Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 153, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF MILD SOLUTIONS FOR A NEUTRAL
FRACTIONAL EQUATION WITH FRACTIONAL NONLOCAL
CONDITIONS
NASSER-EDDINE TATAR
Abstract. The existence of mild solutions in an appropriate space is established for a second-order abstract problem of neutral type with derivatives of
non-integer order in the nonlinearity as well as in the initial conditions. We
introduce some new spaces taking into account the minimum requirements of
regularity.
1. Introduction
In this article we study the neutral second-order abstract differential problem
d 0
[u (t) + g(t, u(t))] = Au(t) + f (t, u(t), Dα u(t)), t ∈ I = [0, T ]
dt
(1.1)
u(0) = u0 + p(u, Dβ u(t)),
u0 (0) = u1 + q(u, Dγ u(t))
with 0 ≤ α, β, γ ≤ 1. Here the prime denotes time differentiation and Dκ , κ =
α, β, γ denotes fractional time differentiation (in the sense of Riemann-Liouville).
The operator A is the infinitesimal generator of a strongly continuous cosine family
C(t), t ≥ 0 of bounded linear operators in the Banach space X and f , g are nonlinear
functions from R+ × X × X to X and R+ × X to X, respectively, u0 and u1 are
given initial data in X. The functions p : [C(I; X)]2 → X, q : [C(I; X)]2 → X are
given continuous functions.
This problem has been studied in case α, β, γ are 0 or 1 (see [1, 2, 3, 4, 10,
11, 22]). Well-posedness has been established using different fixed point theorems
and the theory of strongly continuous cosine families in Banach spaces. We refer
the reader to [6, 20, 21] for a good account on the theory of cosine families. A
similar problem to this one with Caputo derivative has been studied by the present
author in [19]. Here the situation with the Riemann-Liouville fractional derivative
is completely different. The singularity at zero inherent to the Riemann-Liouville
derivative brings new challenges and difficulties. Moreover the underlying spaces
are different. Indeed, Caputo derivative needs more regularity as it uses the first
2000 Mathematics Subject Classification. 26A33, 34K40, 35L90, 35L70, 35L15, 35L07.
Key words and phrases. Cauchy problem; cosine family; fractional derivative;
mild solutions; neutral second-order abstract problem.
c
2012
Texas State University - San Marcos.
Submitted May 5, 2012. Published September 7, 2012.
1
2
N.-E. TATAR
EJDE-2012/153
derivative of the function in question in its definition whereas the Riemann-Liouville
does not require as much smoothness.
Problems with fractional derivatives are very convenient to model hereditary
phenomena in many fields of sciences and engineering [5, 7, 8, 9, 12, 14, 15, 16, 17].
They can be used, for instance, as damping to reduce the effect of vibrations in
mechanical structures or to reduce noise in signals.
Here we consider the neutral case (g 6 ≡0) and prove existence and uniqueness
of mild solutions under different conditions on the different data. Even in the case
g ≡ 0 (and p = q = 0) these conditions are different from the ones assumed in [18].
In particular, this work may be viewed as an extension of the works in [10, 11] to
the fractional order case and of (18) to the neutral case.
The next section of this paper contains some notation and preliminary results
needed in our proofs. Section 3 treats the existence of a mild solution in an appropriate “fractional” space.
2. Preliminaries
In this section we present some notation, assumptions and results needed in our
proofs later.
Definition 2.1. The integral
κ
(Ia+
h)(x) =
Z
1
Γ(α)
a
x
h(t)dt
,
(x − t)1−κ
x>a
is called the Riemann-Liouville fractional integral of h of order κ > 0 when the
right side exists.
Here Γ is the usual Gamma function
Z ∞
Γ(z) :=
e−s sz−1 ds,
z > 0.
0
Definition 2.2. The (left hand) Riemann-Liouville fractional derivative of h of
order κ > 0 is defined by
Z x
d
1
h(t)dt
( )n
, x > a, n = [κ] + 1
(Daκ h)(x) =
Γ(n − κ) dx
(x
−
t)κ−n+1
a
whenever the right side is pointwise defined. In particular
Z x
1
h(t)dt
d
κ
(Da h)(x) =
, x > a, 0 < κ < 1
Γ(1 − κ) dx a (x − t)κ
and
(Daκ h)(x) =
1
d
( )2
Γ(2 − κ) dx
Z
x
a
h(t)dt
,
(x − t)κ−1
x > a, 1 < κ < 2.
