Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 296353, 13 pages
doi:10.1155/2011/296353
Research Article
Employing of Some Basic Theory for Class of
Fractional Differential Equations
Azizollah Babakhani1 and Dumitru Baleanu2, 3
1
Faculty of Basic Science, Babol University of Technology, Babol 47148-71167, Iran
Department of Mathematics and Computer Science, Cankaya University, 06530 Ankara, Turkey
3
Institute of Space Sciences, P.O. Box MG-23, Magurele, 077125 Bucharest, Romania
2
Correspondence should be addressed to Azizollah Babakhani, babakhani@nit.ac.ir and
Dumitru Baleanu, dumitru@cankaya.edu.tr
Received 19 September 2010; Accepted 15 November 2010
Academic Editor: J. J. Trujillo
Copyright q 2011 A. Babakhani and D. Baleanu. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Basic theory on a class of initial value problem of some fractional differential equation involving
Riemann-Liouville differential operators is discussed by employing the classical approach from
the work of Lakshmikantham and A. S. Vatsala 2008. The theory of inequalities, local existence,
extremal solutions, comparison result and global existence of solutions are considered. Our work
employed recent literature from the work of Lakshmikantham and A. S. Vatsala, 2008.
1. Introduction
Differential equations of fractional order have recently proven to be valuable tools in the
modeling of many phenomena in various fields of science and engineering. Indeed, we
can find numerous applications in viscoelasticity, electrochemistry, control, porous media,
electromagnetism, and so forth 1–5. There has been a significant development in the study
of fractional differential equations and inclusions in recent years; see the monographs of
Kilbas et al. 6, Lakshmikantham et al. 7, Podlubny 4, and the survey by Agarwal et al.
8. For some recent contributions on fractional differential equations, see 9–20 and the
references therein. Very recently in 10, 11, 21, 22, the author and other researchers studied
the existence and uniqueness of solutions of some classes of fractional differential equations
with delay. For more details on the geometric and physical interpretation for fractional
derivatives of both the Caputo types see 5, 23. Băleanu and Mustafa 16 allowed for
immediate applications of a general comparison-type result from the recent literature 24.
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Advances in Difference Equations
Lakshmikantham and Vatsal have discussed basic theory of fractional differential equations
for initial value problem fractional differential equations type 24
Dα xt ft, xt,
x0 x0 , t ∈ 0, T .
1.1
This paper deals with the basic theory of initial value problem IVP for a generalized class
of fractional order differential equation of the form
LDxt − x0 ft, xt,
x0 x0 , t ∈ 0, T ,
1.2
β
where LD D0α − tn D0 , f ∈ C0, T , R and 0 < β ≤ α < 1. Recall that C0, T , R is the
Banach space of continuous functions from the interval 0, T into R endowed with uniform
norm.
We begin in this section with the recall of some definitions and results for fractional
calculus which are used throughout this paper 4, 19, 20.
The left-sided Riemann-Liouville fractional integral of a function x of order α > 0 is
defined as
I0α xt 1
Γα
t
t − sα−1 xsds,
t > 0,
1.3
0
and the left sided Riemann-Liouville fractional derivative operator of order 0 < α < 1 is
defined by
D0α xt d 1−α
I0 xt ,
dt
1.4
∞
where Γα 0 e−t tα−1 dt. We denote D0α xt by D0α xt and I0α xt by I0α xt. Also Dα xt
and I α xt refer to D0α xt and I0α xt, respectively.
β
Assume that 0 < β < 1, β ≤ α, if the fractional derivative D0 xt is integrable, then 4,
page 72
β
1−β
α−β
I0α D0 xt I0 xt − I0 xt
tα−1
.
t0 Γα
1.5
β
If xt is continuous on 0, T then D0 xt is integrable, I 1−β xt|t0 0 25, and 1.5 reduces
to
β
α−β
I0α D0 xt I0 xt,
0 < β < 1, β ≤ α.
