Generalized fractional calculus with applications to the calculus of variations

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by  Tatiana Odzijewicz

GENERALIZED FRACTIONAL CALCULUS WITH
APPLICATIONS TO THE CALCULUS OF VARIATIONS
2
1 TATIANAODZIJEWICZ,AGNIESZKAB.MALINOWSKA,ANDDELFIMF.M.TORRES
0
2
Abstract. WestudyoperatorsthataregeneralizationsoftheclassicalRiemann–

n Liouville fractional integral, and of the Riemann–Liouville and Caputo frac-
a tional derivatives. Ausefulformularelatingthegeneralizedfractionalderiva-
J tivesisproved,aswellasthreerelationsoffractionalintegrationbypartsthat

7 change the parameter set of the given operator into its dual. Such results
2 areexploredinthecontextofdynamicoptimization,byconsideringproblems
ofthecalculusofvariationswithgeneralfractionaloperators. Necessaryopti-

] malityconditionsofEuler–Lagrangetypeandnaturalboundaryconditionsfor
A
unconstrained and constrained problems are investigated. Interesting results
C are obtained even in the particular case when the generalized operators are
reduced to be the standard fractional derivatives in the sense of Riemann–
.
h LiouvilleorCaputo. Asanapplicationweprovideaclassofvariationalprob-
t lems with an arbitrary kernel that give answer to the important coherence
a
embeddingproblem. Illustrativeoptimizationproblemsareconsidered.
m
[

1
v 1. Introduction
7
4 Fractional calculus studies derivatives (and integrals) of non-integer order. It is
7 aclassicalmathematicalfieldasoldascalculusitself[25]. Duringalmost300years,
5 fractional calculus was considered as pure mathematics, with nearly no applica-
.
1 tions. In recentyears,however,the situation changeddramatically,with fractional
0 calculus becoming an interesting and useful topic among engineers and applied
2
scientists,andanexcellenttoolfordescriptionofmemoryandheredity effects [30].
1
One of the earliest applications of fractional calculus was to construct a com-
:
v plete mechanicaldescriptionofnonconservativesystems,including Lagrangianand
i
X Hamiltonian mechanics [42, 43]. Riewe’s results [42, 43] mark the beginning of the
fractional calculus of variations and are of upmost importance: nonconservative
r
a and dissipative processes are widespread in the physical world. Fractional calculus
provide the necessary tools to apply variational principles to systems character-
ized by friction or other dissipative forces, being even possible to deduce fractional
conservation laws along the nonconservative extremals [18].
The theory of the calculus of variations with fractional derivatives is nowadays
under strong current development, and the literature is already vast. We do not
try to make here a review. Roughly speaking, available results in the literature
2010MathematicsSubjectClassification. Primary: 26A33,34A08;Secondary: 49K05,49K21.
Key words and phrases. Fractional operators; calculus of variations; generalized fractional
calculus;integrationbyparts;necessaryoptimalityconditions;coherentembedding.
Submitted 22-Dec-2011; revised 26-Jan-2012; accepted 27-Jan-2012; for publication in Com-
putersand Mathematics with Applications.
Part of the first author’s Ph.D., which is carried out at the University of Aveiro under the
DoctoralProgrammeMathematics and Applications ofUniversitiesofAveiroandMinho.
1
2 T.ODZIJEWICZ,A.B.MALINOWSKA,ANDD.F.M.TORRES
use different notions of fractional derivatives, in the sense of Riemann–Liouville
[8, 19, 37], Caputo [20, 32, 36], Riesz [1, 21, 41], combined fractional derivatives
[27, 28, 33], or modified/generalized versions of the classical fractional operators
[3, 13, 24, 31, 38], in order to describe different variational principles. Here we
develop a more general perspective to the subject, by considering three fractional
operatorsthatdependonageneralkernel. Bychoosingspecialcasesforthekernel,
oneobtainsthestandardfractionaloperatorsandpreviousresultsintheliterature.
Moreimportant,thegeneralapproachhereconsideredbringsnewinsightsandgive
answers to some important questions.
