Table Of ContentPrinciples of Delta and Nabla Fractional Differences
Thabet Abdeljawad a’b
a DepartmentofMathematics, C¸ankayaUniversity,06530, Ankara,Turkey
b DepartmentofMathematics andPhysicalSciences
1
PrinceSultanUniversity,P.O.Box66833, Riyadh11586,SaudiArabia
1
0
2
Abstract. We define nabla right fractional sum and difference and obtain a nabla integration by
c
e parts formula. Some properties of nabla and delta fractional sums and differences are obtained to
D derivedualidentitiesbetweenthenablaanddeltaonesandotherQ-dualidentitiestorelateleftand
rightones. Thesedualidentitiesgivetheimpressionthat thedefinitionofthedeltarightfractional
5 difference presented by Abdeljawad and Baleanu in [1] and [12] are more appropriate than those
2 introduced by other authors. Hence, the delta integration by parts formulaformulated in [12] and
thenablaonepresentedinthisarticleareconsistent.
]
S Keywords: right (left) delta and nabla fractional sums, right (left) delta and nabla Riemann
D andCaputofractionaldifferences. Integration byparts,Q-operator,dualidentity.
.
h 1 Introduction
t
a
m During the last two decades, due to its widespread applications in different fields of science and
engineering,fractionalcalculushasattractedtheattentionofmanyresearchers[9,10,11,23,24,25].
[
StartingfromtheideaofdiscretizingtheCauchyintegralformula,MillerandRoss[8]andGray
and Zhang [18] obtained discrete versions of left type fractional integrals and derivatives, called
1
fractional sums and differences. Fifteen years later, several authors started to deal with discrete
v
fractionalcalculus[1,3,4,5,6,14,12,20,21,22],benefitingfromthetheoryoftimescalesoriginated
5
byHilgerin1988(see[15]).
9
In this article, we summarize some of the results mentioned in the above references and add
7
more in the Caputo type and right type fractional differences. Throughout the article we almost
5
agreewiththepreviouslypresenteddefinitionsexceptforthedefinitionofrightfractionaldifference.
2. Weshallfigureoutthatthesedefinitionsseemtobemoreconvenientthanthepreviouslypresented
1 onesbyprovingsomedualidentities. Theseidentitiesfallintotwokinds. Thefirstrelatenablatype
fractionaldifferencesandsumstodeltaones. ThesecondkindrepresentedbytheQ-operatorrelate
1
leftandrightfractionalsumsanddifferences. Also,inthedefinitionofleftandrightfractionalsums
1
weusethe ones that agreewiththe general theoryfortimescalesandnot asin[5]orin[18]. This
:
v setting enables us to get identities resembling better the ordinary fractional case. Along with the
i previouslymentionedpointsweareabletofitareasonablenablaintegrationbypartsformulawhich
X
remainsinaccordancewiththeoneobtainedin[12]butdifferentfromthoseobtainedin[6]and[14].
r Thearticleisorganizedasfollows: Theremainingpartofthissectioncontainssummarytosome
a ofthebasicnotations anddefinitions indeltaandnablacalculus. Section2contains thedefinitions
intheframeofdeltaandnablafractionalsumsanddifferencesintheRiemannsense. Moreoversome
essential lemmas about the commutativity of the different fractional sum operators with the usual
difference operators are established. These lemmas are vital to proceed in the next sections. The
thirdsection contains some dual identities relating nabla and delta fractional sums and differences
in the left and right cases. Using these dual identities, power formulae for nabla left and right
fractionalsums andacommutative lawfornablaleftandrightsumsareobtained. InSection 4an
integrationbypartsformulafornablafractionalsumsanddifferencesisobtainedwithanaccordance
totheoneobtainedin[12]. Section5isdevotedtodeltaandnablaCaputofractionaldifferencesand
theirrelationswiththeRiemannones. Finally,Section6containsCaputotypefractionaldynamical
equations where anonhomogeneous nabla Caputo fractional difference equation issolved to obtain
nabladiscreteversionsforMittag-Lefflerfunctions. Forthecaseα=1weobtainthediscretenabla
exponentialfunction[15]. Inadditiontothis,theQ-operatorisusedtorelateleftandrightfractional
sumsinthenablaanddeltacase. TheQ-dualidentitiesobtainedinthissectionexposethevalidity
ofthedefinitionofdeltaandnablarightfractionaldifferences.