Proposition 1.1. Let x ∈ C0, T , R and 0 < β < 1, β ≤ α. Then nk0
k n αk
n−k
i I α tn xt nk0 −α
xt nk0 −α
/n − k!I αk xt,
k D t I
k n!t
n−k
/n − k!I α−βk xt,
ii I α tn Dβ xt nk0 −α
k n!t
1.6
Advances in Difference Equations
3
where n is a nonnegative integer, and
−α
k
−1k
Γα k
.
k! Γα
1.7
Proof. i can be found in 19, page 53 and ii is an immediate consequence of 1.6 and
i.
2. Strict and Nonstrict Inequalities
Since f is assumed to be continuous, the initial value problem 1.2 is equivalent to the
following Volterra fractional integral 10, 25:
n
−α n!tn−k α−βk
xt x0 I
{xt − x0} I α ft, xt
−
k!
n
k
k0
x0 n
−α
k
k0
1
Γα
t
n!tn−k
n − k! Γ α − β k
t − sα−1 fs, xsds,
t
t − sα−βk−1 xsds
2.1
0
0 ≤ t ≤ T.
0
Note. Let us consider the notation of Ixt for second term in 2.1 which is used in
throughout of text and so that
n
−α n!tn−k α−βk
I
Ixt {xt − x0}
k n − k!
k0
n
−α
k0
k
n!tn−k
n − k! Γ α − β k
t
2.2
t − sα−βk−1 xsds.
0
We now first discuss a fundamental result relative to the fractional inequalities.
Theorem 2.1. Let v, w ∈ C0, T , R, f ∈ C0, T × R, R, and
t
i vt ≤ v0 Ivt 1/Γα 0 t − sα−1 fs, vsds,
t
ii wt ≥ w0 Iwt 1/Γα 0 t − sα−1 fs, wsds,
one of the foregoing inequalities being strict. Moreover, if ft, x is nondecreasing in x for each t and
v0 < w0, then one has vt < wt, t ∈ 0, T .
Proof. Suppose that for each t ∈ 0, T , the conclusion vt < wt is not true. Then, because of
the continuity of the functions involved and v0 < w0 it follows that there exists a t1 such
that 0 < t1 ≤ T and
vt1 wt1 ,
vt < wt,
0 < t ≤ t1 .
2.3
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Advances in Difference Equations
Let us suppose that the inequality ii is strict. Then using the nondecreasing nature of f and
2.3 we get
t1
1
t − sα−1 fs, wsds
Γα 0
t1
1
≥ v0 Ivt1 t − sα−1 fs, vsds ≥ vt1 ,
Γα 0
wt1 > w0 Iwt1 2.4
which is a contradiction in view of 2.3. Hence the conclusion of this theorem holds and the
proof is complete.
The next result is for nonstrict inequalities, which require a one-sided Lipschitz type
condition.
Theorem 2.2. Assume that the conditions of Theorem 2.1 hold with nonstrict inequalities (i) and (ii).
Suppose further that
ft, x − f t, y ≤
L x−y ,
2
1t
2.5
whenever x ≥ y and L > 0. Then, v0 ≤ w0, and L < Γα 1 implies
vt ≤ wt,
0 ≤ t ≤ T.
2.6
Proof. Set w t wt 1 tα , for small > 0 so that we have,
w 0 w0 > w0,
w t > wt, 0 ≤ t ≤ T.
2.7
Now,
n
−α n!tn−k α−βk
I
w t ≥ w0 {w t − ω0}
k n − k!
k0
1
Γα
t
2.8
t − sα−1 fs, wsds 1 tα .
0
In view of 2.7, using one-sided Lipschitz condition 2.5 we see that
t
L1 sα ds tα
t − sα−1 fs, w s − 1 sα 0
2.9
t
1
L t
α−1
α−1
w 0 Iw t t − s fs, w sds −
t − s ds tα .
Γα 0
Γα 0
w t ≥ w 0 Iw t 1
Γα
Advances in Difference Equations
Now, since
t
0
t − sα−1 ds tα
1
0
5
1 − τα−1 dτ Γα/Γα 1tα , we arrive at
w t > w 0 Iw t 1
Γα
t
t − sα−1 fs, w sds,
2.10
0
in view of the condition L < Γα 1. We now apply Theorem 2.1 to the inequalities i, 2.9,
and 2.10 to get vt < w t, t ∈ 0, T . Since > 0 is arbitrary, we conclude that 2.6 is true,
and we are done.
