Memoirs on Differential Equations and Mathematical Physics
Volume 50, 2010, 129–138
T. S. Aleroev and H. T. Aleroeva
SOME APPLICATIONS OF THE PERTURBATION
THEORY TO FRACTIONAL CALCULUS
Abstract. Spectral analysis of a class of integral operators associated
with fractional order differential equations arising in mechanics is carried
out. The connection between the eigenvalues of these operators and zeros of
Mittag–Leffler type functions is established. Sufficient conditions for complete non-self-adjointness and completeness of the systems of eigenfunctions
are given.
2010 Mathematics Subject Classification. 34K.
Key words and phrases. Caputo’s derivatives, Riemann–Liouville
derivatives, fractional differential equation, two-point boundary value problem.
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Some Applications of the Perturbation Theory to Fractional Calculus
131
The spectral analysis of operators of the kind
Z1
Zx
1
1
1
−1
[α,β]
Aγ u(x) = cα (x − t) α u(t) dt + cβ,γ x β −1 (1 − t) γ −1 u(t) dt
0
0
has been carried out in [1]. Here α, β, γ, cα , cβ,γ are real numbers, while α,
β, γ are positive ones (similar operators were considered by G. M. Gubreev
in [2]). These operators arise in studying boundary value problems for
differential equations of fractional order (see [3] and references therein).
In particular, it is shown in [1] that the operator
u(x) =
Aρ u(x) = A[ρ,ρ]
ρ
1
=
Γ(ρ−1 )
Zx
(x − t)
0
1
ρ −1
1
u(x) dt −
Γ(ρ−1 )
Z1
1
1
x ρ −1 (1 − t) ρ −1 u(t) dt
0
is almost non-self-conjugate ([4]) if ρ > 1), and if 0 < ρ < 1, the kernel of
the operator Aρ coincides with the set of roots of the entire Mittag–Leffler
type function
∞
X
λk
Eρ (λ; ρ−1 ) =
.
Γ(ρ−1 + kρ−1 )
k=0
Thus it follows from the above that all eigenvalues of the operator Aρ are
complex if ρ > 1, while if 0 < ρ < 1 the operator Aρ has real eigenvalues
(in fact, if 21 < ρ < 1, then the set of real eigenvalues is finite). All zeros of
the function Eρ (λ; ρ−1 ) are complex if ρ > 1, and if 0 < ρ < 1 the function
Eρ (λ; ρ−1 ) has real zeros. This confirms the hypothesis about the existence
of real zeros of the function Eρ (λ; ρ−1 ) if 12 < ρ < 1, as is stated in the
monograph ([5], p. 248).
The given paper is also devoted to investigation of boundary value problems for differential equations of fractional order and associated integral
[α;β]
operators of the kind Aγ .
To state the corresponding problems, we use some concepts from the
fractional calculus.
Let f (x) ∈ L1 (0, 1). Then the function
Zx
d−α
1
f (x) ≡
(x − t)α−1 f (t) dt ∈ L1 (0, 1)
dx−α
Γ(α)
0
is called the fractional integral of order α > 0 with the origin at the point
x = 0 [6] while the function
1
d−α
f (x) ≡
d(1 − x)−α
Γ(α)
Z1
(t − x)α−1 f (t) dt ∈ L1 (0, 1)
x
is called the fractional integral of order α > 0 with the end at the point
x = 1 [6]. Here Γ(α) is Euler’s gamma-function. In case α = 0, it is natural
132
T. S. Aleroev and H. T. Aleroeva
to put both fractional integrals equal to the function f (x). As is known [6],
the function g(x) ∈ L1 (0, 1) is called a fractional derivative of the function
f (x) ∈ L1 (0, 1) of order α > 0 with the origin at the point x = 0, if
f (x) =
d−α
g(x).
dx−α
g(x) =
dα
f (x).
dxα
Denote
α
d
In what follows, under the symbol dx
α , we will mean the fractional integral for α < 0 and the fractional derivative for α > 0. The fractional
dα
derivative d(1−x)
α of the function f (x) ∈ L1 (0, 1) of order α > 0, with the
end at the point x = 1 is defined similarly.
n
Let {γk }0 be an arbitrary set of real numbers satisfying the condition
0 < γj ≤ 1 (j = 0, 1, . . . , n). Denote
σk =
k
X
µk = σk + 1 =
j=0
k
X
γj (k = 0, 1, . . . , n)
j=0
and assume that
n
1 X
=
γj − 1 = σn = µn − 1 > 0.
