Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 690653, 12 pages
doi:10.1155/2011/690653
Research Article
Existence of Solutions to Fractional Mixed
Integrodifferential Equations with Nonlocal
Initial Condition
A. Anguraj,1 P. Karthikeyan,2 and J. J. Trujillo3
1
Department of Mathematics, PSG College of Arts and Science, Coimbatore 641 014, India
Department of Mathematics, KSR College of Arts and Science, Tiruchengode 637 215, India
3
Departamento de Análisis Matemático, Universidad de La Laguna, Tenerife 38271 La Laguna, Spain
2
Correspondence should be addressed to J. J. Trujillo, jtrujill@ullmat.es
Received 14 November 2010; Accepted 6 January 2011
Academic Editor: Dumitru Baleanu
Copyright q 2011 A. Anguraj 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 study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition dα xt/dtα ft, xt,
t
1
1
kt, s, xsds, 0 ht, s, xsds, x0 0 gsxsds, where t ∈ 0, 1, 0 < α < 1, and f is a
0
given function. We point out that such a kind of initial conditions or nonlocal restrictions could
play an interesting role in the applications of the mentioned model. The results obtainded are
applied to an example.
1. Introduction
Recently it have been proved that the differential models involving nonlocal derivatives
of fractional order arise in many engineering and scientific disciplines as the mathematical
modeling of systems and processes in many fields, for instance, physics, chemistry,
aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth see 1–
6. In fact, such models can be considered as an efficient alternative to the classical nonlinear
differential models to simulate many complex processes see 7. For instance, fractional
differential equations are an excellent tool to describe hereditary properties of viscoelastic
materials and, in general, to simulate the dynamics of many processes on anomalous media.
Theory of fractional differential equations has been extensively studied by several authors
as Delbosco and Rodino 8, Kilbas et al. 6, Lakshmikantham et al. 9–11, and also see
2, 12–16.
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Recently Mophou and N’Guérékata 17, studied the Cauchy problem with nonlocal
conditions
Dq xt Axt tn ft, xt, Bxt,
t ∈ 0, T , n ∈ Z ,
x0 gx x0 ,
1.1
in general Banach space X with 0 < q < 1, and A is the infinitesimal generator of a C0 semigroup of bounded linear operators. By means of the Krasnoselskii’s Theorem, existence
of solutions was also obtained.
Subsequently several authors have investigated the problem for different types
of nonlinear differential equations and integrodifferential equations including functional
differential equations in Banach spaces.
Very recently N’Guérékata 2, 18 discussed the existence of solutions of fractional
abstract differential equations with nonlocal initial condition. The nonlocal Cauchy problem
is discussed by authors in 15 using the fixed-point concepts. Tidke 19 studied
the nonfractional mixed Volterra-Fredholm integrodifferential equations with nonlocal
conditions using Leray-Schauder Theorem.
Motivated by the above mentioned works in this manuscript we discuss the existence
and the uniqueness of the solution for the following fractional integrodifferential equation
with nonlocal integral initial condition in Banach Space:
dα xt
f
dtα
t
t, xt,
1
kt, s, xsds,
0
x0 ht, s, xsds ,
0
1.2
1
gsxsds,
0
where t ∈ J 0, 1, 0 < α < 1, gt ∈ 0, 1, g ∈ L1 0, 1, R , x ∈ Y CJ, E is a continuous
function on J with values in the Banach space E and xY maxt∈J xtE , and f : J × E ×
E × E → E, k : D × E → E, and h : D0 × E → E are continuous E-valued functions. Here
D {t, s ∈ R2 : 0 ≤ s ≤ t ≤ 1}, and D0 J × J. The operator dα /dtα denotes the Caputo
fractional derivative of order α.
For the sake of the shortness let
Kxt t
kt, s, xsds,
0
Hxt 1
ht, s, xsds.
A
0
The paper is organized as follows. In Section 2, some definitions, lemmas and preliminary
results are introduced to be used in the sequel. Section 3 will involve the assumptions, main
results and proofs of existence problem of 1.2, together with a nonlocal initial condition.
Finally an example is presented.
2. Preliminaries
Let E be a real Banach space and θ the zero element of E. Let L1 0, 1, E be the Banach space
of measurable functions x : 0, 1 → E which are Lebesgue integrable, equipped with the
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1
norm xL1 0 xsds. We will use the following notation R 0, ∞, and R 0, ∞. A
function x ∈ C0, 1, E is called a solution of 1.2 if it satisfies 1.2.