1
Foranaturalnumbern,thefractionalpolynomialisdefinedby,
n−1 Γ(t+1)
t(n)= Y(t−j)= , (1)
Γ(t+1−n)
j=0
whereΓdenotes thespecialgammafunctionandtheproductiszerowhent+1−j=0forsomej.
Moregenerally,forarbitraryα,define
Γ(t+1)
t(α)= , (2)
Γ(t+1−α)
wheretheconvention thatdivisionatpoleyieldszero. Giventhattheforwardandbackwarddiffer-
enceoperatorsaredefinedby
∆f(t)=f(t+1)−f(t), ∇f(t)=f(t)−f(t−1) (3)
respectively, wedefineiterativelytheoperators ∆m=∆(∆m−1)and∇m=∇(∇m−1),wheremis
anaturalnumber.
Herearesomepropertiesofthefactorialfunction.
Lemma 1.1. ([3])Assume the following factorial functions are well defined.
(i) ∆t(α)=αt(α−1).
(ii) (t−µ)t(µ) =t(µ+1), where µ∈R.
(iii) µ(µ)=Γ(µ+1).
(iv) If t≤r, then t(α)≤r(α) for any α>r.
(v) If 0<α<1, then t(αν)≥(t(ν))α.
(vi) t(α+β)=(t−β)(α)t(β).
Also,forourpurposeswelistdownthefollowingtwoproperties,theproofsofwhicharestraight-
forward.
∇s(s−t)(α−1) =(α−1)(ρ(s)−t)(α−2). (4)
∇t(ρ(s)−t)(α−1) =−(α−1)(ρ(s)−t)(α−2). (5)
Forthesakeofthenablafractionalcalculuswehavethefollowingdefinition
Definition 1.1. ([15, 16, 17, 19])
(i) For a natural number m, the m rising (ascending) factorial of t is defined by
m−1
tm= Y(t−k), t0=1. (6)
k=0
(ii) For any real number the α rising functionis defined by
Γ(t+α)
tα= , t∈R− {...,−2,−1,0}, 0α=0 (7)
Γ(t)
Regardingtherisingfactorialfunctionweobservethefollowing:
(i)
∇(tα)=αtα−1 (8)
(ii)
(tα)=(t+α−1)(α). (9)
(iii)
∆t(s−ρ(t))α =−α(s−ρ(t))α−1 (10)
Notation:
(i) Forarealα>0,wesetn=[α]+1,where[α]isthegreatestinteger lessthanα.
(ii) Forrealnumbersaandb,wedenote Na={a,a+1,...}and bN={b,b−1,...}.
(iii) Forn∈Nandreala,wedenote
a∆nf(t),(−1)n∆nf(t).
(iv) Forn∈Nandrealb,wedenote
∇nf(t),(−1)n∇nf(t).
b
2
2 Definitions and essential lemmas
Definition 2.1. Letσ(t)=t+1 and ρ(t)=t−1be the forward and backward jumping operators,
respectively. Then
(i) The (delta) left fractional sum of order α>0 (starting from a) isdefined by:
t−α
∆−aαf(t)= Γ(1α) X(t−σ(s))(α−1)f(s), t∈Na+α. (11)
s=a
(ii) The (delta) right fractional sum of order α>0 (ending at b) is defined by:
b b
b∆−αf(t)= 1 X (s−σ(t))(α−1)f(s)= 1 X (ρ(s)−t)(α−1), t∈ b−αN. (12)
Γ(α) Γ(α)
s=t+α s=t+α
(iii) The (nabla) left fractional sum of order α>0 (starting from a) isdefined by:
t
∇−aαf(t)= Γ(1α) X (t−ρ(s))α−1f(s), t∈Na+1. (13)
s=a+1
(iv)The (nabla) right fractional sum of order α>0 (ending at b) isdefined by:
b∇−αf(t)= Γ(1α)bX−1(s−ρ(t))α−1f(s)= Γ(1α)bX−1(σ(s)−t)α−1f(s), t∈ b−1N. (14)
s=t s=t
Regardingthedeltaleftfractionalsumweobservethefollowing:
(i)∆−aα mapsfunctions definedonNa tofunctions definedonNa+α.