Lemma 2.3. Let 0 < α, β < 0 and ϕ ∈ L1 (a, b) be such that
I n−α ϕ ∈ AC n ([a, b]) := φ : [a, b] → R and (Dn−1 φ)(x) ∈ AC[a, b] .
Then
α+β
α β
Ia+
Ia+ ϕ = Ia+
ϕ−
n−1
X
k=0
where ϕn+β (x) =
n+β
Ia+
ϕ(x)
(n−k−1)
ϕn+β
(a)
Γ(α − k)
and n = [−β] + 1.
(x − a)α−k−1
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EXISTENCE OF MILD SOLUTIONS
3
If 0 < κ < 1, n = 1 and γ ≤ κ then
κ−γ κ
γ
Ia+
Da+ ϕ = Da+
ϕ−
ϕ1−κ (a)
(x − a)κ−γ−1 .
Γ(κ − γ)
See [12, 13, 14, 16, 17] for more on fractional derivatives and fractional integrals.
We will also need the following lemmas. The first one can be found in [17].
Lemma 2.4. If h(x) ∈ AC n [a, b], α > 0 and n = [α] + 1, then
Z x
n−1
X
h(k) (a)
1
h(n) (t)dt
α
k−α
(Da h)(x) =
(x − a)
+
Γ(1 + k − α)
Γ(n − α) a (x − t)α−n+1
k=0
=:
n−1
X
k=0
h(k) (a)
(x − a)k−α + (C Daα h)(x),
Γ(1 + k − α)
x > a.
The expression (C Daα h)(x) is known as the fractional derivative of h of order α
in the sense of Caputo.
We will assume the following condition.
(H1) A is the infinitesimal generator of a strongly continuous cosine family C(t),
t ∈ R, of bounded linear operators in the Banach space X.
The associated sine family S(t), t ∈ R is defined by
Z t
S(t)x :=
C(s)x ds, t ∈ R, x ∈ X.
0
It is known (see [20, 21, 22]) that there exist constants M ≥ 1 and ω ≥ 0 such that
Z t
ω|t|
|C(t)| ≤ M e , t ∈ R and |S(t) − S(t0 )| ≤ M |
eω|s| ds|, t, t0 ∈ R.
t0
For simplicity we will write |C(t)| ≤ M̃ and |S(t)| ≤ Ñ on I = [0, T ] (of course
M̃ ≥ 1 and Ñ ≥ 1 depend on T ).
Let us define
E := {x ∈ X : C(t)x is once continuously differentiable on R}.
Lemma 2.5 ([20, 21, 22]). Assume (H1) is satisfied. Then
(i) S(t)X ⊂ E, t ∈ R,
(ii) S(t)E ⊂ D(A), t ∈ R,
d
(iii) dt
C(t)x = AS(t)x, x ∈ E, t ∈ R,
d2
(iv) dt2 C(t)x = AC(t)x = C(t)Ax, x ∈ D(A), t ∈ R.
Lemma 2.6 ([20, 21, 22]). Suppose that (H1) holds, v : R → X is a continuously
Rt
differentiable function, and q(t) = 0 S(t − s)v(s) ds. Then, q(t) ∈ D(A), q 0 (t) =
Rt
Rt
C(t − s)v(s) ds and q 00 (t) = 0 C(t − s)v 0 (s) ds + C(t)v(0) = Aq(t) + v(t).
0
Now we make clear what we mean by a mild solution of (1.1).
Definition 2.7. A continuous function u, such that Dη u (η = max{α, β, γ}) exists
and is continuous on I, satisfying the integro-differential equation
u(t) = C(t)[u0 + p(u, Dβ u(t))]
+ S(t)[u1 + q(u, Dγ u(t)) − g(0, u0 + p(u, Dβ u(t)))]
4
N.-E. TATAR
Z
−
t
Z
EJDE-2012/153
t
C(t − s)g(s, u(s))ds +
0
S(t − s)f (s, u(s), Dα u(s)) ds,
t∈I
0
is called a mild solution of problem (1.1).