Definition 2.1. A real function ft is said to be in the space Cα , α ∈ R if there exists a real
number p > α, such that ft tp gt, where g ∈ C0, ∞, while ft is said to be in the space
Cαm if and only if f m ∈ Cα , m ∈ N.
Definition 2.2. The fractional arbitrary order Riemann-Liouville integral on the right and
on the left of the function f ∈ L1 a, b, R of order α, a, b ∈ R is defined by
Iaα ft 1
Γα
t
a
t − sα−1
fsds,
Γα
α
Ib−
ft 1
Γα
b
t
s − tα−1
fsds,
Γα
2.1
where Γ is the Gamma function of Euler.
α
When a 0, we write I α ft I0
ft f ∗ φα t, where φα t tα−1 /Γα for t > 0,
φα t 0 for t ≤ 0, and ∗ represents the Convolution of Laplace. Then, it is well known that
φα t → δt as α → 0, where δ is the Delta function.
Definition 2.3. The Riemann-Liouville fractional integral operator of order α > 0, of a function
f ∈ Cμ , μ ≥ −1 is defined as
I α ft φα ∗ ft,
α > 0, t > 0,
I 0 fx fx.
2.2
Definition 2.4. The Caputo’s derivative of fractional order α > 0 for a suitable function ft is
defined by
c
1
D f t Γn − α
α
t
0
f n s
t − sα−n1
ds,
n − 1 < α < n, n α 1,
2.3
where α denotes the integer part of real number α.
It is obvious that the Caputo’s derivative of a constant is equal to 0.
Lemma 2.5. Let α > 0 and n α 1. Then
n−1 k
f 0 k
t .
I α c Dα f t ft −
k!
k0
2.4
1
α
Lemma 2.6. If Qτ, α ΓαI1−
gτ τ gss − τα−1 ds for τ ∈ 0, 1, and if g ∈ L1 0, 1, R satisfies 0 ≤ gs ≤ 1 for 0 ≤ s ≤ 1 and α > 0, then
Qτ, α
< e,
Γα
t
0
t − sα−1 ds
< e.
Γα
2.5
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Proof. A direct computation shows
Qτ, α
Γα
1
τ
1
gss − τα−1 ds
s − τα−1 ds
∞
τ ∞
sα−1 e−s ds
sα−1 e−s ds
0
0
1−τ
t
0
sα−1 ds
∞0
sα−1 e−s ds
0
t
e
1−τ
≤ 0∞
0
sα−1 e−s ds
sα−1 e−s ds
< e,
2.6
t
sα−1 ds
e sα−1 e−s ds
t − sα−1 ds
∞0
≤ ∞0
< e.
Γα
sα−1 e−s ds
sα−1 e−s ds
0
0
Theorem 2.7 Krasnoselkii. Let X be a Banach space, let S be a bounded closed convex subset of X
and let A, B be maps of S into X such that Ax By ∈ S for every pair x, y ∈ S. If A is completely
continuous and B is a contraction then the equation Ax Bx x has a solution on S.
3. Main Results
We assume the following.
A1 If f ∈ C0, 1 × E × E × E, E and a nonnegative, bounded pf ∈ L1 0, 1, R , there
exist M > 0, pf t ≤ M for t ∈ 0, 1 such that
ft, x, Kx, Hx ≤ pf tx
for x ∈ E.
3.1
A2 There exist positive constants L1 , L2 , and L such that
f t, x1 , y1 , z1 − f t, x2 , y2 , z2 ≤ L1 x1 − x2 y1 − y2 z1 − z2 3.2
for all x1 , y1 , z1 , x2 , y2 , z2 ∈ Y , L2 maxt∈J ft, 0, 0, 0, and L max{L1 , L2 }.
A3 There exist positive constants N1 , N2 , and N such that
kt, s, x1 − kt, s, x2 ≤ N1 x1 − x2 3.3
for all x1 , x2 ∈ Y , N2 maxt,s∈D kt, s, 0, and N max{N1 , N2 }.
A4 There exist positive constants C1 , C2 , and C such that
ht, s, x1 − ht, s, x2 < C1 x1 − x2 3.4
for all x1 , x2 ∈ Y , C2 maxt,s∈D0 ht, s, 0, and C max{C1 , C2 }.