(ii)u(t)=∆−anf(t), n∈N,satisfiestheinitialvalueproblem
∆nu(t)=f(t), t∈Na, u(a+j−1)=0, j=1,2,...,n. (15)
(iii)TheCauchyfunction (t−σ(s))(n−1) vanishes ats=t−(n−1),...,t−1.
(n−1)!
Regardingthedeltarightfractionalsumweobservethefollowing:
(i) b∆−α mapsfunctions definedonbNtofunctionsdefinedonb−αN.
(ii)u(t)= b∆−nf(t), n∈N,satisfiestheinitialvalueproblem
∇nbu(t)=f(t), t∈ bN, u(b−j+1)=0, j=1,2,...,n. (16)
(iii)theCauchyfunction (ρ(s)−t)(n−1) vanishesats=t+1,t+2,...,t+(n−1).
(n−1)!
Regardingthenablaleftfractionalsumweobservethefollowing:
(i)∇−aα mapsfunctions definedonNa tofunctions definedonNa+1.
(ii)∇−anf(t)satisfiesthen-thorderdiscreteinitialvalueproblem
∇ny(t)=f(t), ∇iy(a)=0, i=0,1,...,n−1 (17)
(iii)TheCauchyfunction (t−ρ(s))n−1 satisfies∇ny(t)=0.
Γ(n)
Regardingthenablarightfractionalsumweobservethefollowing:
(i) b∇−α mapsfunctions definedon bNtofunctions definedon b−1N.
(ii)b∇−nf(t)satisfiesthen-thorderdiscreteinitialvalueproblem
a∆ny(t)=f(t), a∆iy(a)=0, i=0,1,...,n−1. (18)
Theproofcanbedoneinductively. Namely,assumingitistrueforn,wehave
a∆n+1 b∇−(n+1)f(t)= a∆n[−∆b∇−(n+1)f(t)]. (19)
Bythehelpof(10),itfollowsthat
a∆n+1 b∇−(n+1)f(t)= a∆n b∇−nf(t)=f(t). (20)
Theotherpartisclearbyusingtheconvention that t =0, s>t.
Pk=s
(iii)TheCauchyfunction (s−ρΓ((tn)))n−1 satisfiesa∆ny(t)=0.
3
Definition 2.2. (i)[8] The (delta) left fractional sum of order α>0 (starting from a ) is defined
by:
t−(n−α)
∆αaf(t)=∆n∆−a(n−α)f(t)= Γ(n1−α) X (t−σ(s))(n−α−1)f(s), t∈Na+(n−α) (21)
s=a
(ii) [12] The (delta) right fractional difference of order α>0(ending at b) is defined by:
b
b∆αf(t)=(−1)n∇n b∆−(n−α)f(t)= Γ(n1−α) X (s−σ(t))(n−α−1)f(s), t∈ b−(n−α)N
s=t+(n−α)
(22)
(iii) The (nabla) left fractional difference of order α>0 (starting from a ) isdefined by:
t
∇αaf(t)=∇n∇−a(n−α)f(t)= Γ(n1−α) X (t−ρ(s))n−α−1f(s), t∈Na+1 (23)
s=a+1
(iv) The (nabla) right fractional difference of order α>0 (ending at b )is defined by:
b∇αf(t)=(−1)n∆n b∇−(n−α)f(t)= Γ(n1−α)bX−1(s−ρ(t))n−α−1f(s), t∈ b−1N (24)
s=t
Regardingthedomainsofthefractionaltypedifferences weobserve:
(i)Thedeltaleftfractionaldifference∆αa mapsfunctionsdefinedonNa tofunctionsdefinedon
Na+(n−α).