3. Existence of mild solutions
In this section we prove the existence and uniqueness of a mild solution in the
space
CηRL ([0, T ]) := {v ∈ C([0, T ]) : Dη v ∈ C([0, T ])}
(3.1)
equipped with the norm kvkη := kvkC + kDη vkC where k.kC is the sup norm in
C([0, T ]) and η = max{α, β, γ}. For the initial data we define
Eη := {x ∈ X : Dη C(t)x is continuous on R+ }.
1−ν
(3.2)
1
Lemma 3.1. If R(t) is a linear operator such that I
R(t)x ∈ C ([0, T ]), T > 0,
then, for 0 < ν < 1, we have
Z t
Z t
Dν
R(t − s)x ds =
Dν R(t − s)x ds + lim+ I 1−ν R(t)x, x ∈ X, t ∈ [0, T ].
0
t→0
0
Proof. By Definition 2.2 and Fubini’s theorem we have
Z t
Dν
R(t − s)x ds
0
Z
Z τ
dτ
d t
1
R(τ − s)x ds
=
Γ(1 − ν) dt 0 (t − τ )ν 0
Z
Z t
d t
R(τ − s)x
1
dτ
ds
=
Γ(1 − ν) dt 0
(t − τ )ν
s
Z t
Z t
Z t
1
1
∂
R(τ − s)x
R(τ − s)x
=
dτ
+
dτ.
ds
lim
Γ(1 − ν) 0
∂t s (t − τ )ν
Γ(1 − ν) s→t− s (t − τ )ν
These steps are justified by the assumption I 1−ν R(t)x ∈ C 1 ([0, T ]). Moreover, a
change of variable σ = τ − s leads to
Z t
Dν
R(t − s)x ds
0
Z t
Z t−s
Z t
1
∂
R(σ)x
1
R(σ)x
=
dσ
+
lim
dσ.
ds
Γ(1 − ν) 0
∂t 0
(t − s − σ)ν
Γ(1 − ν) t→0+ 0 (t − σ)ν
This is exactly the formula stated in the lemma.
Lemma 3.2. If g is a continuous function such that I 1−ν g(t) ∈ C 1 ([0, T ]), T > 0
and R(t) is continuous, then, for 0 < ν < 1, we have
Z t
Z t
ν
D
R(t − s)g(s) ds =
R(t − s)Dν g(s) ds, t ∈ [0, T ].
0
0
Proof. By Definition 2.2, we have
Z t
Z
d 1−ν t
ν
D
R(t − s)g(s) ds = I
R(t − s)g(s) ds
dt
0
0
Z t
d
= I 1−ν
R(s)g(t − s) ds
dt
0
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EXISTENCE OF MILD SOLUTIONS
=
d
1
Γ(1 − ν) dt
t
Z
0
dτ
(t − τ )ν
Z
5
τ
R(s)g(τ − s) ds.
0
Then, Fubini’s theorem and the continuity of g allow us to write
Z t
Z
Z t
d t
1
g(τ − s)
ν
dτ ds
D
R(t − s)g(s) ds =
R(s)
ν
Γ(1
−
ν)
dt
0
0
s (t − τ )
Z t
Z t−s
1
d
g(σ)
=
dσds
R(s)
Γ(1 − ν) dt 0
(t − s − σ)ν
0
Z t
d
=
R(s) I 1−ν g(t − s) ds
dt
0
Z t
=
R(s)Dν g(t − s) ds
0
which is the desired relation. Note that we have used the continuity of g to deduce
that the value (or the limit) of the inner integral in the second line of the relation
is zero at s = t.
Corollary 3.3. For the sine family S(t) associated with the cosine family C(t),
x ∈ X, t ∈ [0, T ] and 0 < ν < 1, we have
Z t
Z t
Z t
Dν
S(t − s)x ds =
Dν S(t − s)x ds =
I 1−ν C(t − s)x ds.
0
0
0
Proof. First, from Lemma 2.4 as S(t)x is absolutely continuous on [0, T ], we have
h S(0)x Z t
i
d 1−ν
1
−ν dS(s)
(t
−
s)
I
S(t)x = Dν S(t)x =
+
x
ds
dt
Γ(1 − ν)
tν
ds
0
Z t
1
(t − s)−ν C(s)x ds
=
Γ(1 − ν) 0
= I 1−ν C(t)x.