Thetextisorganizedasfollows. InSection2thegeneralizedfractionaloperators
Kα, Aα and Bα are introducedandbasicresults given. The maincontributions of
P P P
thepaperappearinSection3: weproveausefulrelationbetweenAα andBα (The-
P P
orem3.1),severalformulasofintegrationbypartsthatchangetheparametersetP
intoitsdualP∗ (Theorems3.4,3.7and3.8),andnewfractionalnecessaryoptimal-
ityconditionsforgeneralizedvariationalproblemswithmixedintegerandfractional
orderderivativesandintegrals(Theorems3.11,3.17and3.22). Weseethatevenfor
an optimization problem that does not depend on generalized Riemann–Liouville
fractionalderivatives,suchderivativesappearnaturallyinthenecessaryoptimality
conditions. This is connected with duality of operators in the formulas of inte-
gration by parts and explains no-coherence of the fractional embedding [13]. This
is addressed in Section 3.4, where we give an answer to the important question
of coherence, by providing a class of fractional variational problems that does not
depend on the kernel, for which the embedded Euler–Lagrange equation coincides
with the one obtained by the least action principle (Theorem 3.26). Finally, some
concrete examples of optimization problems are discussed in Section 4.
2. Basic notions
Throughout the text, α denotes a positive real number between zero and one,
and ∂ F the partial derivative of a function F with respect to its ith argument.
i
WeconsiderthegeneralizedfractionaloperatorsKα, Aα andBα asdenotedin[2].
P P P
The study ofgeneralizedfractionaloperatorsandtheir applicationshas a long and
rich history. We refer the reader to the book [26].
Definition 2.1 (Generalized fractional integral). The operator Kα is given by
P
t b
Kαf(t)=p k (t,τ)f(τ)dτ +q k (τ,t)f(τ)dτ,
P α α
Z Z
a t
where P = ha,t,b,p,qi is the parameter set (p-set for brevity), t ∈ [a,b], p,q are
real numbers, and k (t,τ) is a kernel which may depend on α. The operator Kα
α P
is referred as the operator K (K-op for simplicity) of order α and p-set P.
Theorem 2.2. Let α∈(0,1)and P =ha,t,b,p,qi. If k (·,·) is a square-integrable
α
function on ∆ = [a,b]×[a,b], then Kα : L ([a,b]) → L ([a,b]) is a well defined
P 2 2
bounded linear operator.
Proof. Let α∈(0,1) and P =ha,t,b,p,qi. Define
pk (t,τ) if τ <t,
G(t,τ):= α
qk (τ,t) if τ ≥t.
α
(cid:26)
GENERALIZED FRACTIONAL CALCULUS WITH APPLICATIONS 3
For all f ∈L ([a,b]) one has Kαf(t)= bG(t,τ)f(τ)dτ with G(t,τ) ∈L (∆). It
2 P a 2
is not difficult to see that Kα is linear and Kαf ∈L ([a,b]) for all f ∈L ([a,b]).
P R P 2 2
Moreover,applyingtheCauchy–SchwarzinequalityandFubini’stheorem,weobtain
2
b b
kKαfk2 = G(t,τ)f(τ)dτ dt
P 2
Za (cid:12)(cid:12)Za (cid:12)(cid:12)
(cid:12) (cid:12)
b(cid:12) b (cid:12) b
≤ (cid:12) |G(t,τ)|2dτ(cid:12) |f(τ)|2dτ dt
Za " Za ! Za !#
b b
=kfk2 |G(t,τ)|2dτdt.
2
Za Za
1
For f ∈L ([a,b]) such that kfk ≤1 we have kKαfk ≤ b b|G(t,τ)|2dτdt 2.
2 2 P 2 a a
Therefore, kKαk ≤ b b|G(t,τ)|2dτdt 21. (cid:16)R R (cid:17)(cid:3)
P 2 a a
(cid:16)R R (cid:17)
Theorem 2.3. Let k be a difference kernel, i.e., k (t,τ) = k (t−τ) and k ∈
α α α α
L ([a,b]). Then Kα : L ([a,b]) → L ([a,b]) is a well defined bounded linear
1 P 1 1
operator.