3. Local Existence and Extremal Conditions
In this section, we will consider the local existence and the existence of extremal solutions for
the IVP 1.2. We now first discuss Peano’s type existence result.
Theorem 3.1. Assume that f ∈ Ω0 , R, where Ω0 {t, x : 0 ≤ t ≤ r and |x − x0 | ≤ b}, and let
|ft, x| ≤ M on Ω. Then the IVP 1.2 has at least one solution xt on 0, λ, where
1/α−βk
b Γα 1 1/α b Γα − β k 1
λ min r,
,
, k 0, n ,
n 2 M
n 2 ρ b
3.1
so that ρ and x0 will be observing in the proof of this theorem.
Proof. Let x0 t be a continuous function on −δ, 0, δ > 0 such that x0 0 x0 , |x0 t − x0 | ≤ b.
For ≤ δ, we define a function x t x0 t on −δ, 0 and
x t x0 Ix t − I α ft, x t − 3.2
on 0, λ1 , where λ1 min{λ, }. We observe that
t
n −α n−k
1
n!t
α−βk
I
−
x
−
−
x
≤
t
t
t − sα−1 fs, x s − ds,
|x
|x
0|
0|
Γα
−
k!
n
k
0
k0
t
n
1
≤ ρ I α−βk |x t − − x0 | t − sα−1 fs, x s − ds,
Γα 0
k0
3.3
where
−α n!μn−k −α n!
1
ρ max ,
: k 0, n .
k n − k! k n − k!
3.4
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Advances in Difference Equations
Note that, 0 < s ≤ t ≤ λ1 ≤ , as λ1 min{λ, }. Then −δ ≤ − < s − ≤ 0, and hence we can
consider x s − t x0 s − in the last above inequality. Thus |x s − − x0 | ≤ b.
n
ρb
|x t − x0 | ≤
k0 Γ α − β k
t
t − s
α−βk−1
0
M
Γα
t
t − sα−1 ds
0
n
ρb
M
tα
tα−βk Γα
1
k0 Γ α − β k 1
3.5
n
ρb
M
λα ≤ b,
λα−βk Γα
1
Γ
α
−
β
k
1
k0
because of the choice of λ1 . If λ1 < λ, we can employ 3.2 to extend as a continuous function
on −δ, λ2 , λ2 min{λ, 2}, such that |x − x0 | ≤ b holds. Continuing this process, we can
define x t over −δ, λ so that |x t − x0 | ≤ b is satisfied on −δ, λ. Furthermore, letting
0 ≤ t1 ≤ t2 ≤ λ we see that
|x t1 − x t2 | |Ix t1 − Ix t2 |
t
t2
1
fs, x s − fs, x s − 1
ds −
ds
Γα 0
t1 − s1−α
t2 − s1−α
0
t1 1
|Ix t1 − Ix t2 | t1 − s1−α − t2 − s1−α fs, x s − ds
Γα 0
1 t2
1−α
t2 − s fs, x s − ds
Γα t1
≤ |Ix t1 − Ix t2 |
M t1 M t2
1−α
1−α
1−α
ds −
t2 − s ds.
t1 − s − t2 − s
Γα 0
Γα t1
3.6
Notice that
−α 2k
> 0 and
−α
2k1
< 0, when 0 < α < 1. Let,
n
n!λn−2k −α
2k
>0
Λ1 : max
, k 0, 1, 2, . . . ,
2
n − 2k! Γ α − β 2k
Λ2 : min
n!λ
n−2k−1
−α
2k1
n
, k 0, 1, . . . , 1 2
n − 2k − 1! Γ α − β 2k 1
3.7
< 0,
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firstly we have
Ix t1 − Ix t2 n
−α
k
k0
−
n!tn−k
1
n − k! Γ α − β k
n
−α
k
k0
Λ1
n/2
t
Λ2
t1 − sβ−α−2k1
0
k0
t
n/21
1
≤ bΛ1
n/2
t
1
k0
0
bΛ1
t1 − s
n/2
t2
k0
bΛ2
bΛ2
≤ bΛ1
k0
t2 − sβ−α−2k1
t2
0
− t2 − s
ds
x s − − x0
t2 − sβ−α−2k
α−β2k−1
ds
3.8
ds
t1 − s
α−β2k
− t2 − s
α−β2k
ds
0
n/21
t2
n/2
ds −
x s − − x0
t2 − sα−β2k−1 ds
t
1
k0
t2
t2 − sα−βk−1 {x s − − x0 }ds
0
t1
n/21
k0
α−β2k−1
t2
0
x s − − x0
t1 − sβ−α−2k
0
k0
ds −
t1 − sα−βk−1 {x s − − x0 }ds
0
n!t2 n−k
n − k! Γ α − β k
x s − − x0
1
t1
t2 − sα−β2k ds
t1
n/21
t2 − t1 α−β2k1
t2 − t1 α−β2k
b|Λ2 |
.