Now from the continuity of C(t) it is clear that I 1−ν C(t)x is continuous on [0, T ]
and therefore I 1−ν S(t)x ∈ C 1 ([0, T ]). We can therefore apply Lemma 2.6 to obtain
Z t
Z t
Dν
S(t − s)x ds =
Dν S(t − s)x ds + lim+ I 1−ν S(t)x, x ∈ X, t ∈ [0, T ].
0
t→0
0
1−ν
S(t)x = 0. This follows easily from
Next, we claim that limt→0+ I
Z t
1
t1−ν
|I 1−ν S(t)x| ≤
(t − s)−ν |S(s)x|ds ≤
sup |S(t)x|.
Γ(1 − ν) 0
Γ(2 − ν) 0≤t≤T
On the functins f, g, p and q we assume the following conditions.
(H2) (i) f (t, ., .) : X × X → X is continuous for a.e. t ∈ I.
(ii) For every (x, y) ∈ X × X, the function f (., x, y) : I → X is strongly
measurable.
(iii) There exists a nonnegative continuous function Kf (t) and a continuous
nondecreasing positive function Ωf such that
kf (t, x, y)k ≤ Kf (t)Ωf (kxk + kyk)
for (t, x, y) ∈ I × X × X.
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N.-E. TATAR
EJDE-2012/153
(iv) For each r > 0, the set f (I × Br (0, X 2 )) is relatively compact in X.
(H3 (i) g ∈ CηRL ([0, T ]).
(ii) There exist a nonnegative continuous function Kg (t) and a continuous
nondecreasing positive function Ωg such that
kg(t, x)kη ≤ Kg (t)Ωg (kxk)
for (t, x) ∈ I × X.
(iii) The family of functions {t → g(t, u) u ∈ Br (0, C(I; X))} is equicontinuous on I.
(iv) For each r > 0, the set g(I × Br (0, X)) is relatively compact in X.
(H4) u0 + p : [C(I; X)]2 → Eη (takes its values in Eη , see (3.2)) and q :
[C(I; X)]2 → X are completely continuous.
The positive constants Np , Nq , Ñp and Ng will denote bounds for ku0 + p(u, v)k,
kq(u, v)k, kDη C(t)[u0 + p(u, I 1−β v(t))]k and the term kg(0, u0 + p(u, I η−β v(t)))k,
respectively. Note that kDη C(t)[u0 + p(u, I 1−β v(t))]k is finite by the assumption
in the next theorem. By Br (x, X) we will denote the closed ball in X centered at
x and of radius r. Let also
M̃ T 1−η 1
M̃ T 1−η Nq + Ñ +
(ku k + Ng ) + Ñp
(3.3)
C1 := M̃ Np + Ñ +
Γ(2 − η)
Γ(2 − η)
n
o
T η−α
C3 := max 1,
, C2 := C1 C3
(3.4)
Γ(η − α + 1)
H(t) := 2M̃ Kg (t) + Ñ Kf (t) + M̃ (I 1−η Kf )(t).
(3.5)
We are now ready to state and prove our main result.
Theorem 3.4. Assume that (H1)–(H4) hold. If 0 < η < 1 and
Z ∞
Z t
1
ds
H(s) ds <
C
Ω
(s)
+ Ωg (s)
3
f
0
C2
then problem (1.1) admits a mild solution u ∈ CηRL ([0, T ]).