(ii)Thedeltarightfractionaldifference b∆αmapsfunctionsdefinedon bNtofunctionsdefined
on b−(n−α)N.
(iii) The nabla left fractional difference ∇αa maps functions defined on Na to functions defined
onNa+1−n.
(iv)Thenablarightfractionaldifference b∇αmapsfunctionsdefinedon bNtofunctionsdefined
on b−1+nN.
Lemma 2.1. [3] For any α>0, the following equality holds:
∆−α∆f(t)=∆∆−αf(t)− (t−a)α−1f(a).
a a Γ(α)
Lemma 2.2. [12] For any α>0, the following equality holds:
b∆−α∇bf(t)=∇b b∆−αf(t)− (b−t)α−1f(b).
Γ(α)
Lemma 2.3. [7] For any α>0, the following equality holds:
∇−α ∇f(t)=∇∇−α− (t−a+1)α−1f(a) (25)
a+1 a Γ(α)
The resultof Lemma2.3was obtained in[7] byapplying the nabla leftfractional sum starting
fromanotfroma+1. NextwillprovidetheversionofLemma2.3byapplyingthedefinitioninthis
article.
Lemma 2.4. For any α>0, the following equality holds:
∇−α∇f(t)=∇∇−α− (t−a)α−1f(a). (26)
a a Γ(α)
Proof. Bythehelpofthefollowingbypartsidentity
∇s[(t−s)α−1f(s)]=∇s(t−s)α−1f(s)+(t−ρ(s))α−1∇sf(s) (27)
=−(α−1)(t−ρ(s))α−2f(s)+(t−ρ(s))α−1∇sf(s)
wehave
t
∇−aα∇f(t)= Γ(1α) X (t−ρ(s))α−1∇sf(s)=
s=a+1
4
t
Γ(1α)[(t−s)α−1f(s)|ta+(α−1) X (t−ρ(s))α−2f(s)]= (28)
s=a+1
− (t−a)α−1f(a)+ 1 Xt (t−ρ(s))α−2f(s) (29)
Γ(α) Γ(α−1)
s=a+1
Ontheother hand
∇∇−αf(t)=
a
t t
1 X ∇t(t−ρ(s))α−1f(s)= 1 X (t−ρ(s))α−2f(s) (30)
Γ(α) Γ(α−1)
s=a+1 s=a+1
Remark 2.1. Let α>0 and n=[α]+1. Then, by the help of Lemma 2.4 we have
∇∇αaf(t)=∇∇n(∇−a(n−α)f(t))=∇n(∇∇−a(n−α)f(t)). (31)
or
∇∇αaf(t)=∇n[∇−a(n−α)∇f(t)+ (t−Γ(an)−n−αα)−1f(a)] (32)
Then, using the identity
(t−a)n−α−1 (t−a)−α−1
∇n = (33)
Γ(n−α) Γ(−α)
we infer that (26) is valid for any real α.
By the help of Lemma 2.4, Remark 2.1 and the identity ∇(t−a)α−1 =(α−1)(t−a)α−2, we
arriveinductivelyatthefollowinggeneralization.
Theorem 2.5. For any real number α and any positive integerp, the following equality holds:
∇−aα ∇pf(t)=∇p∇−aαf(t)−pX−1Γ((αt−+ak)α−−pp++k1)∇kf(a). (34)
k=0
where f isdefined on Na .