ρ j=0
Following [6], we introduce the differential operators
D(σ0 ) f (x) ≡
d−(1−γ0 )
f (x)
dx−(1−γ0 )
of, generally speaking, fractional order:
d−(1−γ1 ) dγ0
f (x),
dx−(1−γ1 ) dxγ0
d−(1−γn ) dγn−1
dγ0
D(σn ) f (x) ≡ −(1−γ ) γn−1 · · · γ0 .
n dx
dx
dx
D(σ1 ) f (x) ≡
Note that if γ0 = γ1 = · · · = γn = 1, then it is obvious that
D(σk ) f (x) = f (k) (x) (k = 0, 1, 2, . . . , n).
Now,
a) a two-point Dirichlet boundary value problem
u(0) = 0, u(1) = 0,
(1)
for the fractional oscillatory equation
α
u00 + λD0x
u = 0,
(2)
Some Applications of the Perturbation Theory to Fractional Calculus
133
α
(D0x
is the operator of fractional differentiation of order 0 < α < 1 [2])
investigated by many authors is equivalent to the equation [3], [4]
λ
u(x) +
Γ(ρ−1 )
· Zx
Z1
(x − t)
1
ρ −1
1
x(1 − t) ρ −1 u(t) dt = 0,
u(t) dt −
0
(∗)
0
1
= 2 − α,
ρ
α=2−
1
.
ρ
b) the operator, inverse to the operator generated by the differential
expression
Zx
1
dn−1
u0 (t)
lu =
dt,
n−1
Γ(1 − γ) dx
(x − t)γ
0
0 < γ < 1, and the natural boundary conditions
¯
¯
¯
¯
u(0) = 0, u(1) = 0, D(σ1 ) u¯
= 0, . . . , D(σn−2 ) u¯
t=0
t=0
=0
can be represented in the form
u(x) =
A[ρ,ρ]
ρ
1
Γ(ρ−1 )
Zx
1
(x−t) ρ −1 u(t) dt−
0
1
Γ(ρ−1 )
Z1
1
1
x ρ −1 (1 − t) ρ −1 u(t) dt
0
´
= n − 1 + γ, n = 1, 2, 3, . . . .
³1
ρ
A remarkable work of M. K. Gubreev in which similar operators are stud[ρ,ρ]
(∗) is investigated by
ied is worth mentioning. In [4], the operator Aρ
methods of the perturbation theory. Assume that the operator
Zx
Au =
Z1
(x − t)u(t) dt −
0
x(1 − t)u(t) dt
0
is non-perturbed, and the operator
1
e =
Au
Γ(ρ−1 )
· Zx
Z1
(x − t)
0
1
ρ −1
u(t) dt −
¸
1
x(1 − t) ρ −1 u(t) dt
0
is perturbed. Towards this end, the use was made of the concept of opening
between the closed operators. In [4], no explicit expression for the perturbation of the operator A was written. In the present work we obtain the
corresponding expression and it seems to the authors that the present work
will become a factor allowing one to insert the theory of differential equations of fractional order in a general scheme of the theory of perturbations
[6], [7].
134
T. S. Aleroev and H. T. Aleroeva
Thus using the methods of the theory of perturbations in L1 (0, 1), we
study the operator
Z1
Zx
1+ε
Aε u =
(x − t)
x(1 − t)1+ε u(t) dt.
u(t) dt −
0
0
Theorem 1. The representation
Aε u = A + εA1 + ε2 A2 + · · · + εn An + · · ·
(ε > 0)
holds, where
Zx
Au =
Z1
(x − t)u(t) dt −
0
x(1 − t)u(t) dt,
0
1
An u =
n!
· Zx
Z1
n
(x − t)[ln(x − t)] dt −
0
¸
x(1 − t)[ln(x − t)]n dt
0
are operators with power-logarithmic kernels.
Proof. We rewrite the operator Aε as follows:
Aε u = Mε u + Nε u,
where
Zx
Kε (x, t)u(t) dt,
Mε u =
0
(
(x − t)1+ε , t < x
Kε (x, t) =
0,
t≥x
and
Z1
e ε (x; t)u(t) dt,
K
Nε =
0
(
e ε (x, t) =
K
x(1 − t)1+ε , t 6= 1
.