A5 p L/Γα 11 C N/α 1 is such that 0 ≤ p < 1.
Firstly, we obtain the following lemmas to prove the main results on the existence of solutions
to 1.2.
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Lemma 3.1. If (A1) holds with μ 0 gsds, then the problem 1.2 is equivalent to the following
equation:
1
1
Qτ, αfτ, xτ, Kxτ, Hxτdτ φα t ∗ ft, xt, Kxt, Hxt.
xt 1 − μ Γα 0
3.5
Proof. By Lemma 2.5 and 1.2, we have
1
xt x0 Γα
t
3.6
t − sα−1 fs, xs, Kxs, Hxsds.
0
Therefore,
x0 1
gsxsds
0
1
0
1
gsdsx0 Γα
1
s
gs
0
3.7
s − τα−1 fτ, xτ, Kxτ, Hxτdτ ds.
0
So,
1
x0 1 − μ Γα
1
1 − μ Γα
1
1
fτ, xτ, Kxτ, Hxτ
0
s − τ
τ
1
α−1
gsds dτ
3.8
Qτ, αfτ, xτ, Kxτ, Hxτdτ,
0
and then
1
xt 1 − μ Γα
1
Qτ, αfτ, xτ, Kxτ, Hxτdτ φα t ∗ ft, xt, Kxt, Hxt.
0
3.9
Conversely, if x is a solution of 3.5, then for every t ∈ 0, 1, according to Definition 2.4 we
have
1
1
c α
c α
Qτ, αfτ, xτ, Kxτ, Hxτdτ
D xt D 1 − μ Γα 0
φα t ∗ ft, xt, Kxt, Hxt
3.10
θ c Dα I α ft, xt, Kxt, Hxt
ft, xt, Kxt, Hxt.
It is obvious that x0 1
0
gsxsds. This completes the proof.
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Advances in Difference Equations
Lemma 3.2. If (A3) and (A4) are satisfied, K, H are defined in A, then the conditions
Kxt ≤ tN1 x N2 ,
Kx1 t − Kx2 t ≤ N1 tx1 − x2 ,
Hxt ≤ C1 x C2 ,
3.11
Hx1 t − Hx2 t ≤ C1 x1 − x2 ,
are satisfied for any t ∈ J, and x, x1 , x2 ∈ Y .
Proof. By A3, we have
Kxt ≤
t
kt, s, xsds
0
t
kt, s, xs − kt, s, 0 kt, s, 0ds
0
≤
t
kt, s, xs − kt, s, 0ds 0
t
3.12
kt, s, 0ds
0
≤ N1 tx N2 t ≤ N1 x N2 .
On the other hand,
Kx1 t − Kx2 t ≤
t
kt, s, x1 s − kt, s, x2 sds
0
≤ N1
t
x1 s − x2 sds
3.13
0
≤ N1 tx1 − x2 .
Similarly, for the other conditions, we use assumption A4, to get
Hxt ≤
1
ht, s, xsds ≤ C1 x C2 ,
0
3.14
Kx1 t − Kx2 t ≤ C1 x1 − x2 .
Theorem 3.3. If (A1)–(A5) are satisfied, then the fractional integrodifferential equation 1.2 has a
unique solution continuous in J.
Proof. We use the Banach contraction principle to prove the existence and uniqueness of the
solution to 1.2. Let Br {x ∈ Y : x ≤ r} ⊆ Y , where r ≥ eM/1 − μ L/Γα 11 C N/α 1 and define the operator Ψ on the Banach space Y by
1
1
Qτ, αfτ, xτ, Kxτ, Hxτdτ
1 − μ Γα 0
t
1
t − sα−1 fs, xs, Kxs, Hxsds.
Γα 0
Ψxt 3.15
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Firstly, we show that the operator Ψ maps Br into itself. By using A1 and triangle inequality,
we have
t
1 eM
α−1
x Ψxt ≤
t − s fs, xs, Kxs, Hxsds
1−μ
Γα 0
t
eM
1
≤
x t − sα−1 fs, xs, Kxs, Hxsds
1−μ
Γα 0
eM
x
1−μ
t
1
t − sα−1 fs, xs, Kxs, Hxs − fs, 0, 0, 0 fs, 0, 0, 0ds
Γα 0
t
eM
1
≤
x t − sα−1 fs, xs, Kxs, Hxs − fs, 0, 0, 0ds
1−μ
Γα 0
t
1
t − sα−1 fs, 0, 0, 0ds.
