Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 510314, 11 pages
doi:10.1155/2011/510314
Research Article
Semilinear Volterra Integrodifferential Problems
with Fractional Derivatives in the Nonlinearities
Mokhtar Kirane,1 Milan Medved’,2 and Nasser-eddine Tatar3
1
Université de la Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics,
Comenius University, Mlynska Doliná, 84248 Bratislava, Slovakia
3
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
2
Correspondence should be addressed to Nasser-eddine Tatar, tatarn@kfupm.edu.sa
Received 29 January 2011; Accepted 7 April 2011
Academic Editor: Allan C. Peterson
Copyright q 2011 Mokhtar Kirane 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.
A second-order semilinear Volterra integrodifferential equation involving fractional time derivatives is considered. We prove existence and uniqueness of mild solutions and classical solutions in
appropriate spaces.
1. Introduction
In this work we discuss the following problem:ď
t u t Aut ft g t, s, us, Dβ1 us, . . . , Dβn us ds,
0
u0 u0 ∈ X,
t > 0,
1.1
u 0 u1 ∈ X,
where 0 < βi ≤ 1, i 1, . . . , n. Here the prime denotes time differentiation and Dβi , i 1, . . . , n
denotes fractional time differentiation in the sense of Riemann-Liouville or Caputo. The
operator A is the infinitesimal generator of a strongly continuous cosine family Ct, t ≥ 0
of bounded linear operators in the Banach space X, f and g are nonlinear functions from
R to X and R × R × X × · · · × X to X, respectively, u0 and u1 are given initial data in X.
The problem with β1 · · · βn 0 or 1 has been investigated by several authors see 1–
7 and references therein, to cite a few. Well-posedness has been proved using fixed point
2
Abstract and Applied Analysis
theorems and the theory of strongly continuous cosine families in Banach spaces developed
in 8, 9. This theory allows us to treat a more general integral or integrodifferential equation,
the solutions of which are called “mild” solutions. In case of regularity of the initial data and
the nonlinearities, the mild solutions are shown to be classical. In case β1 β2 · · · βn 1,
the underlying space is the space of continuously differentiable functions.
In this work, when 0 < βi < 1, i 1, . . . , n, we will see that mild solutions need not
be that regular especially when dealing with Riemann-Liouville fractional derivatives. It
is the objective of this paper to find the appropriate space and norm where the problem is
solvable. We first consider the problem with a fractional derivative in the sense of Caputo
and look for a mild solution in C1 . Under certain conditions on the data it is shown that this
mild solution is classical. Then we consider the case of fractional derivatives in the sense of
Riemann-Liouville. We prove existence and uniqueness of mild solution under much weaker
regularity conditions than the expected ones. Indeed, when the nonlinearity involves a term
of the form
1
Γ 1−β
t
0
u sds
t − sβ
,
1.2
0 < β < 1,
then one is attracted by u s in the integral and therefore it is natural to seek mild solutions
in the space of continuously differentiable functions. This is somewhat surprising if instead
of this expression one is given C Dβ ut the latter is exactly the definition of the former.
However, this is not the case when we deal with the Riemann-Liouville fractional derivative.
Solutions are only β-differentiable and not necessarily once continuously differentiable. It will
be therefore wise to look for solutions in an appropriate “fractional” space. We will consider
the new spaces Eβ and FSβ see 4.1 instead of the classical ones E and C1 see 1–7.
To simplify our task we will treat the following simpler problem
t u t Aut ft g t, s, us, Dβ us ds,
0
t > 0,
1.3
u 0 u1 ∈ X,
u0 u0 ∈ X,
with 0 < β < 1. The general case can be derived easily.
The rest of the paper is divided into three sections. In the second section we prepare
some material consisting of notation and preliminary results needed in our proofs. The next
section treats well-posedness when the fractional derivative is taken in the sense of Caputo.
Section 4 is devoted to the Riemann-Liouville fractional derivative case.
2. Preliminaries
In this section we present some assumptions and results needed in our proofs later. This will
fix also the notation used in this paper.