0,
t=1
Find the operator Aε as follows:
(A − Aε )u = (M − Mε )u − (N − Nε )u.
First, we find (M − Mε )u. Obviously,
Z1
(M − Mε )u =
0
£
¤
K(x, t) − Kε (x, t) u(t) dt.
Some Applications of the Perturbation Theory to Fractional Calculus
Since
£
¤
K(x, t) − Kε (x, t) =
we have
135
(
(x − t) − (x − t)1+ε , t < x
,
0,
t≥x
(
(x − t)[1 − (x − t)ε ], t < x
K(x, t) − Kε (x, t) =
.
0,
t≥x
As far as
ax = 1 +
x
(x)2
(x)n
+
+ ··· +
+ · · · (a > 0, x < ∞),
1!
2!
n!
we find
K(x, t) − Kε (x, t) =

h ln(x−t)
i
(ln(x−t))2
(ln(x−t))n

+ε2
+· · · +εn
+ · · · , t<x
(x−t) ε
=
.
1!
2!
n!
0,
t≥x
From the last expression we find that
Z1
Z1
n
(M − Mε )u = ε
K1 (x, t)u(t) dt + · · · + ε
0
where
Kn (x, t)u(t) dt + · · · ,
0

 (x − t)[ln(x − t)]n
, t<x
Kn (x, t) =
.
n!
0,
t≥x
It follows from (4) that
K1 (x, t)u(t) dt − · · · − ε
K(x, t)u(t) dt − ε
Mε =
Z1
Z1
Z1
Kn (x, t)u(t) dt.
0
0
0
n
In the same way we can obtain the representation
Z1
Z1
e
K(x,
t)u(t) dt−ε
Nε u =
0
where
Z1
e 1 (x, t)u(t) dt− · · · −ε
K
e n (x, t)u(t) dt+ · · · ,
K
n
0
0

 x(1 − t)[ln(x − t)]n
, t<1
e n (x, t) =
K
n!
0,
t=1
which was to be proved.
Theorem 2. All eigenvalues λn (ε) of the operator A(ε) are real.
¤
136
T. S. Aleroev and H. T. Aleroeva
Proof. We have
λn (ε) = πn2 + ελ1 + ε2 λ2 + · · · ,
2
ϕn (ε) = sin nx + εϕ1 + ε ϕ2 + · · · ,
(3)
(4)
where
λn =
n
X
(Ak ϕn−k , sin nx),
k=1
n
X
ϕn = R
(λk − Ak )ϕn−k .
(5)
(6)
k=1
Here, R is the reduced resolvent of the operator A corresponding to the
eigenvalue πn2 . This resolvent is an integral operator with the kernel
h y
i
1−x
1
S(x, y) = − cos ny sin nx+
sin ny cos nx+ 2 sin ny sin nx , y ≤ x
n
n
2n
(if y > x, in the middle part of this expression y and x should be interchanged).
Clearly, R is a one-to-one transformation of H0 into itself which annuls
sin πnx (H0 is the orthogonal complement of the function sin πnx).
It follows from [8] that
λ1 = (A1 sin nx, sin x),
because the kernel of the operator A1 , and sin λ1 = 0 is real-valued.
Next, from (6) it follows
ϕ1 = R(nk 2 − A1 ) sin nx,
and the kernels of the operators R and A1 are real-valued. Hence Imϕ1 = 0.
Consequently, we can successively prove that all λi are real, and since ε is
real, therefore λu(ε) is real too.
¤
Theorem 3. For the eigenvalues λn (x) and eigenfunctions ϕn (ε) of the
operator A(ε) the estimates
¯
¯
¯λn (ε) − πn2 ¯ < π(2n − 1) ,
2
¯
¯ 1
¯ϕn (ε) − sin nx¯ <
2
hold.
Proof. From (5) and (6), under the assumption that
©
ª
1
1
kAn uk ≤ pn−1 ukuk + bkA0 uk , m = kA0 k,
= kRk, |ε| < ,
d
c
where
n 8(a + mb)
a + mb o
, 8p + 4
,
c = max
d
d
Some Applications of the Perturbation Theory to Fractional Calculus
137
we obtain the following simple formulas [8]:
¯
¯
¯λ(ε) − λ0 − ελ1 − εn λn ¯ ≤ d (|ε|c)n+1 ,
(7)
2
¯
¯
¯ϕ(ε) − ϕ0 − εϕ1 − εn ϕn ¯ ≤ 1 (|ε|c)n+1 .