Lemma 2.6. For any α>0, the following equality holds:
b∇−α a∆f(t)= a∆ b∇−αf(t)− (b−t)α−1f(b) (35)
Γ(α)
Proof. Bythehelpofthefollowingdiscretebypartsformula:
∆s[(ρ(s)−ρ(t))α−1f(s)]=
(α−1)(s−ρ(t))α−2f(s)+(s−ρ(t))α−1∆f(s) (36)
wehave
b−1
b∇−α a∆f(t)=− 1 X(s−ρ(t))α−1∆f(s)=
Γ(α)
s=t
b−1 b−1
1 [−X∆s((ρ(s)−ρ(t))α−1f(s))+(α−1)X(s−ρ(t))α−2f(s)]= (37)
Γ(α)
s=t s=t
1 bX−1(s−ρ(t))α−2f(s)− (b−t)α−1f(b). (38)
Γ(α−1) Γ(α)
s=t
Ontheother hand,
a∆b∇−αf(t)=
b−1 b−1
− 1 X∆t(s−ρ(t))α−1f(s)= 1 X(s−ρ(t))α−2f(s) (39)
Γ(α) Γ(α−1)
s=t s=t
wheretheidentity
∆t(s−ρ(t))α−1 =−(α−1)(s−ρ(t))α−2
andtheconvention that(0)α−1 =0areused.
5
Remark 2.2. Let α>0 and n=[α]+1. Then, by the help of Lemma 2.6 we can have
a∆ b∇αf(t)= a∆ a∆n( b∇−(n−α)f(t))= a∆n( a∆ b∇−(n−α)f(t)) (40)
or
a∆ b∇αf(t)= a∆n[ b∇−(n−α) a∆f(t)+ (b−t)n−α−1f(b)] (41)
Γ(n−α)
Then, using the identity
(b−t)n−α−1 (b−t)−α−1
a∆n = (42)
Γ(n−α) Γ(−α)
we infer that (35) is valid for any real α.
By the help of Lemma 2.6, Remark 2.2 and the identity ∆(b−t)α−1 =−(α−1)(b−t)α−2, if
wefollowinductivelywearriveatthefollowinggeneralization
Theorem 2.7. For any real number α and any positive integerp, the following equality holds:
b∇−α a∆pf(t)= a∆p b∇−αf(t)−pX−1 (b−t)α−p+k a∆kf(b) (43)
Γ(α+k−p+1)
k=0
where f isdefined on bN and we remind that a∆kf(t)=(−1)k∆kf(t).
3 Dual identities for right fractional sums and dif-
ferences
The dual relations for left fractional sums and differences were investigated in [5]. Indeed, the
followingtwolemmasaredualrelationsbetweenthedeltaleftfractionalsums(differences)andthe
nablaleftfractionalsums(differences).
Lemma3.1. [5]Let0≤n−1<α≤nandlety(t)bedefinedonNa. Thenthefollowingstatements
are valid.
((ii)i)(∆(∆αa)−ayα(t)y−(tα+)=α)∇=αa∇y(−at)αyfo(rt)tf∈orNtn∈+aN.a.
Lemma 3.2. [5] Let 0 ≤ n−1 < α ≤ n and let y(t) be defined on Nα−n. Then the following
statementsare valid.
(i)∆αα−ny(t)=(∇αα−ny)(t+α) for t∈N−n.
(ii) ∆−α−(nn−α)y(t)=(∇−α−(nn−α)y)(t−n+α) for t∈N0.
Weremindthattheabovetwoduallemmasforleftfractionalsumsanddifferenceswereobtained
whenthenablaleftfractionalsumwasdefinedby
t
∇−aαf(t)= Γ(1α) X(t−ρ(s))α−1f(s), t∈Na (44)
s=a
Now, in analogous to Lemma 3.1 and Lemma 3.2, for the right fractional summations and
differencesweobtain
Lemma 3.3. Let y(t) be defined on b+1N. Then the following statementsare valid.
(i)( b∆α)y(t+α)= b+1∇αy(t) for t∈ b−nN.
(ii) ( b∆−α)y(t−α)= b+1∇−αy(t) for t∈ bN.
Proof. Weproveonly(i). Theproofof(ii)issimilarandeasier.