Proof. Obviously, the operator is linear. Let α ∈ (0,1), P = ha,t,b,p,qi, and
f ∈L ([a,b]). Define
1
p|k (t−τ)|·|f(τ)| if τ ≤t
F(τ,t):= α
q|k (τ −t)|·|f(τ)| if τ >t
α
(cid:26)
for all (τ,t)∈∆=[a,b]×[a,b]. Since F is measurable on the square ∆ we have
b b b b τ
F(τ,t)dt dτ = |f(τ)| p|k (t−τ)|dt+ q|k (τ −t)|dt dτ
α α
Za Za ! Za " Zτ Za !#
b
≤ |f(τ)|(p−q)kk kdτ
α
Za
=(p−q)kk k·kfk.
α
It follows from Fubini’s theorem that F is integrable on the square ∆. Moreover,
b t b
kKαfk= p k (t−τ)f(τ)dτ +q k (τ −t)f(τ)dτ dt
P α α
Za (cid:12)(cid:12) Za Zt (cid:12)(cid:12)
(cid:12) (cid:12)
b(cid:12) t b (cid:12)
≤ (cid:12) p |k (t−τ)|·|f(τ)|dτ +q |k (τ −t)|(cid:12)·|f(τ)|dτ dt
α α
Za Za Zt !
b b
= F(τ,t)dτ dt
Za Za !
≤(p−q)kk k·kfk.
α
Hence, Kα :L ([a,b])→L ([a,b]) and kKαk≤(p−q)kk k. (cid:3)
P 1 1 P α
Theorem 2.4. Let k1−α be a difference kernel, i.e., k1−α(t,τ) =k1−α(t−τ) and
k1−α ∈ L1([a,b]). If f ∈ AC([a,b]), then the K-op of order 1 − α and p-set
4 T.ODZIJEWICZ,A.B.MALINOWSKA,ANDD.F.M.TORRES
P =ha,t,b,p,qi, i.e.,
t b
KP1−αf(t)=p k1−α(t−τ)f(τ)dτ +q k1−α(τ −t)f(τ)dτ,
Z Z
a t
belongs to AC([a,b]).
Proof. Let P =ha,t,b,p,0i and P =ha,t,b,0,qi. Then, K1−α =K1−α+K1−α.
1 2 P P1 P2
First we show that K1−αf ∈AC([a,b]). The condition f ∈AC([a,b]) implies
P1
x
f(x)= g(t)dt+f(a), where g ∈L ([a,b]).
1
Za
Let s=x−a and
s
h(s)= k1−α(τ)g(s+a−τ)dτ.
Z0
Integrating,
s s θ
h(θ)dθ = dθ k1−α(τ)g(θ+a−τ)dτ,
Z0 Z0 Z0
and changing the order of integration we obtain
s s s s s
h(θ)dθ = dτ k1−α(τ)g(θ+a−τ)dθ = k1−α(τ)dτ g(θ+a−τ)dθ.
Z0 Z0 Zτ Z0 Zτ
Putting ξ =θ+a−τ and dξ =dθ, we have
s s x−τ
h(θ)dθ = k1−α(τ)dτ g(ξ)dξ.
Z0 Z0 Za
x−τ
Because g(ξ)dξ =f(x−τ)−f(a), the following equality holds:
Za
s s s
h(θ)dθ = k1−α(τ)f(x−τ)dτ −f(a) k1−α(τ)dτ,
Z0 Z0 Z0
that is,
s s s
k1−α(τ)f(x−τ)dτ = h(θ)dθ+f(a) k1−α(τ)dτ.
Z0 Z0 Z0
Both functions on the right-hand side of the equality belong to AC([a,b]). Hence,
s
k1−α(τ)f(x−τ)dτ ∈AC([a,b]).
Z0
Substituting t=x−τ and dt=−dτ, we get
x
k1−α(x−t)f(t)dt∈AC([a,b]).