(8)
2
Let us calculate the values of the parameters a, b, c, d, m. First, we find m.
m = kAk = sup A, where sup A is the spectral radius of the operator A.
Since
sup A = π −1 ,
we have m = π −1 . Further, we find d.
00
d = dist(πn2 ; Σ )
00
(d is an isolating distance). Here Σ is the spectrum of the operator A−1
with the unique excluded point πn2 . Clearly, d = π(2n − 1). To find the
remaining parameters a, b, c, we have to estimate the norm of the operator An ,
Z1 Z1
|Kn (x, t)| |ϕ(t)| dt dx.
kAn ϕkL(0,1) ≤
0
0
Here, Kn (x, t) is the kernel of the operator An ,
Z1 Z1
0
0
Z1 Zx ¯
¯
¯ (x − t)[ln(x − t)]n ¯
|Kn (x, t)| |ϕ(t)| dt dx =
¯ |ϕ(t)| dt dx =
¯
n!
0
Z1−t
Z1
|ϕ(t)|
=
0
0
0
z|(ln z)n |
1
dz dt ≤
n!
n!
We now calculate the integral
R
Z
Z1
z|(ln z)n | dzkϕkL1 (0,1) .
0
n
z ln z dz. It is equal to
z ln z n dz =
=
z 2 (ln z)n nx2 (ln x)n−1
(−1)n−1 n(n−1)(n−2) · · · 2 h x2 1 i
−
+·
·
·
+
−
,
2
22
2n−1
2 22
whence kAn k ≤
b = 0. Since
1
2n+1 .
It is now obvious that we can take α = 14 , p = 12 , and
n a + mb
a + mb o
c = max 8
+ 8p + 4
,
d
d
we have
c = max
n2
d
;4 +
Thus
|λ(ε) − λ| ≤
1o
1
= 4 + = 5.
d
d
1 π(2n − 1)
.
2
2
138
T. S. Aleroev and H. T. Aleroeva
In a similar way we find
|ϕε − ϕ0 | ≤
which proves Theorem 3.
1
,
2
¤
References
1. T. S. Aleroev, On a class of operators associated with differential equations of
fractional order. (Russian) Sibirsk. Mat. Zh. 46 (2005), No. 6, 1201–1207; English
transl.: Siberian Math. J. 46 (2005), No. 6, 963–968.
2. M. M. Dzhrbashyan, Ein Randwertproblem fur einen Differentialoperator gebrochener Ordnung von Sturm-Liouvilleschen Typ. (Russian) Izv. Akad. Nauk Arm. SSR,
Mat. 5 (1970), 71–96.
3. T. S. Aleroev, Dissertation, Phys.-Math. Dr., 2000, Moscow, Moscow State University.
4. T. S. Aleroev, On a boundary value problem for a fractional-order differential operator. (Russian) Differ. Uravn. 34 (1998), No. 1, 123, 144; English transl.: Differential
Equations 34 (1998), No. 1, 126.
5. G. M. Gubreev, Regular Mittag-Leffler kernels and spectral decomposition of a class
of nonselfadjoint operators. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005),
No. 1, 17–60; English transl.: Izv. Math. 69 (2005), No. 1, 15–57.
6. K. O. Friedrichs, Perturbation of spectra in Hilbert space. Marc Kac, editor. Lectures in Applied Mathematics (Proceedings of the S ummer Seminar, Boulder, Colorado, (1960), Vol. III American Mathematical Society, Providence, R.I., 1965.
7. T. Kato, Perturbation theory for linear operators. (Russian) Translated from the
English by G. A. Voropaeva, A. M. Stepin and I. A. Sismarev. Edited by V. P.
Maslov. Izdat. Mir, Moscow, 1972.
8. B. V. Logunov, About estimations for eigennumbers and eigenvectors of the disturb
operator. (Russian) Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat., No. 5, 1963, 131–139.
(Received 9.02.2009)
Authors’ addresses:
T. S. Aleroev
Moscow Institute of a Municipal Services and Construction
Faculty of Higher Mathematics
Russia
The Academy of National Economy
under the Government of the Russian Federation
Moscow 119571, Russia
E-mail: aleroev@mail.ru
Moscow Institute of Municipal Economy and Construction
H. T. Aleroeva
Moscow Technical University of Communications and Informatics
Russia
E-mail: BinSabanur@gmail.com