Proof. Note that by our assumptions and for u ∈ CηRL ([0, T ]) (see (3.1)), the maps
Φ(u, v)(t) := C(t)[u0 + p(u, I η−β v(t))]
+ S(t)[u1 + q(u, I η−γ v(t)) − g(0, u0 + p(u, I η−β v(t)))]
(3.6)
Z t
Z t
η−α
C(t − s)g(s, u(s))ds +
S(t − s)f (s, u(s), I
v(s)) ds,
−
0
0
for t ∈ I, and
Ψ(u, v)(t) := Dη C(t)[u0 + p(u, I η−β v(t))]
+ Dη S(t)[u1 + q(u, I η−γ v(t)) − g(0, u0 + p(u, I η−β v(t)))]
Z t
C(t − s)Dη g(s, u(s)) ds
−
0
Z t
+
I 1−η C(t − s)f (s, u(s), I η−α v(s)) ds, t ∈ I
(3.7)
0
are well-defined, and map [C([0, T ])]2 into C([0, T ]). We obtained these mappings
from the definition of a mild solution taking into account Lemma 3.2 (note that
EJDE-2012/153
EXISTENCE OF MILD SOLUTIONS
7
I 1−η g ∈ C 1 ([0, T ]) by our assumption (H3)(i)) and Corollary 3.3. In addition to
that we passed from Dκ u, κ = α, β, γ to Dη u =: v through the formula
η−κ η
κ
Da+
u(t) = Ia+
Da+ u(t)(t) +
u1−η (0) η−κ−1
t
Γ(η − κ)
and noticing that u1−η (0) = 0.
We would like to apply the Leray-Schauder Alternative (which states that either
the set of solutions (below) is unbounded or we have a fixed point in D (containing
zero) a convex subset of X provided that the mappings Φ and Ψ are completely
continuous). To this end we first prove that the set of solutions (uλ , vλ ) of
(uλ , vλ ) = λ(Φ(uλ , vλ ), Ψ(uλ , vλ )), 0 < λ < 1
(3.8)
is bounded. Then, we prove that this map is completely continuous. Therefore
there remains the alternative which is the existence of a fixed point. We observe
first from (3.6) that
Z t
kuλ (t)k ≤ M̃ Np + Ñ (ku1 k + Nq + Ng ) + M̃
Kg (s)Ωg (kuλ (s)k) ds
0
Z t
sη−α
+ Ñ
Kf (s)Ωf (kuλ (s)k +
sup kvλ (τ )k) ds
Γ(η − α + 1) 0≤τ ≤s
0
and from (3.7),
kvλ (t)k
Z t
M̃ t1−η
1
≤ Ñp +
[ku k + Nq + Ng ] + M̃
Kg (s)Ωg (kuλ (s)k) ds
Γ(2 − η)
0
Z t
sη−α
+ M̃
(I 1−η Kf )(s)Ωf sup kuλ (τ )k +
sup kvλ (τ )k ds
Γ(η − α + 1) 0≤τ ≤s
0≤τ ≤s
0
where Ñp and Ng are bounds for the expressions Dη C(t)[u0 + p(u, I η−β v(t))] and
g(0, u0 + p(u, I η−β v(t))), respectively. Clearly
kuλ (t)k + kvλ (t)k
Z t
Z t
≤ C1 + 2M̃
Kg (s)Ωg (kuλ (s)k) ds +
(Ñ Kf (s) + M̃ (I 1−η Kf )(s))
0
× Ωf
0
sη−α
sup kvλ (τ )k ds
sup kuλ (τ )k +
Γ(η − α + 1) 0≤τ ≤s
0≤τ ≤s
where C1 is given in (3.3). If we put
Θλ (t) = max{1, T η−α /Γ(η − α + 1)} sup (kuλ (τ )k + kvλ (τ )k),
0≤τ ≤t
(3.9)
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N.-E. TATAR
EJDE-2012/153
then (3.9) yields
Z t
n
Θλ (t) ≤ max{1, T η−α /Γ(η − α + 1)} C1 + 2M̃
Kg (s)Ωg (Θλ (s)) ds
0
Z t
o
+
Ñ Kf (s) + M̃ (I 1−η Kf )(s) Ωf (Θλ (s)) ds
0
Z t
o
n
T η−α
≤ max 1,
C1 +
H(s)[Ωg (Θλ (s)) + Ωf (Θλ (s))]ds
Γ(η − α + 1)
0
Z t
≤ C2 + C3
H(s)[Ωg (Θλ (s)) + Ωf (Θλ (s))]ds
0
(3.10)
where H(s), C2 and C3 are as in the paragraph preceding the statement of the
theorem (see (3.4) and (3.5)). Let us denote by ϕλ (t) the right hand side of (3.10).
Then ϕλ (0) = C2 , Θλ (t) ≤ ϕλ (t) and
ϕ0λ (t) ≤ C3 H(t)[Ωg (ϕλ (t)) + Ωf (ϕλ (t))],
t ∈ I.
