Using the following identity Δ t ( s ρ t ) α α ( s ρ t ) α 1, 6.2 nd the definition of the nbl right frctionl difference ii of Definition 2.3,forα>0, α/ N we hve b α f t 1 n Δ n b n α f t 1 n Δ n Γ n α ( ) n α 1f s s ρ t 1 n Δ n 1 Γ n α Δ t ( ) n α 1f s s ρ t [ ] 1 n Δ n 1 ( ) n α 2f s n α 1 s ρ t t t n α 1 Γ n α 6.3 1 n Δ n 1 Γ n α 1 ( ) n α 2f s. s ρ t By pplying the Leibniz s Rule 6.1, n 1 number of times we get b α 1 ( ) α 1f s. f t s ρ t 6.4 Γ α Inthebove,itistobeinsistedthtα/ N is required due the fct tht the term 1/Γ α is undefined for negtive integers. Therefore we cn proceed nd unify the definitions of nbl right frctionl sums nd differences similr to Definition 5.3 in 18. Also, the lterntive formul 6.4 cn be employed, similr to Theorem 5.4 in 18, to show tht the nbl right frctionl difference b α f is continuous with respect to α 0. 7. Conclusions In frctionl clculus there re two pproches to obtin frctionl derivtives. The first pproch is by iterting the integrl nd then defining frctionl order by using Cuchy formul to obtin Riemnn frctionl integrls nd derivtives. The second pproch is by iterting the derivtive nd then defining frctionl order by mking use of the binomil theorem to obtin Grünwld-Letnikov frctionl derivtives. In this pper we followed the discrete form of the first pproch vi the nbl difference opertor. However, we noticed tht in the right frctionl difference cse we used both the nbl nd delt difference opertors. This setting enbles us to obtin resonble summtion by prts formuls for nbl frctionl sums nd differences in Section 4 nd to obtin n lterntive definition for nbl right frctionl differences through the delt Leibniz s Rule in Section 6. While following the discrete form of the first pproch, two types of frctionl sums nd hence frctionl differences ppered; one type by strting from nd the other type,
12 Abstrct nd Applied Anlysis which obeys the generl theory of nbl time scle clculus, by strting from 1intheleft cse nd ending t b 1 in the right cse. Section 3 discussed the reltion between the two types of opertors, where certin properties of one opertor re obtined by using the second opertor. An initil vlue problem discussed in Section 5 is n importnt ppliction exposing the derived properties of the two types of opertors discussed throughout the pper, where the solution representtion ws obtined explicitly for order 0 < α < 1. Regrding this exmple we remrk the following. In frctionl clculus, Initil vlue problems usully mke sense for functions not necessrily continuous t left cse so tht the initil conditions re given by mens of. Since sequences re nice continuous functions then in Theorem 4.2, which is the tool in solving our exmple, the identity 4.4 ppers without ny initil condition. To crete n initil condition in our exmple we shifted the frctionl difference opertor so tht it strted t 1. Acknowledgment The second uthor is grteful to the Kentucky Science nd Engineering Foundtion for finncil support through reserch Grnt KSEF-2488-RDE-014. References 1 H. L. Gry nd N. F. Zhng, On new definition of the frctionl difference, Mthemtics of Computtion, vol. 50, no. 182, pp. 513 529, 1988. 2 F. M. Atıcı nd P. W. Eloe, Discrete frctionl clculus with the nbl opertor, Electronic Journl of Qulittive Theory of Differentil Equtions, no. 3, pp. 1 12, 2009. 3 K. S. Miller nd B. Ross, Frctionl difference clculus, in Proceedings of the Interntionl Symposium on Univlent Functions, Frctionl Clculus nd Their Applictions, pp. 139 152, Nihon University, Koriym, Jpn, 1989. 4 T. Abdeljwd, On Riemnn nd Cputo frctionl differences, Computers & Mthemtics with Applictions, vol. 62, no. 3, pp. 1602 1611, 2011. 5 T. Abdeljwd nd D. Blenu, Frctionl differences nd integrtion by prts, Journl of Computtionl Anlysis nd Applictions, vol. 13, no. 3, pp. 574 582, 2011. 6 F. Jrd, T. Abdeljwd, D. Blenu, nd K. Biçen, On the stbility of some discrete frctionl nonutonomous systems, Abstrct nd Applied Anlysis, vol. 2012, Article ID 476581, 9 pges, 2012. 7 J. Fhd, T. Abdeljwd, E. Gündoğdu, nd D. Blenu, On the Mittg-Leffler stbility of q-frctionl nonliner dynmicl systems, Proceedings of the Romnin Acdemy, vol. 12, no. 4, pp. 309 314, 2011. 8 T. Abdeljwd nd D. Blenu, Cputo q-frctionl initil vlue problems nd q-nlogue Mittg- Leffler function, Communictions in Nonliner Science nd Numericl Simultion, vol. 16, no. 12, pp. 4682 4688, 2011. 9 F. M. Atici nd P. W. Eloe, A trnsform method in discrete frctionl clculus, Interntionl Journl of Difference Equtions, vol. 2, no. 2, pp. 165 176, 2007. 10 F. M. Atici nd P. W. Eloe, Initil vlue problems in discrete frctionl clculus, Proceedings of the Americn Mthemticl Society, vol. 137, no. 3, pp. 981 989, 2009. 11 F. M. Atıcı nd S. Şengül, Modeling with frctionl difference equtions, Journl of Mthemticl Anlysis nd Applictions, vol. 369, no. 1, pp. 1 9, 2010. 12 F. M. Atici nd P. W. Eloe, Liner systems of frctionl nbl difference equtions, The Rocky Mountin Journl of Mthemtics, vol. 41, no. 2, pp. 353 370, 2011. 13 F. M. Atıcı nd P. W. Eloe, Gronwll s inequlity on discrete frctionl clculus, Computer nd Mthemtics with Applictions. In press. 14 R. A. C. Ferreir nd D. F. M. Torres, Frctionl h-difference equtions rising from the clculus of vritions, Applicble Anlysis nd Discrete Mthemtics, vol. 5, no. 1, pp. 110 121, 2011.
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