Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 163648, 16 pages
doi:10.1155/2012/163648
Research Article
Existence Results for the Distributed Order
Fractional Hybrid Differential Equations
Hossein Noroozi, Alireza Ansari,
and Mohammad Shafi Dahaghin
Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran
Correspondence should be addressed to Alireza Ansari, alireza 1038@yahoo.com
Received 22 July 2012; Accepted 7 October 2012
Academic Editor: Yongfu Su
Copyright q 2012 Hossein Noroozi et al. 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.
We introduce the distributed order fractional hybrid differential equations DOFHDEs involving
the Riemann-Liouville differential operator of order 0 < q < 1 with respect to a nonnegative
density function. Furthermore, an existence theorem for the fractional hybrid differential equations
of distributed order is proved via a fixed point theorem in the Banach algebras under the mixed
Lipschitz and Caratheodory conditions.
1. Introduction
The differential equations involving Riemann-Liouville differential operators of fractional
order 0 < q < 1 are very important in the modeling of several physical phenomena 1, 2.
In recent years, quadratic perturbations of nonlinear differential equations and first-order
ordinary functional differential equations in Banach algebras, have attracted much attention
to researchers. These type of equations have been called the hybrid differential equations 3–
8. One of the important first-order hybrid differential equations HDE is defined as 4, 9
xt
d
gt, xt,
dt ft, xt
t ∈ J,
1.1
xt0 x0 ,
where J t0 , t0 a is a bounded interval in R for some t0 and a ∈ R with a > 0. Also,
f ∈ CJ × R, R \ {0} and g ∈ CJ × R, such that CJ × R, R is the class of continuous
2
Abstract and Applied Analysis
functions and CJ × R is called the Caratheodory class of functions g : J × R → R which are
Lebesgue integrable bounded by a Lebesgue integrable function on J. Moreover
i the map t → gt, x is measurable for each x ∈ R,
ii the map x → gt, x is continuous for each t ∈ J.
For the above hybrid differential equation, Dhage and Lakshmikantham 9 established
existence, uniqueness, and some fundamental differential inequalities. Also, they stated
some theoretical approximation results for the extremal solutions between the given lower
and upper solutions 10. Later, Zhao. et al. 11 developed the following fractional hybrid
differential equations involving the Riemann-Liouville differential operators of order 0 < q < 1,
D
q
xt
gt, xt,
ft, xt
t ∈ J,
1.2
x0 0,
where J 0, T is bounded in R for some T ∈ R and f ∈ CJ × R, R \ {0}, g ∈ CJ × R.
They established the existence, uniqueness, and some fundamental fractional
differential inequalities to prove existence of the extremal solutions of 1.2. Also, they
considered necessary tools under the mixed Lipschitz and Caratheodory conditions to prove
the comparison principle.
Now, in this article in view of the distributed order fractional derivative 12–14,
we develop the distributed order fractional hybrid differential equations DOFHDEs with
respect to a nonnegative density function.
In this regard, in Section 2 we introduce the distributed order fractional hybrid
differential equation. Section 3 is about some main theorems which are used in this paper.
In Section 4, we prove the existence theorem for this class of equations, and we express
some special cases for the density function used in the distributed order fractional hybrid
differential equation. Finally, the main conclusions are set.
2. The Fractional Hybrid Differential Equation of Distributed Order
In this section, we recall some definitions which are used throughout this paper.
Definition 2.1 see 1, 2. The fractional integral of order q with the lower limit t0 for the
function f is defined as
1
I q ft Γ q
t
t0
fs
t − s1−q
ds,
t > t0 , q > 0.
2.1
Definition 2.2 see 1, 2. The Riemann-Liouville derivative of order q with the lower limit t0
for the function f : t0 , ∞ → R can be written as
dn
1
Dq ft n
Γ n − q dt
t
t0
fs
t − sq1−n
ds,
t > t0 , n − 1 < q < n.