Γα 0
3.16
≤
Now, if A2 is satisfied, then
L1 t
eM
x t − sα−1 xs Kxs Hxsds
1−μ
Γα 0
L2 t
t − sα−1 ds
Γα 0
L1 t
eM
L1 t
x ≤
t − sα−1 xsds t − sα−1 Kxsds
1−μ
Γα 0
Γα 0
L1 t
L2 t
t − sα−1 Hxsds t − sα−1 ds.
Γα 0
Γα 0
Ψxt ≤
3.17
Using Lemma 3.2 and A3, we have
Ψxt ≤
eM
L1
L1
x tα x N1 x N2 1−μ
Γα 1
Γα
≤
t
L2
L1
tα
C1 x C2 tα Γα 1
Γα 1
L1
eM
L1
x tα x tα1 N1 x N2 1−μ
Γα 1
Γα 2
L2
L1
tα
C1 x C2 tα Γα 1
Γα 1
t − sα−1 sds
0
L1 N2 α1
eM
L1 C2 α
L2
x t t tα
1−μ
Γα 2
Γα 1
Γα 1
L1
N1
α
t 1
t C1 x,
Γα 1
α1
3.18
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if x ∈ Br , we have
eM
L
N
x 1
C
1−μ
Γα 1
α1
Lr
N
1
C
Γα 1
α1
≤ r.
Ψxt ≤
3.19
Thus ΨBr ⊂ Br . Next, we prove that Ψ is a contraction mapping. For this, let x1 , x2 ∈ Y .
Applying A2, we have
Ψx1 t − Ψx2 t
Qτ, α 1 fs, x1 τ, Kx1 τ, Hx1 τ − fs, x2 τ, Kx2 τ, Hx2 τ dτ 1 − μ Γα 0
t
1
t − sα−1 fs, x1 s, Kx1 s, Hx1 sds
Γα 0
t
1
α−1
−
t − s fs, x2 s, Kx2 s, Hx2 sds
Γα 0
1
e
≤
L1 x1 τ − x2 τ Kx1 τ − Kx2 τ Hx1 τ − Hx2 τdτ
1−μ 0
t
L1
t − sα−1 x1 s − x2 sds
Γα
0
t
t
α−1
α−1
t − s Kx1 s − Kx2 sds t − s Hx1 s − Hx2 sds
0
0
3.20
then using A3, A4 and Lemma 3.2, one gets
Ψx1 t − Ψx2 t
1
1
1 L1 e
dτ N1 τdτ C1 dτ
≤
x1 − x2 1−μ
0
0
0
t
t
t
L1
α−1
α−1
α−1
x1 − x2 t − s ds N1 t − s sds C1 t − s ds
Γα
0
0
0
α
α1
t
N1 Γαt
C1 tα
L1
N1
L1 e
C1 x1 − x2 x1 − x2 ≤
1
2
Γα α
Γα 2
α
1−μ
3.21
L1 e
N1
L1
N1
α
C1 1 C1 1
t t x1 − x2 2
Γα 1
α1
1−μ
N
L
N
Le
x1 − x2 C1 1C
≤ 1
2
Γα 1
α1
1−μ
N
Le
C p x1 − x2 ≤ 1
2
1−μ
≤ ΩC,N,α,μ,p x1 − x2 ,
Advances in Difference Equations
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where ΩC,N,α,μ,p Le/1 − μ1 N/2 C p < 1 depends on the parameter of the
problem. Therefore Ψ has a unique fixed-point x Ψx ∈ Br , which is a solution of 3.5,
and hence is a solution of 1.2.
Theorem 3.4. Assume (A1)–(A4) holds. If eM < 1 − μ, then 1.2 has at least one solution on I.
Proof. Choose r ≥ eM/1 − μ pf L1 /Γα 1 and consider Br : {x ∈ C : x ≤ r}. Now
define on Br the operators A, B by
Axt : φα t ∗ ft, xt, Kxt, Hxt,
1
1
Qτ, αfτ, xτ, Kxτ, Hxτdτ.
Bxt : 1 − μ Γα 0
3.22
Let us observe that if x, y ∈ Br then Ax By ∈ Br . Indeed it is easy to check the inequality
pf 1
eM
L
Ax By ≤ ≤ r.