We infer that
Z t
ds
≤ C3
H(s) ds.
(3.11)
Ωf (s) + Ωg (s)
0
C2
This relation, together with our hypotheses, shows that Θλ (t) and thereafter the
set of solutions of (3.8) is bounded in [C(I; X)]2 .
It remains to show that the maps Φ and Ψ are completely continuous. From our
hypotheses it is immediate that
Z
ϕλ (t)
Φ1 (u, v)(t) := C(t)[u0 + p(u, I η−β v(t))]
+ S(t)[u1 + q(u, I η−γ v(t)) − g(0, u0 + p(u, I η−β v(t)))]
is completely continuous. To apply Ascoli-Arzela Theorem we need to check that
(Φ − Φ1 )(Br2 ) := {(Φ − Φ1 )(u, v) : (u, v) ∈ Br2 }
is equicontinuous on I. Let us observe that
k(Φ − Φ1 )(u, v)(t + h) − (Φ − Φ1 )(u, v)(t)k
Z t
≤
k(C(t + h − s) − C(t − s))g(s, u(s))kds
0
Z
t+h
kC(t + h − s)g(s, u(s))kds
+
t
Z
+
t
k(S(t + h − s) − S(t − s))f (s, u(s), I η−α v(s))kds
0
Z
+
t+h
kS(t + h − s)f (s, u(s), I η−α v(s))kds.
t
By (H1) and (H3), for t ∈ I and ε > 0 given, there exists δ > 0 such that
k(C(s + h) − C(s))g(t − s, u(t − s))k < ε
for s ∈ [0, t] and u ∈ Br , when |h| < δ. This together with (H2), (H3) and the fact
that S(t) is Lipschitzian imply that
k(Φ − Φ1 )(u, v)(t + h) − (Φ − Φ1 )(u, v)(t)k
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EXISTENCE OF MILD SOLUTIONS
Z
≤ εt + M̃ Ωg (r)
9
t+h
Kg (s) ds
t
Z
T η−α r t
Kf (s) ds
+ N1 hΩf r +
Γ(η − α + 1) 0
Z
T η−α r t+h
Kf (s) ds
+ Ñ Ωf r +
Γ(η − α + 1) t
for some positive constant N1 . The equicontinuity is therefore established.
On the other hand, for t ∈ I, as (s, ξ) → C(t − s)ξ is continuous from [0, t] ×
g(I × X) to X and [0, t] × g(I × X) is relatively compact in X. The set
Z t
Φ2 u(t) :=
C(t − s)g(s, u(s)) ds, u ∈ Br (0, X)
0
is relatively compact in X as well. We infer that Φ2 is completely continuous. As
for Φ3 := Φ − Φ1 + Φ2 we decompose it as follows
k−1
X Z si+1
Φ3 (u, v)(t) =
(S(s) − S(si ))f (t − s, u(t − s), I η−α v(t − s)) ds
i=1
+
si
k−1
X Z si+1
i=1
S(si )f (t − s, u(t − s), I η−α v(t − s)) ds
si
and select the partition {si }ki=1 of [0, t] in such a manner that, for a given ε > 0
k(S(s) − S(s0 ))f (t − s, u(t − s), I η−α v(t − s))k < ε,
for (u, v) ∈ Br2 (0, X), when s, s0 ∈ [si , si+1 ] for some i = 1, . . . , k − 1. This is
possible in as much as
{f (t − s, u(t − s), I η−α v(t − s)), s ∈ [0, t], (u, v) ∈ Br2 (0, X)}
is bounded (by (H2) (iii)) and the operator S is uniformly Lipschitz on I. This
leads to
k−1
X
Φ3 (u, v)(t) ∈ εBT (0, X) +
(si+1 − si )co(U (t, si , r))
i=1
where
U (t, si , r)
:= S(si )f (t − s, u(t − s), I η−α v(t − s)), s ∈ [0, t], (u, v) ∈ Br2 (0, X)
and co U (t, si , r)) designates its convex hull. Therefore Φ3 (Br2 )(t) is relatively compact in X. By Ascoli-Arzela Theorem, Φ3 (Br2 ) is relatively compact in C(I; X)
and consequently Φ3 is completely continuous. Similarly we may prove that Ψ is
completely continuous.