2.2
Abstract and Applied Analysis
3
Definition 2.3. The distributed order fractional hybrid differential equation DOFHDEs, involving the Riemann-Liouville differential operator of order 0 < q < 1 with respect to the
nonnegative density function bq > 0, is defined as
1
b q Dq
0
xt
dq gt, xt,
ft, xt
t ∈ J,
1
0
b q dq 1,
2.3
x0 0.
Moreover, the function t → x/ft, x is continuous for each x ∈ R, where J 0, T is
bounded in R for some T ∈ R. Also, f ∈ CJ × R, R \ {0} and g ∈ CJ × R.
3. The Main Theorems
In this section, we state the existence theorem for the DOFHDE 2.3 on J 0, T . For this
purpose, we define a supremum norm of · in CJ, R as
x sup|xt|,
t∈J
3.1
and for x, y ∈ CJ, R
xy t xtyt,
3.2
is a multiplication in this space. We consider CJ, R is a Banach algebra with respect to norm
· and multiplication 3.2. Moreover the norm · L1 for x ∈ CJ, R is defined by
xL1 T
|xs|ds.
3.3
0
Now, for expressing the existence theorem for the DOFHDE 2.3, we state a fixed point
theorem in the Banach algebra.
Theorem 3.1 see 15. Let S be a nonempty, closed convex, and bounded subset of the Banach
algebra X and let A : X → X and B : S → X be two operators such that
a A is Lipschitz constant α,
b B is completely continuous,
c x AxBy ⇒ x ∈ S for all y ∈ S,
d αM < 1, where M BS sup{Bx : x ∈ S}.
Then the operator equation AxBx x has a solution in S.
At this point, we consider some hypotheses as follows.
A0 The function x → x/ft, x is increasing in R almost everywhere for t ∈ J.
4
Abstract and Applied Analysis
A1 There exists a constant L > 0 such that
ft, x − f t, y ≤ Lx − y,
3.4
for all t ∈ J and x, y ∈ R.
A2 There exists a function h ∈ L1 J, R and a real nonnegative upper bound h∗
such that
gt, x ≤ ht ≤ h∗ ,
3.5
for all t ∈ J and x ∈ R.
Theorem 3.2 Titchmarsh theorem 16. Let Fs be an analytic function which has a branch cut
on the real negative semiaxis. Furthermore, Fs has the following properties:
Fs O1, |s| −→ ∞,
1
Fs O
, |s| −→ 0,
|s|
3.6
for any sector | args| < π − η, where 0 < η < π. Then, the Laplace transform inversion ft can be
written as the Laplace transform of the imaginary part of the function Fre−iπ as follows:
ft L−1 {Fs; t} 1
π
∞
e−rt F re−iπ dr.
3.7
0
Definition 3.3. Suppose that X, d be a metric space and let B ⊆ CX, R. Then, B is
equicontinuous if for all > 0 there exists δ > 0 such that for all f ∈ B and a, x ∈ X
dx, a < δ ⇒ fx − fa < .
3.8
Theorem 3.4 Arzela-Ascoli theorem 17. Let X, d be a compact metric space and let B ⊂
CX, R. Then, B is compact if and only if B is closed, bounded, and equicontinuous.
Theorem 3.5 Lebesgue dominated convergence theorem 18. Let {fn } be a sequence of realvalued measurable functions on a measure space S, Σ, μ. Also, suppose that the sequence converges
pointwise to a function f and is dominated by some integrable function g in the sense that
fn x ≤ gx,
3.9
for all numbers n in the index set of the sequence and all points x in S. Then, f is integrable and
fn dμ lim
S
fdμ.
S
3.10
Abstract and Applied Analysis
5
4. Existence Theorem for the DOFHDEs
We apply the following lemma to prove the main existence theorem of this section.
Lemma 4.1. Assume that hypothesis (A0 ) holds in pervious section, then for any h ∈ L1 J, R and
0 < q < 1, the function x ∈ CJ, R is a solution of the DOFHDE 2.3 if and only if x satisfies the
following equation
xt ft, xt
π
t
L 0
1
;
t
−
τ
gτ, xτdτ
Bre−iπ 4.1
such that 0 ≤ τ ≤ t ≤ T and
Bs 1
b q sq dq.