Definition 2.1. The integral
I α hx 1
Γα
x
a
htdt
x − t1−α
,
x>a
2.1
Abstract and Applied Analysis
3
is called the Riemann-Liouville fractional integral of h of order α > 0 when the right side
exists.
Here Γ is the usual Gamma function
Γz :
∞
e−s sz−1 ds,
2.2
z > 0.
0
Definition 2.2. The left hand Riemann-Liouville fractional derivative of order 0 < α < 1 is
defined by
Daα hx d
1
Γ1 − α dx
x
a
htdt
,
x − tα
2.3
x > a,
whenever the right side is pointwise defined.
Definition 2.3. The fractional derivative of order 0 < α < 1 in the sense of Caputo is given by
C
Daα h
1
x Γ1 − α
x
a
h tdt
,
x − tα
2.4
x > a.
Remark 2.4. The fractional integral of order α is well defined on Lp , p ≥ 1 see 10. Further,
from Definition 2.2, it is clear that the Riemann-Liouville fractional derivative is defined for
any function h ∈ Lp , p ≥ 1 for which k1−α ∗ h is differentiable where k1−α t : t−α /Γ1 − α
and ∗ is the incomplete convolution. In fact, as domain of D0α Dα we can take
DDα h ∈ Lp 0, T : k1−α ∗ h ∈ W 1,p 0, T ,
2.5
where
W
1,p
0, T :
u : ∃ϕ ∈ L 0, T : ut C t
p
ϕsds .
2.6
0
In particular, we know that the absolutely continuous functions p 1 are differentiable
almost everywhere and therefore the Riemann-Liouville fractional derivative exists a.e. In
this case for an absolutely continuous function the derivative is summable 10, Lemma 2.2
and the fractional derivative in the sense of Caputo exists. Moreover, we have the following
relationship between the two types of fractional derivatives:
x ha
h tdt
1
α
Γ1 − α t − aα
a x − t
1
ha
C α
Da h x,
x > a.
α Γ1 − α t − a
Daα hx See 10–15 for more on fractional derivatives.
2.7
4
Abstract and Applied Analysis
We will assume the following.
H1 A is the infinitesimal generator of a strongly continuous cosine family Ct, t ∈ R,
of bounded linear operators in the Banach space X.
The associated sine family St, t ∈ R is defined by
t
Csx ds,
Stx :
2.8
t ∈ R, x ∈ X.
0
It is known see 9, 16 that there exist constants M ≥ 1 and ω ≥ 0 such that
|Ct| ≤ Me
ω|t|
t ∈ R,
,
t
ω|s|
|St − St0 | ≤ M e ds,
t0
t, t0 ∈ R.
2.9
If we define
E : x ∈ X : Ctx is once continuously differentiable on R
2.10
then we have the following.
Lemma 2.5 see 9, 16. Assume that (H1) is satisfied. Then
i StX ⊂ E, t ∈ R,
ii StE ⊂ DA, t ∈ R,
iii d/dtCtx AStx, x ∈ E, t ∈ R,
iv d2 /dt2 Ctx ACtx CtAx, x ∈ DA, t ∈ R.
Lemma 2.6 see 9, 16. Suppose that (H1) holds, v : R → X a continuously differentiable function
t
t
t
and qt 0 St − svsds. Then, qt ∈ DA, q t 0 Ct − svsds and q t 0 Ct −
sv sds Ctv0 Aqt vt.
Definition 2.7. A function u· ∈ C2 I, X is called a classical solution of 1.3 if ut ∈ DA,
satisfies the equation in 1.3 and the initial conditions are verified.
In case of Riemann-Liouville fractional derivative then we require additionally that
Dβ ut be continuous.
Definition 2.8. A continuously differentiable solution of the integrodifferential equation
ut Ctu0 Stu1 t
t
0
St − sfsds
s St − s g s, τ, uτ,C Dβ uτ dτ ds
0
is called mild solution of problem 1.3.
0
2.11
Abstract and Applied Analysis
5
In case of Riemann-Liouville fractional derivative the continuous solution is merely
β-differentiable i.e., Dβ ut exists and is continuous.