Za
This means that K1−αf ∈ AC([a,b]). The proof that K1−αf ∈ AC([a,b]) is
P1 P2
analogous, and since the sum of two absolutely continuous functions is absolutely
continuous, it follows that K1−αf ∈AC([a,b]). (cid:3)
P
GENERALIZED FRACTIONAL CALCULUS WITH APPLICATIONS 5
Remark 2.5. The K-op reduces to the classical left or right Riemann–Liouville
fractionalintegral(see, e.g.,[25,39]) for a suitably chosenkernel k (t,τ) and p-set
α
P. Indeed, let k (t−τ)= 1 (t−τ)α−1. If P =ha,t,b,1,0i, then
α Γ(α)
t
1
(2.1) Kαf(t)= (t−τ)α−1f(τ)dτ =: Iαf(t)
P Γ(α) a t
Z
a
istheleftRiemann–Liouvillefractionalintegraloforderα;ifP =ha,t,b,0,1i,then
b
1
(2.2) Kαf(t)= (τ −t)α−1f(τ)dτ =: Iαf(t)
P Γ(α) t b
Z
t
is the right Riemann–Liouville fractional integral of order α. Theorem 2.3 with
k (t−τ)= 1 (t−τ)α−1 asserts the well-knownfact that the Riemann–Liouville
α Γ(α)
fractional integrals Iα, Iα : L ([a,b]) → L ([a,b]) given by (2.1) and (2.2) are
a t t b 1 1
well defined bounded linear operators.
ThefractionalderivativesAα andBα aredefinedwiththehelpofthegeneralized
P P
fractional integral K-op.
Definition 2.6 (GeneralizedRiemann–Liouville fractionalderivative). LetP be a
given parameter set. The operator Aα, 0 < α < 1, is defined by Aα = D◦K1−α,
P P P
where D denotes the standard derivative. We refer to Aα as operator A (A-op) of
P
order α and p-set P.
A different fractional derivative is obtained by interchanging the order of the
operators in the composition that defines Aα.
P
Definition 2.7 (Generalized Caputo fractional derivative). Let P be a given pa-
rameter set. The operator Bα, α ∈ (0,1), is defined by Bα = K1−α ◦D and is
P P P
referred as the operator B (B-op) of order α and p-set P.
Remark 2.8. The operator Bα is defined for absolute continuous functions f ∈
P
AC([a,b]), while the operator Aα acts on the bigger class of functions f such that
P
K1−αf ∈AC([a,b]).
P
Remark 2.9. The standard Riemann–Liouville and Caputo fractional derivatives
(see, e.g., [25, 39]) are easily obtained from the generalized operators Aα and Bα,
P P
respectively. Let k1−α(t−τ) = Γ(11−α)(t−τ)−α, α ∈ (0,1). If P = ha,t,b,1,0i,
then
t
1 d
Aαf(t)= (t−τ)−αf(τ)dτ =: Dαf(t)
P Γ(1−α)dt a t
Z
a
is the standard left Riemann–Liouville fractional derivative of order α while
t
1
Bαf(t)= (t−τ)−αf′(τ)dτ =:CDαf(t)
P Γ(1−α) a t
Z
a
isthestandardleftCaputofractionalderivativeoforderα;ifP =ha,t,b,0,1i,then
b
1 d
−Aαf(t)=− (τ −t)−αf(τ)dτ =: Dαf(t)
P Γ(1−α)dt t b
Z
t
6 T.ODZIJEWICZ,A.B.MALINOWSKA,ANDD.F.M.TORRES
is the standard right Riemann–Liouville fractional derivative of order α while
b
1
−Bαf(t)=− (τ −t)−αf′(τ)dτ =:CDαf(t)
P Γ(1−α) t b
Z
t
is the standard right Caputo fractional derivative of order α.
3. Main results
We begin by proving in Section 3.1 that for a certain class of kernels there ex-
ists a direct relationbetween the fractionalderivatives Aα and Bα (Theorem 3.1).
P P
Section 3.2 gives integration by parts formulas for the generalized fractional set-
ting (Theorems 3.4, 3.7 and 3.8). Section 3.3 is devoted to variational problems
with generalized fractional-order operators. New results include necessary opti-
mality conditions of Euler–Lagrange type for unconstrained (Theorem 3.11) and
constrainedproblems(Theorem3.22),andageneraltransversalitycondition(The-
orem 3.17). Interesting results are obtained as particular cases. Finally, in Sec-
tion 3.4 we provide a class of generalized fractional problems of the calculus of
variations for which one has a coherent embedding, compatible with the least ac-
tion principle (Theorem 3.26). This provides a generalanswer to an open question
posed in [13].