4.2
0
Proof. Applying the Laplace transform on both sides of 2.3 and letting
Y t xt
,
ft, xt
4.3
we have
1
L
b q Dq Y tdq; s
L gt, xt; s
0
1
0
q−1
b q sq Y s − Dt Y 0 dq Gs.
4.4
Since Y 0 0, we have
Y s
1
b q sq dq
Gs,
4.5
0
and hence,
Y s 1
Gs,
Bs
4.6
such that
Bs 1
0
b q sq dq.
4.7
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Abstract and Applied Analysis
Now, using the inverse Laplace transform on both sides of 4.6 and applying the convolution
product, we get
1
xt
L−1
Gs
ft, xt
Bs
t
1
−1
; t − τ gτ, xτdτ,
L
Bs
0
L−1 {Y s} 4.8
or equivalently
xt ft, xt
t
L−1
0
1
; t − τ gτ, xτdτ.
Bs
4.9
Since Bs is an analytic function which has a branch cut on the real negative semiaxis,
according to the Titchmarsh Theorem 3.2 we get
xt ft, xt
π
t ∞
0
e−rt−τ 0
1
gτ, xτdr dτ,
Bre−iπ 4.10
which by the Laplace transform definition, 4.1 is held. Conversely, let x satisfies 4.1,
therefore, x satisfies the equivalent equation 4.9. By t 0 in 4.1, we have
x0
0
0
.
f0, x0
f0, 0
4.11
According to hypothesis A0 , the map x → x/f0, x is injective in R and hence x0 0.
Next, with dividing 4.9 by ft, xt and using the Laplace transform operator on both sides
of this equation, 4.6 also holds. Since Y 0 0, we obtain 4.4 and by applying the inverse
Laplace transform, 2.3 also holds.
Theorem 4.2. Suppose that hypothesis (A0 )–(A2 ) hold. Further, if
LMhL1
< 1,
π
M > 0,
4.12
then, the DOFHDE 2.3 has a solution defined on J.
Proof. We set X CJ, R as a Banach algebra and define a subset S of X by
S {x ∈ X | x ≤ N},
4.13
such that
N
F0 MhL1
,
π − LMhL1
F0 supft, 0.
t∈J
4.14
Abstract and Applied Analysis
7
It is obvious that S is closed and if x1 , x2 ∈ R, then x1 ≤ N and x2 ≤ N, also by properties
of the norm, we get
λx1 1 − λx2 ≤ λx1 1 − λx2 ≤ λN 1 − λN N.
4.15
Therefore, S is a convex and bounded and by applying Lemma 4.1, DOFHDE 2.3 is
equivalent to 4.1.
Define operators A : X → X and B : S → X by
Axt ft, xt,
1
Bxt π
t
0
L t ∈ J,
1
; t − τ gτ, xτdτ,
Bre−iπ 4.16
t ∈ J,
thus, from 4.1, we obtain the operator equation as follows:
AxtBxt xt,
t ∈ J.
4.17
If operators A and B satisfy all the conditions of Theorem 3.1, then the operator equation
4.17 has a solution in S. For this paper, let x, y ∈ X which by hypothesis A1 we have
Axt − Ayt ft, xt − f t, yt ≤ Lxt − yt ≤ Lx − y,
t ∈ J,
4.18
and if for all x, y ∈ X take a supremum over t, then we have
Ax − Ay ≤ Lx − y.
4.19
Therefore, A is a Lipschitz operator on X with the Lipschitz constant L > 0, and the condition
a from Theorem 3.1 is held. Now, for checking the condition b from this theorem, first, we
shall show that B is continuous on S.