α − β 2k
α − β 2k 1
k0
We, therefore, set Λ max{b|Λ1 |, b |Λ2 |}, and get
n/2
t2 − t1 α−β2k n/21
t2 − t1 α−β2k1
.
Ix t1 − Ix t2 ≤ 2Λ
α − β 2k
α − β 2k 1
k0
k0
3.9
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Advances in Difference Equations
Finally, inequality 3.6 from inequality 3.9 becomes
|x t1 − x t2 | ≤ M 2t2 − t1 α tα1 − tα2 Γα 1
n/2
t2 − t1 α−β2k n/21
t2 − t1 α−β2k1 2Λ
α − β 2k
α − β 2k 1 k0
k0
n/2
t2 − t1 α−β2k n/21
t2 − t1 α−β2k1
≤ 2Λ
α − β 2k
α − β 2k 1
k0
k0
3.10
2M
t2 − t1 α < ,
Γα 1
provided that
|t2 − t1 | < δ0
min
Γα 1
2n 1M
n
1/α α − β 2k 1/α−β2k α − β 2k 1 1/α−β2k1
,
,
.
2n 1Λ
2n 1Λ
k0
3.11
It then follows from 3.4 and 3.6 that the family {x t} forms an equicontinuous and
uniformly bounded functions. Ascoli-Arzela theorem implies that the existence of a sequence
{n } such that 1 > 2 > · · · > m → 0 as m → ∞, and xt limm → ∞ xm t exists uniformly
on −δ, λ. Since f is uniformly continuous, we obtain that ft, xn t − m tends uniformly to
ft, xt as m → ∞, and, hence, term by term integration of 3.2 with m , λ1 λ yields
xt x0 Ixt I α ft, xt.
3.12
This proves that xt is a solution of the IVP 1.2, and the proof is complete.
Theorem 3.2. Under the hypothesis of Theorem 3.1, there exists extremal solution for the IVP 1.2
on the interval 0 ≤ t ≤ λ0 , provided ft, x is nondecreasing in x for each t, where λ0 will be observed
in the proof of this theorem.
Proof. We will prove the existence of maximal solution only, since the case of minimal solution
is very similar. Let 0 < ≤ b/2 and consider the fractional differential equation with an initial
condition
LDxt − x0 ft, xt ,
x0 x0 .
3.13
We observe that f t, x ft, x is defined and continuous on
Ω b
,
t, x : 0 ≤ t ≤ λ, |x − x0 | ≤
2
3.14
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Ω ⊂ Ω0 and |f t, x ≤ M b/2 on Ω . We then deduce from Theorem 3.1 that IVP 3.6 has
a solution xt, on the interval o ≤ t ≤ λ0 . Now for 0 < 2 < 1 ≤ , we have
x0, 2 x0 2 < x0 1 x0, 1 ,
3.15
xt, 2 ≤ x0, 2 Ixt, 2 I α f2 t, xt, 2 xt, 1 ≥ x0, 1 Ixt, 1 I α f1 t, xt, 1 ,
where Ix was introduced by 2.2. In view of Theorem 2.1 to get xt, 2 < xt, 1 , t ∈ 0, λ0 .