It follows from 8 that, in case of continuity of the nonlinearities, solutions of 1.3 are
solutions of the more general problem 2.11.
3. Well-Posedness in C1 0, T
For the sake of comparison with the results in the next section we prove here existence and
uniqueness of solutions in the space C1 0, T. This is the space where we usually look for
mild solutions in case the first-order derivative of u appears in the nonlinearity see 1–
7. We consider fractional derivatives in the sense of Caputo. In case of Riemann-Liouville
fractional derivatives we can pass to Caputo fractional derivatives through the formula 2.7
provided that solutions are in C1 0, T in theory, absolute continuity is enough.
Let XA DA endowed with the graph norm xA x Ax. We need the
following assumptions on f and g:
H2 f : R → X is continuously differentiable,
H3 g : R ×R ×XA ×X → X is continuous and continuously differentiable with respect
to its first variable,
H4 g and g1 the derivative of g with respect to its first variable are Lipschitz
continuous with respect to the last two variables, that is
g t, s, x1 , y1 − g t, s, x2 , y2 ≤ Ag x1 − x2 y1 − y2 ,
A
g1 t, s, x1 , y1 − g1 t, s, x2 , y2 ≤ Ag x1 − x2 y1 − y2 ,
1
A
3.1
for some positive constants Ag and Ag1 .
Theorem 3.1. Assume that (H1)–(H4) hold. If u0 ∈ DA and u1 ∈ E then there exists T > 0 and
a unique function u : 0, T → X, u ∈ C0, T; XA ∩ C2 0, T; X which satisfies 1.3 with
Caputo fractional derivative C Dβ u.
Proof. We start by proving existence and uniqueness of mild solutions in the space of
continuously differentiable functions C1 0, T. To this end we consider for t ∈ 0, T
Kut : Ctu0 Stu1 t
0
t
0
St − sfsds
s St − s g s, τ, uτ,C Dβ uτ dτ ds.
3.2
0
Notice that Ctu0 ∈ DA because u0 ∈ DA and we have ACtu0 CtAu0 . Also from
the facts that u1 ∈ E and StE ⊂ DA see ii of Lemma 2.5 it is clear that Stu1 ∈ DA.
6
Abstract and Applied Analysis
Moreover, it follows from Lemma 2.6, H2 and H3 that both integral terms in 3.2 are in
DA. Therefore, Ku ∈ C0, T; DA. In addition to that we have from Lemma 2.6,
AKut CtAu0 AStu1 t
t
Ct − sf sds Ctf0 − ft
0
s Ct − s g s, s, us,C Dβ us g1 s, τ, uτ,C Dβ uτ dτ ds
0
t − g t, τ, uτ,C Dβ uτ dτ,
3.3
0
t ∈ 0, T.
0
Next, a differentiation of 3.2 yields
Ku t StAu0 Ctu1 t
t
Ct − sfsds
0
s Ct − s g s, τ, uτ,C Dβ uτ dτ ds,
0
3.4
t ∈ 0, T.
0
Therefore, Ku ∈ C1 0, T; X remember that u ∈ C1 0, T; X and K maps C1 into C1 .
Now we want to prove that K is a contraction on C1 endowed with the metric
ρu, v : sup ut − vt Aut − vt u t − v t .
0≤t≤T
3.5
For u, v in C1 , we can write
Kut − Kvt
≤
t
t−s
Me
0
0
ωτ
dτ
Ag
s
0
uτ − vτA C Dβ uτ−C Dβ vτ dτ ds,
3.6
and since
τ
1
C β
τ − σ−β u σ − v σdσ
D uτ−C Dβ vτ ≤ Γ 1−β 0
τ 1−β
≤ sup u t − v t,
Γ 2 − β 0≤t≤T
3.7
it appears that
T
MAg T 2
T 1−β
ωτ
max 1, e dτ ρu, v.