Let {xn } be a sequence in S such that
lim xn x
n→∞
4.20
8
Abstract and Applied Analysis
with x ∈ S. By applying the Lebesgue-dominated convergence Theorem 3.5 for all t ∈ J, we
get
1 t
1
lim Bxn t lim
L ; t − τ gτ, xn τdτ
n→∞
n→∞π 0
Bre−iπ 1
1 t
;
t
−
τ
lim gτ, xn τdτ
L n→∞
π 0
Bre−iπ 1
1 t
L ; t − τ gτ, xτdτ
π 0
Bre−iπ 4.21
Bxt.
Thus, B is a continuous operator on S. In next stage, we shall show that B is a compact
operator on S. For this paper, we shall show that Bs is a uniformly bounded and
eqicontinuous set in X. Let x ∈ S, then by hypothesis A2 for all t ∈ J we have
1 t
1
L ;
t
−
τ
gτ,
xτdτ
|Bxt| π 0
Bre−iπ 1
≤
π
t 1
L ; t − τ |hτ|dτ.
Bre−iπ 4.22
0
Let s t − τ such that 0 ≤ τ ≤ t ≤ T . Then by the existence Laplace transform theorem 19,
there exists a constant M > 0 such that for a constant c that s > c,
1
≤ M ecr .
−iπ
Bre 4.23
Hence, we find an upper bound for the integral of 4.22 as follows:
L ∞
1
1
−sr
e ;t − τ dr −iπ
−iπ
Bre Bre 0
∞
1
−sr dr
≤
e −iπ
Bre 0
∞
M
≤
M ec−sr dr ≤
≤ M,
|s − c|
0
4.24
such that
M
.
0≤τ≤t≤T |t − τ − c|
M sup
4.25
Abstract and Applied Analysis
9
Finally, with respect to the inequality 4.22 we obtain
MhL1
,
π
|Bxt| ≤
4.26
which by applying supremum over t, we get for all x ∈ S
Bx ≤
M
hL1 .
π
4.27
Thus, B is uniformly bounded on S.
In this stage, now we show that BS is an equicontinuous set in X. Let t1 , t2 ∈ J, with
t1 < t2 . In this respect, we have for all x ∈ S
1 t1 1
L ; t1 − τ gτ, xτdτ
|Bxt1 − Bxt2 | π 0
Bre−iπ 1
−
π
t2
L 0
1
−
τ
gτ,
xτdτ
;
t
2
Bre−iπ 1 t1 1
≤
L −
τ
gτ, xτdτ
;
t
1
π 0
Bre−iπ 1 t1
1
−
;
t
L −
τ
gτ,
xτdτ
2
π 0
Bre−iπ 1 t1 1
L −
τ
gτ, xτdτ
;
t
2
π 0
Bre−iπ 1
−
π
t2
0
L 4.28
1
; t2 − τ gτ, xτdτ .
−iπ
Bre If we set s1 t1 − τ and s2 t2 − τ, then by Laplace transform definition and 4.23, for s1 > c
and s2 > c we can write
1
1
L −
L
;
s
; s2 1
−iπ
−iπ
Bre Bre ∞
e−s1 r ∞
1
1
−s2 r
dr
−
dr e
−iπ
−iπ
Bre Bre 0
0
∞
−s r
1
dr
e 1 − e−s2 r ≤
Bre−iπ 0
∞
1
1
c−s1 r
c−s2 r
≤M
−
.
e
dr M
−e
s1 − c s2 − c
0
4.29
10
Abstract and Applied Analysis
Therefore, we have
1 t1
1
1
;
s
;
s
L −
L
gτ,
xτdτ
1
2
π 0
Bre−iπ Bre−iπ h∗ t1 1
1
≤
−
dτ
M
π 0
t1 − τ − c t2 − τ − c
M h∗
c t1 − t2 c − t1 ln
.
π
cc − t2 4.30
Also, by 4.24 we have
1 t1 1
L −
τ
gτ,
xτdτ
;
t
2
π t2
Bre−iπ h∗ t1
M h∗
c
M
≤
dτ ln
.