Γα 1
1−μ
3.23
We can easily show that that B is a contraction mapping. Let u, v ∈ Br . Then
But − Bvt
1
1 − μ Γα
: e
≤
1−μ
1
1
Qτ, αfτ, uτ, Kuτ, Huτ − fτ, vτ, Kvτ, Hvτdτ
0
L1 uτ − vτ Kuτ − Kvτ Huτ − Hvτdτ
0
L1 e
N1
C1 u − v
1
2
1−μ
≤
≤ ΛN1 ,C1 ,L1 u − v,
3.24
where ΛN1 ,C1 ,L1 L1 e/1 − μ1 N1 /2 C1 < 1 depends only on the parameter of the
problem and hence B is contraction. Since x is continuous, then Axt is continuous in view
of A1. Let us now note that A is uniformly bounded on Br . This follows from the inequality
Axt ≤
pf 1
L
Γα 1
Now let us prove that Axt is equicontinuous.
.
3.25
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Let t1 , t2 ∈ J and x ∈ Br . Using the fact that f is bounded on the compact set J ×Br thus
supt,s∈J×Br ft, xs, Kxs, Hxs : c0 < ∞, we will get
s
t
1 1
Axt1 − Axt2 t1 − sq−1 fs, xs, Kxs, Hxsds
Γ q 0
0
−
t2
t2 − s
q−1
s
0
0
fs, xs, Kxs, Hxsds
s
t
1 1
q−1
t1 − s
fs, xs, Kxs, Hxsds
Γ q t2
0
−
t2 t2 − s
q−1
q−1
− t1 − s
s
0
0
1
≤ Γ q
fs, xs, Kxs, Hxsds
t
s
1
q−1
fs, xs, Kxs, Hxsds
t1 − s
t2
0
1
Γ q
t
s
2
q−1
q−1
fs, xs, Kxs, Hxsds
t2 − s − t1 − s
0
0
c0
q
q
≤ 2t1 − t2 q t2 − t1 Γ q1
2c0
≤ |t1 − t2 |q ,
Γ q1
3.26
which does not depend on x. So ABr is relatively compact. By the Arzela-Ascoli Theorem,
A is compact. We now conclude the result of the theorem based on the Krasnoselkii’s theorem
above.
4. Example
Consider the following fractional integrodifferential equation:
c
Dα xt e−t xt
1
t
9 e 1 xt 10
t
e−1/2xs ds 0
x0 1
0
1
10
1
xsds,
2
t
0
e−1/49xs ds,
t ∈ J 0, 1,
4.1
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t
t
where 0 < α ≤ 1. Take E R . Set Kxt 0 e−1/2xs ds, Hxt 0 e−1/49xs ds,
ft, x, Kx, Hx e−t x/9 et 1 x Kx Hx, gs 1/2, pf t e−t xt/9 et . Then it
is clear that
f ∈ C0, 1 × E × E × E, E,
pf ∈ L0, 1, R ,
pf t ≤
1
M,
10
4.2
ft, x, Kx, Hx ≤ pf x.
So, A1 is satisfied. Let x, y ∈ E and t ∈ J. Then we have
t
1
t −1/2xs
−1/2ys
Kx − Ky e
ds − e
ds ≤ x − y,
0
2
0
t
t −1/49xs
1 −1/49ys
Hx − Hy e
x − y,
ds − e
ds ≤
0
49
0
ft, x, Kx, Hx − f t, y, y, Hy e−t y
e−t x
1 −
Kx − Ky Hx − Hy 9 et 1 x 9 et 1 y
10
e−t x − y
1
≤
Kx − Ky Hx − Hy
t
10
9 e 1 x 1 y
4.3
1
e−t x − y Kx − Ky Hx − Hy
t
10
9e
1 x − y Kx − Ky Hx − Hy .
≤
10
≤
Hence the conditions A1–A4 hold with M 1/10, L1 1/10, N1 1/2, C1 1/49.
Choose r 1 and μ 1/2. Indeed
149e
L1 e
N1
C1 < 1,
1
2
490
1−μ
eM
e
< 1.
1−μ 5
4.4
Further A5 is satisfied by a suitable choice of α. Then by Lemma 3.2 the problem 1.2 has
a unique solution on 0,1.
Acknowledgment
This paper has been partially supported by MICINN project MTM2010-16499 to which the
authors are very thankful.
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