Kut − Kvt ≤
2
Γ 2−β
0
3.8
Abstract and Applied Analysis
7
Moreover,
AKut − AKvt
t
≤ Meωt−s Ag us − vsA C Dβ us−C Dβ vs ds
0
t
Meωt−s Ag1
0
t
0
s
0
uτ − vτA C Dβ uτ−C Dβ vτ dτ ds
3.9
Ag us − vsA C Dβ us−C Dβ vs ds
implies that
AKut − AKvt
T
T 1−β
ωT −s
≤ max 1, A g T M A g A g1 T
e
ds ρu, v.
Γ 2−β
0
3.10
In addition to that, we see that
Ku t − Kv t
≤
t
Meωt−s Ag
0
s
0
uτ − vτA C Dβ uτ−C Dβ vτ dτ ds
s
τ 1−β
ωt−s
≤ MAg e
sup u σ − v σ
dτ ds
uτ − vτA Γ 2 − β 0≤σ≤τ
0
0
T
T 1−β
ωT −s
MAg T
e
ds ρu, v.
≤ max 1, Γ 2−β
0
t
3.11
These three relations 3.8, 3.10, and 3.11 show that, for T small enough, K is indeed a
contraction on C1 , and hence there exists a unique mild solution u ∈ C1 . Furthermore, it is
clear from 3.4, Lemmas 1, and 2 that u ∈ C2 0, T; X and satisfies the problem 1.3.
4. Existence of Mild Solutions in Case of R-L Derivative
In the previous section we proved existence and uniqueness of classical solutions provided
that u0 , u1 ∈ DA × E. From the proof of Theorem 3.1 it can be seen that existence and
uniqueness of mild solutions hold when u0 , u1 ∈ E × X. In case of Riemann-Liouville
fractional derivative one can still prove well-posedness in C1 by passing to the Caputo
fractional derivative with the help of 2.7 with a problem of singularity at zero which may
be solved through a multiplication by an appropriate term of the form tγ . This also will
require u0 , u1 ∈ E × X. Moreover, from the integrofractional-differential equation 2.11 it
is clear that the mild solutions do not have to be continuously differentiable. In this section
8
Abstract and Applied Analysis
we will prove existence and uniqueness of mild solutions for the case of Riemann-Liouville
fractional derivative for a less regular space than E × X. Namely, for 0 < β < 1, we consider
Eβ : x ∈ X : Dβ Ctx is continuous on R
FSβ : v ∈ C0, T : Dβ v ∈ C0, T
4.1
equipped with the norm vβ : vC Dβ vC where · C is the uniform norm in C0, T.
We will use the following assumptions:
H5 f : R → X is continuous,
H6 g : R × R × X × X → X is continuous and Lipschitzian, that is
g t, s, x1 , y1 − g t, s, x2 , y2 ≤ Ag x1 − x2 y1 − y2 ,
4.2
for some positive constant Ag .
The result below is mentioned in 15, Lemma 2.10 see also 15 for functions. Here
we state it and prove it for Bochner integral.
Lemma 4.1. If I 1−α Rtx ∈ C1 0, T, T > 0, then one has
t
Dα
Rt − sx ds 0
t
0
Dα Rt − sx ds lim I 1−α Rtx, x ∈ X, t ∈ 0, T.
t→0
4.3
Proof. By Definition 2.2 and Fubini’s theorem we have
t
D
α
τ
dτ
Rτ − sx ds
α
0 t − τ
0
t
1
d t
Rτ − sx
4.4
ds
α dτ
Γ1 − α dt 0
s t − τ
t
t
1
Rτ − sx
∂ t Rτ − sx
1
lim−
ds
α dτ α dτ.
s
→
t
Γ1 − α 0 ∂t s t − τ
Γ1 − α
s t − τ
d
1
Rt − sx ds Γ1
−
α
dt
0
t
These steps are justified by the assumption I 1−α Rtx ∈ C1 0, T. Moreover, a change of
variable σ τ − s leads to
t
D
α
1
Rt − sx ds Γ1 − α
0
t
∂
ds
∂t
0
1
lim
Γ1 − α t → 0
This is exactly the formula stated in the lemma.
t−s
0
t
0
Rσx
dσ
t − s − σα
Rσx
dσ.
t − σα
4.5
Abstract and Applied Analysis
9
Corollary 4.2. For the sine family St associated with the cosine family Ct one has, for x ∈ X and
t ∈ 0, T
t
D
α
St − sx ds 0
t
D St − sx ds t
α
0
I 1−α Ct − sx ds.