π t2 t2 − τ − c
π
c t 1 − t2
4.31
Finally, with respect to 4.28, 4.30, and 4.31 we obtain
c
M h∗
c t1 − t2 c − t1 ln
ln
|Bxt1 − Bxt2 | ≤
π
cc − t2 c t 1 − t2
c − t1
M h∗
ln
.
π
c − t2
4.32
Hence, for > 0, there exists δ > 0 such that if |t1 − t2 | < δ, then for all t1 , t2 ∈ J and all x ∈ S
we have
|Bxt1 − Bxt2 | < ,
4.33
which implies that BS is an equicontinuous set in X and according to the Arzela-Ascoli
Theorem 3.4, B is compact. Therefore B is continuous and compact operator on S into X and
B is a completely continuous operator on S and the condition b from the Theorem 3.1 is
held.
For checking the condition c of Theorem 3.1, let x ∈ X and y ∈ S be arbitrary such
that x AxBy. Then, by hypothesis A1 we get
|xt| |Axt|Byt
1 t
1
ft, xt
L ; t − τ g τ, yτ dτ π 0
Bre−iπ t 1
1
≤ ft, xt − ft, 0 ft, 0
; t − τ g τ, yτ dτ
L π 0
Bre−iπ Abstract and Applied Analysis
t 1
1
≤ L|xt| F0 L ; t − τ hτdτ
π 0
Bre−iπ MhL1
.
≤ L|xt| F0 π
11
4.34
Therefore,
F0 MhL1
,
π − LMhL1
|xt| ≤
4.35
which by taking a supremum over t, we obtain
xt ≤
F0 MhL1
N.
π − LMhL1
4.36
Thus, the condition c of Theorem 3.1 is satisfied. If we consider
M1 Bs sup{Bx : x ∈ S} ≤
αM1 ≤ L
M
hL1
π
M
hL1 ,
π
4.37
< 1,
the hypothesis d of Theorem 3.1 is satisfied.
Hence, all the conditions of Theorem 3.1 are satisfied and therefore the operator
equation AxBx x has a solution in S. As a result, the DOFHDE 2.3 has a solution defined
on J and proof is completed.
5. Some Special Cases
In this section, we discuss some special cases of the density function bq for the DOFHDE
2.3 and we find the operators A and B which introduce in Theorem 4.2. In proof of
Lemma 4.1, the following equation is equivalent to the DOFHDE 2.3,
xt ft, xt
t
L−1
0
1
; t − τ gτ, xτdτ,
Bs
5.1
b q sq dq.
5.2
such that,
Bs 1
0
12
Abstract and Applied Analysis
1 Let bq 1. Then we have
Bs 1
s−1
.
lns
5.3
et−τ Eit − τ,
5.4
sq dq 0
Thus,
L−1
1
;t − τ
Bs
where Eit is the exponential integral defined by
Eit ∞
t
e−u
du.
u
5.5
Therefore, for this case, the DOFHDE 2.3 is
1
Dq
0
xt
dq gt, xt,
ft, xt
t ∈ J,
5.6
x0 0,
and it is equivalent to the following equation:
xt ft, xt
t
et−τ Eit − τgτ, xτdτ,
5.7
0
such that the operators A and B in Theorem 4.2 are
Axt ft, xt,
Bxt t
et−τ Eit − τgτ, xτdτ.
5.8
0
2 Two-term equation: Let bq a1 δq − q1 a2 δq − 0, which 0 < q1 < 1 also, a1
and a2 are nonnegative constant coefficients and δ is the Dirac delta function. Then by the
following inverse Laplace transform 2:
−1
L
1
;t − τ
Bs
L
−1
1
a1 sq1 a2
;t − τ
1
a2
q1 −1
q1
t − τ Eq1 ,q1 − t − τ ,
a1
a1
5.9
where Eλ,μ z is the Mittag-Leffler function in two parameters
Eλ,μ z ∞
zj
j0 Γ λj μ
λ, μ > 0, z ∈ C
5.10
Abstract and Applied Analysis
13
we get the DOFHDE 2.3 as
a1 Dq1
xt
xt
a2
gt, xt,
ft, xt
ft, xt
t ∈ J,
5.11
x0 0.