We conclude that (Φ, Ψ) admits a fixed point in [C([0, T ])]2 .
Remark 3.5. In the same way we may treat the more general case
d 0
[u (t) + g(t, u(t))] = Au(t) + f (t, u(t), Dα1 u(t), . . . , Dαn u(t)),
dt
u(0) = u0 + p(u, Dβ1 u(t), . . . , Dβm u(t)),
u0 (0) = u1 + q(u, Dγ1 u(t), . . . , Dγr u(t))
10
N.-E. TATAR
EJDE-2012/153
where 0 ≤ αi , βj , γk ≤ 1, i = 1, . . . , n, j = 1, . . . , m, k = 1, . . . , r.
Example. Consider the problem
∂
[ut (t, x) + G(t, x, u(t, x))]
∂t
= uxx (t, x) + F (t, x, u(t, x), Dα u(t, x)),
t ∈ I = [0, T ], x ∈ [a, b]
u(t, a) = u(t, b) = 0, t ∈ I
Z T
0
u(0, x) = u (x) +
P (u(s), Dβ u(s))(x) ds,
u0 (0) = u1 (x) +
Z
(3.12)
x ∈ [a, b]
0
T
Q(u(s), Dγ u(s))(x) ds,
x ∈ [a, b]
0
in the space X = L2 ([0, π]). This problem can be reformulated in the abstract
setting (1.1). To this end we define the operator Ay = y 00 with domain
D(A) := {y ∈ H 2 ([0, π]) : y(0) = y(π) = 0}.
2
The operator
p A has a discrete spectrum with −n , n = 1, 2, . . . . as eigenvalues and
zn (s) = 2/π sin(ns), n = 1, 2, . . . , as their corresponding normalized eigenvectors.
So we may write
∞
X
n2 (y, zn )zn , y ∈ D(A).
Ay = −
n=1
Since −A is positive and self-adjoint in L2 ([0, π]), the operator A is the infinitesimal
generator of of a strongly continuous cosine family C(t), t ∈ R which has the form
C(t)y =
∞
X
cos(nt)(y, zn )zn ,
y ∈ X.
∞
X
sin(nt)
(y, zn )zn ,
n
n=1
y ∈ X.
n=1
The associated sine family is
C(t)y =
One can also consider more general non-local conditions by allowing the Lebesgue
measure ds to be of the form dµ(s) and dη(s) for non-decreasing functions µ and η
(or even more general: µ and η of bounded variation); that is,
Z T
u(0, x) = u0 (x) +
P (u(s), Dβ u(s))(x)dµ(s),
0
ut (0, x) = u1 (x) +
T
Z
Q(u(s), Dγ u(s))(x)dη(s).
0
These (continuous) non-local conditions cover, of course, the discrete cases
u(0, x) = u0 (x) +
n
X
αi u(ti , x) +
m
X
i=1
i=1
ut (0, x) = u1 (x) +
r
X
i=1
βi Dβ u(ti , x),
γi u(ti , x) +
k
X
λi Dγ u(ti , x)
i=1
which have been extensively studied by several authors in the integer order case.
EJDE-2012/153
EXISTENCE OF MILD SOLUTIONS
11
For u, v ∈ C([0, T ]; X) and x ∈ [a, b], defining the operators
Z T
p(u, v)(x) :=
P (u(s), v(s))(x) ds,
0
Z
q(u, v)(x) :=
T
Q(u(s), v(s))(x) ds,
0
g(t, u)(x) := G(t, x, u(t, x)),
f (t, u, v)(x) := F (t, x, u(t, x), v(t, x)),
allows us to write (3.12) abstractly as
d 0
[u (t) + g(t, u(t), u0 (t))] = Au(t) + f (t, u(t), Dα u(t)),
dt
u(0) = u0 + p(u, Dβ u(t)),
u0 (0) = u1 + q(u, Dγ u(t)).
Under appropriate conditions on F , G, P and Q which make (H2)–(H4) hold for
the corresponding functions f , g, p and q, Theorem 3.4 ensures the existence of a
mild solution to problem (3.12).
Some special cases of this problem may be found in models of some phenomena
with hereditary properties (see [5, 7, 8, 9, 15]).
Acknowledgments. The author is very grateful for the financial support provided
by King Fahd University of Petroleum and Minerals through the project No. IN
100007.