Consider the family of functions {xt, } on 0 ≤ t ≤ λ0 . We have
|xt, − x0, | Ixt, I α f t, xt, Ixt, 2 I α ft, xt, n
ρ1 x t
≤
s − tα−βk−1
Γ
α
−
β
k
0
k0
≤
1
Γα
t
t − sα−1 ft, xt, ds
3.16
0
n
ρ1 x
M α
t
tα−βk Γα
1
Γ
α
−
β
k
1
k0
n
ρ1 x
2M b α
λ0 ≤ b,
λ0 α−βk 2Γα
1
Γ
α
−
β
k
1
k0
where ρ was denoted by 3.4, x max{|xt, | : t ∈ 0, λ0 , ∈ 0, b/2} and
λ0 min r,
2bΓα 1
n 22M b
1/α bΓα − β k 1 1/α−βk
,
, k 0, n .
ρ x 1n 2
3.17
Showing that the family {xt, } is uniformly bounded. Also, if 0 ≤ t1 ≤ t2 ≤ λ0 then there
exists a constant Λ∗ such that
∗
|xt1 , − xt2 , | ≤ 2Λ
n/2
t2 − t1 α−β2k
k0
α − β 2k
n/21
k0
t2 − t1 α−β2k1
α − β 2k 1
2M b
t2 − t1 α .
2Γα 1
3.18
Following the computation similar to 3.10 with suitable changes. This proves that the family
{xt, } is equicontinuous. Hence there exists a sequence {m } with m → 0 as m → ∞ and
the uniform limit
μt lim xt, m m→∞
3.19
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Advances in Difference Equations
exists on 0, λ0 . Clearly μ0 x0 . The uniform continuity of f gives argument as before as
in Theorem 3.1, that μt is a solution of IVP 1.2.
Next, we show that μt is required maximal solution of 1.2, on 0 ≤ t ≤ λ0 . Let xt
be a any solution of 1.2 on 0 ≤ t ≤ λ0 . Then we have
x0 < x0 x0, t
1
xt < x0 Ixt t − sα−1 f s, xsds,
Γα 0
t
1
xt, ≥ x0, Ixt, t − sα−1 f s, xs, ,
Γα 0
3.20
where x0, x0 , f t, xt ft, xt , and f t, xt, ft, xt, .
Using Theorem 2.1, we get xt < xt, on 0, λ0 as → 0. Therefore, the proof is
complete.
4. Global Existence
We need the following comparison result before we proceed further.
Theorem 4.1. Assume that h : 0, T → R , g : 0, T × R → R are continuous, and gt, u is
nondecreasing with respect to the second argument such that
ht ≤ h0 Iht 1
Γα
t
t − sα−1 gs, hsds, t ∈ 0, T ,
4.1
0
where I is defined by 3.5. Let mt be the maximal solution of
LDut − u0 gt, u,
u0 u0 ≥ 0,
4.2
existing on 0, T such that h0 ≤ u0. Then we have
ht ≤ mt,
t ∈ 0, T .
4.3
Proof. In view of the definition of the maximal solution mt, it is enough to prove, to conclude
4.3, that
ht < ut, ,
t ∈ 0, T ,
4.4
where ut, is any solution of
LDut − u0 gt, u ,
u0 u0 , > 0.
4.5
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11
Now, it follows from 4.4 that
1
ut, > u0, Iut, Γα
t
t − sα−1 f s, us, ds,
4.6
0
where u0, u0 . Then applying Theorem 2.1, we get immediately 4.1 and since
lim → 0 ut, mt uniformly on each t ∈ 0, T0 , T0 < T , the proof is complete.
We are now in position to prove global existence result.
Theorem 4.2. Assume that f : 0, ∞ × R → R continuous. Let there exists function g : 0, ∞ ×
R → R continuous and nondecreasing with respect to the second argument such that
ft, x ≤ gt, |x|
∀t ≥ 0, x ∈ R.
4.7
If the maximal solution of the initial value problem
LDut − u0 gt, ut,
u0 u0 , t > 0
4.8
exists in 0, ∞ then all solutions of the initial value problem
LDxt − x0 ft, xt,
x0 x0
4.9
with |x0 | ≤ u0 exist in 0, ∞.
Proof. Let xt, x0 be any solution of IVP 4.8 such that |x0 | ≤ u0 , which exists on 0, γ for
γ ∈ 0, ∞, and the value of γ cannot be increased further. Set ht |xt, x0 | for 0 ≤ t < γ.