(b∆α)y(t+α)=(−1)n∇n b∆−(n−α)y(t+α)=
(−1)n∇n Xb (s−t−1−α)(n−α−1)y(s)= (−1)n∆n Xb (s−t−1−+n−α)(n−α−1)y(s) (45)
Γ(n−α) Γ(n−α)
s=t+n s=t
6
Usingtheidentitytα=(t+α−1)(α),wearriveat
(b∆α)y(t+α)= (−1)n∆n Xb (s−ρ(t))n−α−1y(s)=
Γ(n−α)
s=t
(−1)n∆n b+1∇−(n−α)y(t)= b+1∇αy(t). (46)
Lemma 3.4. Let 0 ≤ n−1 < α ≤ n and let y(t) be defined on n−αN. Then the following
statementsare valid.
(i)
n−α∆αy(t)= n−α+1∇αy(t−α), t∈ nN
(ii)
n−α∆−(n−α)y(t)= n−α+1∇−(n−α)y(t+n−α), t∈ 0N
Proof. Weprove(i),theproofof(ii)issimilar. Bythedefinitionofrightnabladifferencewehave
n−α
n−α+1∇αy(t−α)= a∆n 1 X (s−ρ(t−α))n−α−1y(s)=
Γ(n−α)
s=t−α
n−α n−α
a∆nΓ(n1−α) X (s−ρ(t−α))n−α−1y(s)=∇nbΓ(n1−α) X (s−ρ(t+n−α))n−α−1y(s)
s=t−α s=t+n−α
(47)
Byusing(9)itfollowsthat
n−α
n−α+1∇αy(t−α)=∇nbΓ(n1−α) X (s−σ(t))(n−α−1)y(s)= n−α∆αy(t) (48)
s=t+n−α
Note that the above two dual lemmas for rightfractional differences can not be obtained ifwe
applythedefinitionofthedeltarightfractionaldifferenceintroducedin[14]and[6].
Lemma 3.5. [12] Let α>0, µ>0. Then,
b−µ∆−α(b−t)(µ) = Γ(µ+1) (b−t)(µ+α) (49)
Γ(µ+α+1)
Thefollowingcommutative propertyfordeltarightfractionalsumsisTheorem9in[12].
Theorem 3.6. Let α>0, µ>0. Then, for all t such that t≡b−(µ+α) (mod 1), we have
b∆−α[ b∆−µf(t)]= b∆−(µ+α)f(t)= b∆−µ[ b∆−αf(t)] (50)
where f isdefined on bN.
Proposition 3.7. Let f be a real valued function defined on bN, and let α,β>0. Then
b∇−α[ b∇−βf(t)]= b∇−(α+β)f(t)= b∇−β[ b∇−αf(t)] (51)
Proof. TheprooffollowsbyapplyingLemma3.3(ii)andTheorem3.6above. Indeed,
b∇−α[b∇−βf(t)]= b∇−α b−1∆−βf(t−β)=
b−1∆−α b−1∆−βf(t−(α+β))= b−1∆−(α+β)f(t−(α+µ))= b∇−(α+β)y(t) (52)
Thefollowingpowerrulefornablarightfractionaldifferencesplaysanimportantrule.
Proposition 3.8. Let α,µ>0. Then, fort∈ bN , we have
b∇−α(b−t)µ= Γ(µ+1) (b−t)α+µ (53)
Γ(α+µ+1)
7
Proof. Bythedualformula(ii)ofLemma3.3,wehave
b∇−α(b−t)µ= b−1∆−α(b−r)µ|r=t−α=
b−1
1 X(s−t+α−1)(α−1)(b−s)µ. (54)
Γ(α)
s=t
Thenbytheidentitytα=(t+α−1)(α−1) andusingthechangeofvariabler=s−µ+1,itfollows
that
b∇−α(b−t)µ=
b−µ
1 X (r−σ(t−α−µ+1))(α−1)(b−r)µ=(b−µ∆−α(b−u)µ)|u=−α−µ+1+t. (55)
Γ(α)
r=t−µ+1
WhichbyLemma3.5leadsto
b∇−α(b−t)µ=
Γ(µ+1) Γ(µ+1)
(b−t+α+µ−1)(α+µ) = (b−t)α+µ (56)
Γ(α+µ+1) Γ(α+µ+1)
Similarly,forthenablaleftfractionalsumwecanhavethefollowingpowerformulaandexponent
law
Proposition 3.9. Let α,µ>0. Then, fort∈Na , we have
∇−α(t−a)µ= Γ(µ+1) (t−a)α+µ (57)
a Γ(α+µ+1)
Proposition 3.10. Let f be areal valued function defined on Na, and let α,β>0. Then
∇−aα[∇−aβf(t)]=∇−a(α+β)f(t)=∇−aβ[∇−aαf(t)] (58)
Proof. The proof can be achieved as in Theorem 2.1 [5], by expressing the left hand side of (58),
interchanging the order of summation and using the power formula (57). Alternatively, the proof
canbedonebyfollowingasintheproofofProposition3.7withthehelpofthedualformulaforleft
fractionalsuminLemma3.1afteritsarrangementaccordingtoourdefinitions.