It is equivalent to the following equation
xt ft, xt
t
0
1
a2
q1 −1
q1
gτ, xτdτ,
t − τ Eq1 ,q1 − t − τ
a1
a1
5.12
such that the operators A and B in Theorem 4.2 are
1
Bxt a1
Axt ft, xt,
t
0
a2
gτ, xτdτ.
t − τq1 −1 Eq1 ,q1 − t − τq1
a1
5.13
3 Three-term equation: Let,
b q a1 δ q − q1 a2 δ q − q2 a3 δ q − 0 ,
5.14
which 1 > q1 > q2 > 0 and a1 , a2 , and a3 are nonnegative constant coefficients and δ is the
Dirac delta function. Then, by virtue of 2
L−1
1
;t − τ
Bs
L−1
1
;t − τ
a1 sq1 a2 sq2 a3
G3 t − τ,
5.15
where
G3 t − τ ∞
a2
1
−1k a3 k
k
t − τq1 k1−1 Eq1 −q2 ,q1 kq2 − t − τq1 −q2 ,
a1 k0 k!
a1
a1
5.16
k
and Eλ,μ z is the kth derivative of the Mittag-Leffler function in two parameters
k
Eλ,μ z ≡
∞
j k !zj
dk
E
,
z
λ,μ
dzk
j0 j!Γ λj λk μ
k 0, 1, 2, . . . .
5.17
We get the DOFHDE 2.3 as
a1 Dq1
xt
xt
xt
a2 D q 2
a3
gt, xt,
ft, xt
ft, xt
ft, xt
x0 0.
5.18
14
Abstract and Applied Analysis
It is equivalent to the following equation:
t
xt ft, xt
5.19
G3 t − τgτ, xτdτ,
0
such that the operators A and B in Theorem 4.2 are
Axt ft, xt,
t
Bxt G3 t − τgτ, xτdτ.
5.20
0
4 General Case: n-term equation: suppose that
b q a0 δ q − q0 a1 δ q − q1 a2 δ q − q2 · · · an δ q − qn ,
5.21
which 1 > qn > qn−1 > · · · > q0 > 0 and ai for i 0, 1, 2, . . . , n are nonnegative constant
coefficients. Therefore, by the following inverse Laplace transform 2, we have
−1
L
1
;t − τ
Bs
L
−1
1
;t − τ
a0 sq0 a1 sq1 · · · an sqn
Gn t − τ,
5.22
where
Gn t − τ ∞
1 −1m
an m0 m! k k ···k
0
×
n−2 ai ki
i0
an
1
m; k0 , k1 , . . . , kn−2 n−2 m
t − τqn −qn−1 mqn m
qn −qn−1 ,qn n−2
j0 qn−1 −qj kj
×E
−
n−2
j0 qn−1 −qj kj −1
5.23
an−1
t − τqn −qn−1 .
an
Thus, for this case, the DOFHDE 2.3 is
a0 Dq0
xt
xt
xt
a1 D q 1
· · · an D q n
gt, xt,
ft, xt
ft, xt
ft, xt
5.24
x0 0.
It is equivalent to the following equation:
xt ft, xt
t
0
Gn t − τgτ, xτdτ
5.25
Abstract and Applied Analysis
15
and the operators A and B in Theorem 4.2 are given by
Axt ft, xt,
Bxt t
Gn t − τgτ, xτdτ.
5.26
0
6. Conclusions
In this paper, we introduced a new class; the fractional hybrid differential equations of
distributed order and stated an existence theorem for it. We pointed out a fixed point theorem
in the Banach algebra for the existence of solution. Basis of this theorem is on finding two
operator equations which in special cases for multiterms fractional hybrid equations are given
with respect to the derivatives of Mittag-Leffler function.
Acknowledgment
The authors have been partially supported by the Center of Excellence for Mathematics,
Shahrekord University.
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