Then using the assumption 4.4, we get
ht ≤ |x0 | Iht 1
Γα
t
t − sα−1 gs, hsds.
4.10
0
Applying the comparison Theorem 4.1, we obtain
ht |xt, x0 | ≤ mt,
0 ≤ t < γ.
4.11
Since mt is assumed to exist on 0, ∞, it follows that
gt, mt ≤ M,
0 ≤ t ≤ γ.
4.12
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Now, let 0 ≤ t1 ≤ t2 < γ. Then employing the arguments similar to estimating 3.19 and using
4.7 and the bounded M of g we arrive at
|xt2 , x0 − xt1 , x0 | ≤
2M
t2 − t1 α
Γα 1
n/2
t2 − t1 α−β2k n/21
t2 − t1 α−β2k1
.
2Λ
α − β 2k
α − β 2k 1
k0
k0
4.13
Letting t1 , t2 → γ − and using Cauchy criterion, it follows that limt → γ − xt, x0 exists. We define
xγ, x0 limt → γ − xt, x0 and consider the new IVP
LDxt − x0 ft, xt,
x γ x γ, x0 .
4.14
By the assumed local existence, we find that xt, x0 can be continued beyond γ, contradicting
our assumption. Hence, every solution xt, x0 of 4.9 exists on 0, ∞, and the proof is
complete.
References
1 W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein
dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.
2 R. Hilfe, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
3 R. Metzler, W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a
fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.
4 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,
Academic Press, San Diego, Calif, USA, 1999.
5 I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional
differentiation,” Fractional Calculus & Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
6 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential
Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The
Netherlands, 2006.
7 V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge
Academic, Cambridge, UK, 2009.
8 R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value
problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae,
vol. 109, no. 3, pp. 973–1033, 2010.
9 V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,”
Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004.
10 A. Babakhani and E. Enteghami, “Existence of positive solutions for multiterm fractional differential
equations of finite delay with polynomial coefficients,” Abstract and Applied Analysis, vol. 2009, Article
ID 768920, 12 pages, 2009.
11 A. Babakhani, “Positive solutions for system of nonlinear fractional differential equations in two
dimensions with delay,” Abstract and Applied Analysis, vol. 2010, Article ID 536317, 16 pages, 2010.
12 D. Baleanu, A. K. Golmankhaneh, R. Nigmatullin, and A. K. Golmankhaneh, “Fractional Newtonian
mechanics,” Central European Journal of Physics, vol. 8, no. 1, pp. 120–125, 2010.
13 D. Baleanu and J. I. Trujillo, “A new method of finding the fractional Euler-Lagrange and Hamilton
equations within Caputo fractional derivatives,” Communications in Nonlinear Science and Numerical
Simulation, vol. 15, no. 5, pp. 1111–1115, 2010.
14 D. Baleanu, “New applications of fractional variational principles,” Reports on Mathematical Physics,
vol. 61, no. 2, pp. 199–206, 2008.
15 M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations
with fractional order,” Surveys in Mathematics and Its Applications, vol. 3, pp. 1–12, 2008.
Advances in Difference Equations
13
16 D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential
equations,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1835–1841, 2010.
17 Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions with
boundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.
18 A. Ouahab, “Some results for fractional boundary value problem of differential inclusions,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3877–3896, 2008.
19 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,
John Wiley & Sons, New York, NY, USA, 1993.
20 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach
Science, Yverdon, Switzerland, 1993.
21 A. Belarbi, M. Benchohra, and A. Ouahab, “Uniqueness results for fractional functional differential
equations with infinite delay in Fréchet spaces,” Applicable Analysis, vol. 85, no. 12, pp. 1459–1470,
2006.
22 M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order
functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications,
vol. 338, no. 2, pp. 1340–1350, 2008.
23 N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential
equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, no. 5, pp. 765–771,
2006.
24 V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008.
25 A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions for multi-term non-autonomous
fractional differential equations with polynomial coefficients,” Electronic Journal of Differential
Equations, vol. 2006, pp. 1291–12, 2006.