4 Integration by parts for fractional sums and dif-
ferences
Wefirststatetheintegrationbypartsfordeltafractionalsumsanddifferences.
Proposition 4.1. [12] Let α>0, a,b∈R such that a<b and b≡a+α (mod 1). If f is defined
on Na and g is defined on bN, thenwe have
b b−α
X (∆−aαf)(s)g(s)= Xf(s) b∆−αg(s). (59)
s=a+α s=a
Proposition 4.2. [12] Let α>0 be non-integerand assume that b≡a+(n−α) (mod 1). If f is
defined on bN and g is defined on Na, then
b−n+1 b−(n−α)+1
X f(s)∆αag(s)= X g(s) b∇αf(s). (60)
s=a+(n−α)−1 s=a+n−1
Proposition 4.3. For α>0, a,b∈R, f defined on Na and g defined on bN, we have
b−1 b−1
X g(s)∇−aαf(s)= X f(s) b∇−αg(s). (61)
s=a+1 s=a+1
8
Proof. Bythedefinitionofthenablaleftfractionalsumwehave
b−1 b−1 s
X g(s)∇−aαf(s)= Γ(1α) X g(s) X (s−ρ(r))α−1f(r). (62)
s=a+1 s=a+1 r=a+1
Ifweinterchange theorderofsummationwereachat(61).
BythehelpofTheorem 2.5,Proposition3.10,(17)andthat ∇−a(n−α)f(a)=0,wecan, forthe
nablaleftsumsanddifferences,obtain
Proposition 4.4. For α>0, and f defined in a suitable domain Na, we have
∇α∇−αf(t)=f(t), (63)
a a
∇−α∇αf(t)=f(t), when α∈/N, (64)
a a
and
∇−aα∇αaf(t)=f(t)−nX−1(t−k!a)k∇kf(a),,when α=n∈N. (65)
k=0
We recall that (64) is valid in the usual fractional case for sufficiently good functions such as
continuous functions. As a result of this it was possible to obtain an integration by parts in the
Riemann fractional derivative case for certain class of functions (see [11] page 76, and for more
details see [10]). Since discrete functions are continuous we see that the term ∇a−(1−α)f(t)|t=a,
for 0 < α < 1 disappears in (64), with the application of the convention that a f(s) = 0.
Ps=a+1
However,intheCaputocasetheinitialtypeconditionsstartingfromaappearasweshallseeinthe
next sections. By this connection, we would like to ask the reader to compare with [7], where the
term∇−a(1−α)f(t)|t=a=f(a)appearsandhenceitwaspossibletoemploytheRiemanntypeinitial
value fractional difference equation to obtain a discrete fractional version of Gronwall’s inequality.
However,weremindthattheauthorsthereobey[18]indefiningthenablaleftfractionalsum. Inour
casetheCaputofractionaldifferencewillbethemoresuitabletooltoobtainsuchfractionalversion
ofGronwall’sinequality.