3.1. A relation between operators A and B. Next theorem gives a useful
relation between A-op and B-op. In the calculus of variations, equality (3.2) can
be used to provide a necessary optimality condition involving the same operators
as in the data of the optimization problem (cf. Remark 3.14 of Section 3.3).
Theorem 3.1. Let 0 < α < 1, P = ha,t,b,p,qi, and y ∈ AC([a,b]). If kernel
k1−α is integrable and there exist functions f and g such that
t τ
(3.1) k1−α(θ,τ)dθ+ k1−α(t,θ)dθ =g(t)+f(τ)
Za Za
for all t,τ ∈[a,b], then the following relation holds:
(3.2) AαPy(t)=py(a)k1−α(t,a)−qy(b)k1−α(b,t)+BPαy(t)
for all t∈[a,b].
τ
Proof. Let h1−α be defined by h1−α(t,τ) := a k1−α(t,θ)dθ−g(t). Then, by hy-
pothesis (3.1), ∂2h1−α = −∂1h1−α = k1−α. We obtain the intended conclusion
R
GENERALIZED FRACTIONAL CALCULUS WITH APPLICATIONS 7
from the definition of A-op and B-op, integrating by parts, and differentiating:
d d t b
AαPy(t)= dtKP1−αy(t)= dt p k1−α(t,τ)y(τ)dτ +q k1−α(τ,t)y(τ)dτ
( Za Zt )
d t d
= dt py(t)h1−α(t,τ)|τ=t−py(a)h1−α(t,a)−p h1−α(t,τ)dτy(τ)dτ
( Za
b d
−qy(b)h1−α(b,t)+ qy(t)h1−α(τ,t)|τ=t+q h1−α(τ,t)dτy(τ)dτ
Zt )
d t d
= py(t) h1−α(t,t+ǫ) −py(a)∂1h1−α(t,a)−p ∂1h1−α(t,τ) y(τ)dτ
dt dτ
(cid:12)ǫ=0 Za
(cid:12)(cid:12) d b d
−qy(b)∂2h1−α(b,t)+(cid:12)qy(t) h1−α(t+ǫ,t) +q ∂2h1−α(τ,t) y(τ)dτ
dt dτ
(cid:12)ǫ=0 Zt
=py(a)k1−α(t,a)−qy(b)k1−α(b,t)+BPαy(t).(cid:12)(cid:12)
(cid:12)
(cid:3)
Example 3.2. Let k1−α(t−τ)= Γ(11−α)(t−τ)−α. Simple calculations show that
(3.1) is satisfied. If P =ha,t,b,1,0i, then (3.2) reduces to the relation
y(a)
CDαy(t)= Dαy(t)− (t−a)−α
a t a t Γ(1−α)
between the left Riemann–Liouville fractional derivative Dα and the left Caputo
a t
fractional derivative CDα; if P =ha,t,b,0,1i, then we get the relation
a t
y(b)
CDαy(t)= Dαy(t)− (b−t)−α
t b t b Γ(1−α)
betweentherightRiemann–Liouvillefractionalderivative DαandtherightCaputo
t b
fractional derivative CDα.
t b
3.2. Fractional integration by parts. The proof of Theorem 3.1 uses one basic
but important technique of classicalintegralcalculus: integrationby parts. In this
section we obtain several formulas of integration by parts for the generalized frac-
tional calculus. Our results are particularly useful with respect to applications in
dynamicoptimization(cf. Section3.3),wherethederivationoftheEuler–Lagrange
equations uses, as a key step in the proof, integration by parts.
Inoursetting,integrationbypartschangesagivenp-setP intoitsdualP∗. The
term duality comes from the fact that P∗∗ =P.
Definition 3.3 (Dual p-set). Given a p-set P = ha,t,b,p,qi we denote by P∗ the
p-set P∗ =ha,t,b,q,pi. We say that P∗ is the dual of P.
Our first formula of fractional integration by parts involves the K-op.
Theorem 3.4 (Fractional integration by parts for the K-op). Let α ∈ (0,1),
P = ha,t,b,p,qi, k be a square-integrable function on ∆ = [a,b] × [a,b], and
α
f,g ∈ L ([a,b]). The generalized fractional integral satisfies the integration by
2
parts formula
b b
(3.3) g(t)KPαf(t)dt= f(t)KPα∗g(t)dt,
Z Z
a a
8 T.ODZIJEWICZ,A.B.MALINOWSKA,ANDD.F.M.TORRES
where P∗ is the dual of P.