4.6
0
Proof. First, from 2.7, we have
t
d 1−α
S0x
1
−α dSs
α
I Stx D Stx x ds
t − s
dt
Γ1 − α
tα
ds
0
1
Γ1 − α
t
−α
t − s Csx ds I
1−α
4.7
Ctx.
0
Notice that this means that d/dtI 1−α Stx I 1−α Ctx which is in accordance with a general
permutation property valid when the function is 0 at 0 see 10, 15. It also shows that in
this case the Riemann-Liouville derivative and the Caputo derivative are equal. Now from
the continuity of Ct it is clear that I 1−α Ctx is continuous on 0, T actually, the operator
I α has several smoothing properties, see 11 and therefore I 1−α Stx ∈ C1 0, T. We can
therefore apply Lemma 4.1 to obtain
t
Dα
St − sx ds 0
t
0
Dα St − sx ds lim I 1−α Stx,
t→0
x ∈ X, t ∈ 0, T.
4.8
Next, we claim that limt → 0 I 1−α Stx 0. This follows easily from the definition of St and
I 1−α . Indeed, we have
1−α
I Stx ≤
1
Γ1 − α
t
t − s−α |Ssx|ds ≤
0
t1−α
sup |Stx|.
Γ2 − α 0≤t≤T
4.9
We are now ready to state and prove our main result of this section.
Theorem 4.3. Assume that (H1), (H5), and (H6) hold. If u0 , u1 ∈ Eβ × X, then there exists T > 0
and a unique mild solution u ∈ FSβ of problem 1.3 with Riemann-Liouville fractional derivative.
Proof. For t ∈ 0, T, consider the operator
Kut : Ctu0 Stu1 t
0
t
0
St − sfsds
s St − s g s, τ, uτ, Dβ uτ dτ ds.
0
4.10
10
Abstract and Applied Analysis
It is clear that Ku ∈ C0, T; X when u ∈ FSβ . From Corollary 4.2, we see that
D Kut D Ctu0 D Stu1 β
β
t
β
I 1−β Ct − sfsds
0
t
I
1−β
s Ct − s g s, τ, uτ, Dβ uτ dτ ds.
0
4.11
0
Therefore Ku ∈ FSβ and maps FSβ to FSβ because u0 ∈ Eβ ,
Dβ Stu1 d 1−β
I Stu1 C Dβ Stu1 I 1−β Ctu1 ,
dt
4.12
and the integral terms are obviously continuous. For u, v ∈ FSβ , we find
Kut − Kvt
t
≤
t−s
Me
0
dτ
Ag
s
0
MAg T
≤
2
≤
ωτ
MAg T
2
uτ − vτ Dβ uτ − Dβ vτ dτ ds
0
T
2
e
ωτ
dτ
0
β
β
sup ut − vt sup D ut − D vt
0≤t≤T
T
2
4.13
0≤t≤T
eωτ dτ
0
ut − vtβ .
Further,
β
D Ku t − Dβ Kv t
≤
s t
1−β
β
β
I Ct − s
uτ
−
g
s,
τ,
vτ,
D
vτ
dτ
ds
g
s,
τ,
uτ,
D
0
≤ MAg
≤
0
t
1−β ωt−s
t − s e
Γ 2−β
0
T
2−β
MAg T
Γ 2−β
0
s ds sup ut − vt Dβ ut − Dβ vt
4.14
0≤t≤T
eωT −s ds ut − vtβ .
Thus, for T sufficiently small, K is a contraction on the complete metric space FSβ and hence
there exists a unique mild solution to 1.3.
Acknowledgment
The third author is very grateful for the financial support provided by King Fahd University
of Petroleum and Minerals through Project no. IN100007.
Abstract and Applied Analysis
11
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