BythehelpofTheorem2.7,Proposition3.7,(18)andthat b∇−(n−α)f(b)=0,wecan,forthe
nablarightsumsanddifferences,obtain
Proposition 4.5. For α>0, and f defined in a suitable domain bN, we have
b∇α b∇−αf(t)=f(t), (66)
b∇−α b∇αf(t)=f(t), when α∈/N, (67)
and
b∇−α b∇αf(t)=f(t)−nX−1(b−t)k a∆kf(b),when α=n∈N. (68)
k!
k=0
Proposition 4.6. Let α>0 be non-integer. If f is defined on bN and g is defined on Na, then
b−1 b−1
X f(s)∇αag(s)= X g(s) b∇αf(s). (69)
s=a+1 s=a+1
Proof. Bythehelpofequation (67)ofProposition4.5wecanwrite
b−1 b−1
X f(s)∆αg(s)= X b∇−α(b∇αf(s))∇αag(s) (70)
s=a+1 s=a+1
andbyProposition4.1wehave
b−1 b−1
X f(s)∆αg(s)= X b∇αf(s)∇−aα∇αag(s). (71)
s=a+1 s=a+1
Thentheresultfollowsbyequation(64)ofProposition4.4.
9
5 Caputo fractional differences
Inanalogous totheusualfractionalcalculuswecanformulatethefollowingdefinition
Definition 5.1. Let α>0, α∈/N. Then,
(i)[1]thedeltaα−orderCaputoleftfractionaldifferenceofafunctionf definedonNa isdefined
by
t−(n−α)
C∆αaf(t),∆−a(n−α)∆nf(t)= Γ(n1−α) X (t−σ(s))(n−α−1)∆nsf(s) (72)
s=a
(ii) [1] the delta α− order Caputo right fractional difference of a function f defined on bN is
defined by
b
Cb∆αf(t), b∆−(n−α)∇nbf(t)= Γ(n1−α) X (s−σ(t))(n−α−1)∇nbf(s) (73)
s=t+(n−α)
where n=[α]+1.
If α=n∈N, then
C∆αf(t),∆nf(t) and C∆αf(t),∇nf(t)
a b b
(iii)thenablaα−orderCaputoleftfractionaldifferenceofafunctionf definedonNa isdefined
by
t−(n−α)
C∇αaf(t),∇−a(n−α)∇nf(t)= Γ(n1−α) X (t−ρ(s))n−α−1∆nf(s) (74)
s=a+1
(iv)thenablaα−orderCaputoleftfractionaldifferenceofafunctionf definedon bNisdefined
by
b−1
Cb∇αf(t), b∇−(n−α) a∆nf(t)= Γ(n1−α)X(s−ρ(t))n−α−1 a∆nf(s) (75)
s=t
Ifα=n∈N,then
C∇αaf(t),∇nf(t) and Cb∇αf(t), a∆nf(t)
Itisclearthat C∆αa mapsfunctionsdefinedonNa tofunctionsdefinedonNa+(n−α),andthat
Cb∆αmapsfunctionsdefinedonbNtofunctionsdefinedonb−(n−α)N. Also,itisclearthatthenabla
leftfractionaldifference ∇αa mapsfunctions defined onNa tofunctions defined onNa+1−n andthe
nablarightfractionaldifference b∇αmapsfunctionsdefinedon bNtofunctionsdefinedon b−1+nN.
RiemannandCaputodeltafractionaldifferences arerelatedbythefollowingtheorem
Theorem 5.1. [1] For any α>0, we have
n−1 (t−a)(k−α)
C∆αaf(t)=∆αaf(t)−X Γ(k−α+1)∆kf(a) (76)
k=0
and
n−1 (b−t)(k−α)
Cb∆αf(t)= b∆αf(t)−X Γ(k−α+1)∇kbf(b). (77)
k=0
In particular, when 0<α<1, we have
(t−a)(−α)
C∆af(t)=∆αaf(t)− Γ(1−α) f(a). (78)
(b−t)(−α)
Cb∆f(t)= b∆αf(t)− Γ(1−α) f(b) (79)
OnecannotethattheRiemannandCaputofractionaldifferences,for0<α<1,coincidewhen
f vanishes attheendpoints.
The following identity is useful to transform delta type Caputo fractional difference equations
intofractionalsummations.
10