Proof. Let α∈(0,1), P =ha,t,b,p,qi, and f,g ∈L ([a,b]). Define
2
|pk (t,τ)|·|g(t)|·|f(τ)| if τ ≤t
F(τ,t):= α
|qk (τ,t)|·|g(t)|·|f(τ)| if τ >t
α
(cid:26)
for all (τ,t)∈∆. Then, applying Holder’s inequality, we obtain
b b
F(τ,t)dt dτ
Za Za !
b b τ
= |f(τ)| |pk (t,τ)|·|g(t)|dt+ |qk (τ,t)|·|g(t)|dt dτ
α α
Za " Zτ Za !#
b b b
≤ |f(τ)| |pk (t,τ)|·|g(t)|dt+ |qk (τ,t)|·|g(t)|dt dτ
α α
Za " Za Za !#
1 1
b b 2 b 2
≤ |f(τ)| |pk (t,τ)|2dt |g(t)|2dt
α
Za   Za ! Za !
 1 1
 b 2 b 2
+ |qk (τ,t)|2dt |g(t)|2dt dτ.
α
Za ! Za ! 


By Fubini’s theorem, functions k (t):=k (t,τ) and kˆ (t):=k (τ,t) belong to
α,τ α α,τ α
L ([a,b]) for almost all τ ∈[a,b]. Therefore,
2
1 1
b b 2 b 2
|f(τ)| |pk (t,τ)|2dt |g(t)|2dt
α
Za   Za ! Za !
 1 1
 b 2 b 2
+ |qk (τ,t)|2dt |g(t)|2dt dτ
α
Za ! Za ! 
b 
=kgk |f(τ)| kpk k + qkˆ dτ 
2 α,τ 2 α,τ
Za hb (cid:16) 21 b(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)2(cid:17)i 2 21
≤kgk |f(τ)|2dτ kpk k + qkˆ dτ
2 α,τ 2 α,τ
Za ! Za (cid:12) (cid:13) (cid:13)2(cid:12) !
≤kgk ·kfk (kpk k +kqk k )(cid:12) <∞. (cid:13) (cid:13) (cid:12)
2 2 α 2 α 2(cid:12) (cid:13) (cid:13) (cid:12)
GENERALIZED FRACTIONAL CALCULUS WITH APPLICATIONS 9
Hence, we can use again Fubini’s theorem to change the order of integration:
b b t b b
g(t)Kαf(t)dt=p g(t)dt f(τ)k (t,τ)dτ +q g(t)dt f(τ)k (τ,t)dτ
P α α
Z Z Z Z Z
a a a a t
b b b τ
=p f(τ)dτ g(t)k (t,τ)dt+q f(τ)dτ g(t)k (τ,t)dt
α α
Z Z Z Z
a τ a a
b
= f(τ)KPα∗g(τ)dτ.
Z
a
(cid:3)
Next example shows that one cannot relax the hypotheses of Theorem 3.4.
Example 3.5. Let P = h0,t,1,1,−1i, f(t) = g(t) ≡ 1, and k (t,τ) = t2−τ2 .
α (t2+τ2)2
Direct calculations show that
1 1 t t2−τ2 1 τ2−t2
Kα1dt= dτ − dτ dt
P (t2+τ2)2 (t2+τ2)2
Z0 Z0 (cid:18)Z0 Zt (cid:19)
1 1 t2−τ2 1 1 π
= dτ dt= dt=
(t2+τ2)2 t2+1 4
Z0 (cid:18)Z0 (cid:19) Z0
and
1 1 τ τ2−t2 1 t2−τ2
KPα∗1dτ = − (t2+τ2)2dt+ (t2+τ2)2dt dτ
Z0 Z0 (cid:18) Z0 Zτ (cid:19)
1 1 τ2−t2 1 1 π
=− dt dτ =− dτ =− .