References
[1] M. Benchohra and S. K. Ntouyas; Existence of mild solutions of second order initial value
problems for delay integrodifferential inclusions with nonlocal conditions, Mathematica Bohemica, 4 (127) (2002), 613-622.
[2] M. Benchohra and S. K. Ntouyas; Existence results for the semi-infinite interval for first and
second order integrodifferential equations in Banach spaces with nonlocal conditions, Acta
Univ. Palacki. Olomuc, Fac. Rer. Nat. Mathematica 41 (2002), 13-19.
[3] M. Benchohra and S. K. Ntouyas; Existence results for multivalued semilinear functional
differential equations, Extracta Mathematicae, 18 (1) (2003), 1-12.
[4] L. Byszewski and V. Laksmikantham; Theorems about the existence and uniqueness of a
solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal. 40 (1) (1991),
11-19.
[5] W. Chen and S. Holm; Modified Szabo’s wave equation models for lossy media obeying frequency power law, J. Acoust. Soc. Am. 114 (5) (2003), 2570-2575.
[6] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland
Mathematics Studies, Vol. 108, North-Holland, Amsterdam, 1985.
[7] M. Fukunaga and N. Shimizu; Role of prehistories in the initial value problem of fractional
viscoelastic equations, Nonlinear Dynamics 38 (2004), 207-220.
[8] H. Haddar, J.-R. Li and D. Matignon; Efficient solution of a wave equation with fractionalorder dissipative terms, J. Comput. Appl. Math. 234 (6) (2010), 2003-2010.
[9] K. S. Hedrich and A. Filipovski; Longitudinal creep vibrations of a fractional derivative order
rheological rod with variable cross section, Facta Universitas, Series: Mechanics, Automatic
Control and Robotics 3 (12) (2002), 327-349.
[10] M. E. Hernandez; Existence results for a second order abstract Cauchy problem with nonlocal
conditions, Electr. J. Diff. Eqs, 2005 No. 73 (2005), 1-17.
[11] E. M. Hernandez, H. R. Henrı́quez and M. A. McKibben; Existence of solutions for second
order partial neutral functional differential equations, Integr. Equ. Oper. Theory 62 (2008),
191-217.
12
N.-E. TATAR
EJDE-2012/153
[12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo; Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Editor: Jan van Mill, Elsevier,
Amsterdam, The Netherlands 2006.
[13] K. S. Miller and B. Ross; An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
[14] K. B. Oldham and J. Spanier; The Fractional Calculus, Academic Press, New York-London,
1974.
[15] E. Orsingher and L. Beghin; Time-fractional telegraph equations and telegraph processes with
Brownian time, Probab. Theory Relat. Fields 128 (2004), 141-160.
[16] I. Podlubny; Fractional Differential Equations, Mathematics in Sciences and Engineering,
198, Academic Press, San-Diego, 1999.
[17] S. G. Samko, A. A. Kilbas and O. I. Marichev; Fractional Integrals and Derivatives. Theory
and Applications, Gordon and Breach, Yverdon, 1993.
[18] N.-e. Tatar; The existence of mild and classical solutions for a second-order abstract fractional problem, Nonl. Anal. T.M.A. 73 (2010), 3130-3139.
[19] N.-e. Tatar; Mild solutions for a problem involving fractional derivatives in the nonlinearity
and in the nonlocal conditions, Adv. Diff. Eqs. 2011 No.18 (2011), 1-12.
[20] C. C. Travis and G. F. Webb; Compactness, regularity and uniform continuity properties of
strongly continuous cosine families, Houston J. Math. 3 (4) (1977), 555-567.
[21] C. C. Travis and G. F. Webb; Cosine families and abstract nonlinear second order differential
equations, Acta Math. Acad. Sci. Hungaricae, 32 (1978), 76-96.
[22] C. C. Travis and G. F. Webb; An abstract second order semilinear Volterra integrodifferential
equation, SIAM J. Math. Anal. 10 (2) (1979), 412-424.
Nasser-eddine Tatar
King Fahd University of Petroleum and Minerals, Department of Mathematics and
Statistics, Dhahran 31261, Saudi Arabia
E-mail address: tatarn@kfupm.edu.sa