(t2+τ2)2 τ2+1 4
Z0 (cid:18)Z0 (cid:19) Z0
Therefore, the integration by parts formula (3.3) does not hold. Observe that in
this case 1 1|k (t,τ)|2dτdt=∞.
0 0 α
FortheRclaRssicalRiemann–Liouvillefractionalintegralsthefollowingresultholds.
Corollary 3.6. Let 1 <α<1. If f,g ∈L ([a,b]), then
2 2
b b
(3.4) g(t) Iαf(t)dt= f(t) Iαg(t)dt.
a t t b
Za Za
Proof. Let k (t,τ) = 1 (t −τ)α−1. For α ∈ 1,1 , k is a square-integrable
α Γ(α) 2 α
function on ∆ (see, e.g., [22, Theorem 4]). Therefore, (3.4) follows from (3.3). (cid:3)
(cid:0) (cid:1)
Theorem 3.7. Let 0 < α < 1 and P = ha,t,b,p,qi. If k (t,τ) = k (t − τ),
α α
k ∈ L ([a,b]), and f,g ∈ C([a,b]), then the integration by parts formula (3.3)
α 1
holds.
Proof. Let α∈(0,1), P =ha,t,b,p,qi, and f,g ∈C([a,b]). Define
|pk (t−τ)|·|g(t)|·|f(τ)| if τ ≤t
F(τ,t):= α
|qk (τ −t)|·|g(t)|·|f(τ)| if τ >t
α
(cid:26)
10 T.ODZIJEWICZ,A.B.MALINOWSKA,ANDD.F.M.TORRES
forall(τ,t)∈∆. Sincef andg arecontinuousfunctionson[a,b],theyarebounded
on[a,b],i.e., thereexistrealnumbersC ,C >0suchthat|g(t)|≤C and|f(t)|≤
1 2 1
C for all t∈[a,b]. Therefore,
2
b b
F(τ,t)dt dτ
Za Za !
b b τ
= |f(τ)| |pk (t−τ)|·|g(t)|dt+ |qk (τ −t)|·|g(t)|dt dτ
α α
Za " Zτ Za !#
b b b
≤ |f(τ)| |pk (t−τ)|·|g(t)|dt+ |qk (τ −t)|·|g(t)|dt dτ
α α
Za " Za Za !#
b b b
≤C C |pk (t−τ)|dt+ |qk (τ −t)|dt dτ
1 2 α α
Za Za Za !
=C C (|p|−|q|)kk k(b−a)<∞.
1 2 α
Hence,wecanuseFubini’stheoremtochangetheorderofintegrationintheiterated
integrals. (cid:3)
The next theorem follows from the classicalformula of integrationby parts and
fractional integration by parts for the K-op.
Theorem 3.8 (Fractional integration by parts for A-op and B-op). Let α∈(0,1)
and P =ha,t,b,p,qi. If f,g ∈AC([a,b]), then
b
b
(3.5) g(t)AαPf(t)dt= g(t)KP1−αf(t) ba− f(t)BPα∗g(t)dt,
Z Za
a (cid:12)
b (cid:12)
b
(3.6) g(t)BPαf(t)dt= f(t)KP1−∗αg(t) ba− f(t)AαP∗g(t)dt.
Z Za
a (cid:12)
(cid:12)
Proof. From Definition 2.6 one has Aαf(t)=DK1−αf(t). Therefore,
P P
b b
g(t)Aαf(t)dt= g(t)DK1−αf(t)dt
P P
Za Za
b
= g(t)K1−αf(t) b − Dg(t)K1−αf(t)dt,
P a P
Za
where the second equality follows by the sta(cid:12)ndard integration by parts formula.
(cid:12)
From (3.3) of Theorem 3.4 it follows the desired equality (3.5):
b b
g(t)Aαf(t)dt= g(t)K1−αf(t) b − f(t)K1−αDg(t)dt.
P P a P∗
Za Za
We now prove (3.6). From Definition 2.7 we(cid:12)(cid:12)know that Bαf(t) = K1−αDf(t). It
P P
follows that
b b
g(t)Bαf(t)dt= g(t)K1−αDf(t)dt.
P P
Za Za
By Theorem 3.4
b b
g(t)Bαf(t)dt= Df(t)K1−αg(t)dt.
P P∗
Za Za