Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 265, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS WITH
INFINITE DELAY
KHALIDA AISSANI, MOUFFAK BENCHOHRA
Abstract. In this article, we apply Bohnenblust-Karlin’s fixed point theorem
to prove the existence of mild solutions for a class of impulsive fractional
equations inclusions with infinite delay. An example is given to illustrate the
theory.
1. Introduction
Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications of fractionalorder derivatives in the mathematical modeling of physical and biological phenomena in various fields of science and engineering. For details, including some applications and recent results, see the monographs of Abbas et al. [1], Baleanu et al. [8],
Diethelm [22], Hilfer [26], Kilbas et al. [27], Lakshmikantham et al. [29], Podlubny
[33], and Tarasov [38].
On the other hand, the theory of impulsive differential equations appear frequently in applications because many evolutionary process from fields as physics,
aeronautic, economics, engineering, population dynamics, etc. (see the monographs
of Bainov and Simeonov [7], Benchohra et al. [13], Lakshmikantham et al. [28], and
Samoilenko and Perestyuk [36] and the papers [15, 35]).
Fractional differential inclusions arise in the mathematical modeling of certain
problems in economics, optimal control, etc. and are widely studied by many authors, see [5, 11, 14, 19, 32] and the references therein. For some recent development
on fractional differential inclusions, we refer the reader to the papers [3, 4, 12]. Recently, Benchohra et al. [10] studied the existence of solutions of differential inclusions with Riemann-Liouville fractional derivative. Cernea [17, 18] established some
Filippov type existence theorems for solutions of fractional semilinear differential
inclusions involving Caputo’s fractional derivative in Banach spaces.
Motivated by the papers cited above, in this paper, we consider the existence of
a class of impulsive fractional differential inclusions with infinite delay described by
the form
C
Dtα x(t) − Ax(t) ∈ F (t, xt , x(t)),
t ∈ J = (tk , tk+1 ], k = 0, . . . , m,
(1.1)
2000 Mathematics Subject Classification. 26A33, 34A08, 34A37, 34A60, 34G20, 34H05, 34K09.
Key words and phrases. Impulsive fractional differential inclusions; α-resolvent family;
Caputo fractional derivative; mild solution; multivalued map; fixed point; Banach space.
c
2013
Texas State University - San Marcos.
Submitted September 10, 2013. Published November 30, 2013.
1
2
K. AISSANI, M. BENCHOHRA
∆x(tk ) = Ik (x(t−
k )),
EJDE-2013/265
k = 1, 2, . . . , m,
t ∈ (−∞, 0],
x(t) = φ(t),
(1.2)
(1.3)
where C Dtα is the Caputo fractional derivative of order 0 < α < 1, T > 0, A :
D(A) ⊂ E → E is the infinitesimal generator of an α-resolvent family (Sα (t))t≥0 ,
F : J × B × E → P(E) is a multivalued map (P(E) is the family of all nonempty
subsets of E). Here, 0 = t0 < t1 < · · · < tm < tm+1 = T , Ik : E → E, k =
−
+
1, 2, . . . , m, are multivalued maps, ∆x(tk ) = x(t+
k )−x(tk ), x(tk ) = limh→0 x(tk +h)
−
and x(tk ) = limh→0 x(tk − h) represent the right and the left limit of x(t) at t = tk ,
respectively. We denote by xt the element of B defined by xt (θ) = x(t + θ), θ ∈
(−∞, 0]. Here xt represents the history of the state from −∞ up to the present
time t. We assume that the histories xt belongs to some abstract phase space B,
to be specified later, and φ ∈ B.
2. Preliminaries
We will briefly recall some basic definitions and facts from multivalued analysis
that we will use in the sequel.
Let (E, k · k) be a complex Banach space. Let C = C(J, E) be the Banach space
of continuous functions from J into E with the norm
kykC = sup{ |y(t)| : t ∈ J }.
Let L(E) be the Banach space of all linear and bounded operators on E. Let
L1 (J, E) be the space of E-valued Bochner integrable functions on J with the norm
Z T
kykL1 =
ky(t)kdt.
0
Denote
Pcl (E) = {Y ∈ P (E) : Y closed},
Pb (E) = {Y ∈ P (E) : Y bounded },
Pcp (E) = {Y ∈ P (E) : Y compact},
Pcp,c (E) = {Y ∈ P (E) : Y compact and convex}.
A multivalued map G : E → P (E) is convex (closed) valued if G(E) is convex
(closed) for all x ∈ E. G is bounded on bounded sets if G(B) = ∪x∈B G(x) is
bounded in E for all B ∈ Pb (E) (i.e. supx∈B {sup{kyk : y ∈ G(x)}} < ∞).
G is called upper semi-continuous (u.s.c.) on E if for each x0 ∈ E the set G(x0 )
is a nonempty, closed subset of E, and if for each open set U of E containing G(x0 ),
there exists an open neighborhood V of x0 such that G(V ) ⊆ U .
A map G is said to be completely continuous if G(B) is relatively compact
for every B ∈ Pb (E). If the multivalued map G is completely continuous with
nonempty compact values, then G is upper semi continuous (u.s.c.) if and only if
G has a closed graph (i.e. xn → x∗ , yn → y∗ , yn ∈ G(xn ) imply y∗ ∈ G(x∗ )). For
more details on multivalued maps see the books of Deimling [21], and Górniewicz
[23].
Definition 2.1. The multivalued map F : J × B × E → P(E) is said to be
Carathéodory if
(i) t 7→ F (t, v, w) is measurable for each (v, w) ∈ B × E;
(ii) (v, w) 7→ F (t, v, w) is upper semicontinuous for almost all t ∈ J.
EJDE-2013/265
IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS
3
We need some basic definitions and properties of the fractional calculus theory
which are used further in this paper.
Definition 2.2. Let α > 0 and f : R+ → E be in L1 (R+ , E). Then the RiemannLiouville integral is given by:
Z t
1
f (s)
α
It f (t) =
ds.
Γ(α) 0 (t − s)1−α
Recall that the Laplace transform of a function f ∈ L1 (R+ , E) is defined by
Z ∞
fb(λ) =
e−λt f (t)dt, Re(λ) > ω,
0
if the integral is absolutely convergent for Re(λ) > ω. For more details on the
Riemann-Liouville fractional derivative, we refer the reader to [20].
Definition 2.3. [33] The Caputo derivative of order α for a function f : [0, +∞) →
R can be written as
Z t
1
f (n)(s)
Dtα f (t) =
ds = I n−α f (n) (t), t > 0, n − 1 ≤ α < n.
Γ(n − α) 0 (t − s)α+1−n
If 0 < α ≤ 1, then
Z t
1
f 0 (s)
ds.
=
Γ(1 − α) 0 (t − s)α
Obviously, The Caputo derivative of a constant is equal to zero. The Laplace
transform of the Caputo derivative of order α > 0 is
Dtα f (t)
L{Dtα f (t), λ} = λα fb(λ) −
n−1
X
λα−k−1 f (k) (0),
n − 1 ≤ α < n, n ∈ N.
k=0
To define the mild solution of the problems (1.1)–(1.3) we recall the following
definition.
Definition 2.4. A closed and linear operator A is said to be sectorial if there are
constants ω ∈ R, θ ∈ [ π2 , π], M > 0, such that the following two conditions are
satisfied:
P
(1) ρ(A) ⊂ (θ,ω) := {λ ∈ C : λ 6= ω, |arg(λ − ω)| < θ}.
P
M
(2) kR(λ, A)kL(E) ≤ |λ−ω|
, λ ∈ (θ,ω) .
Sectorial operators are well studied in the literature. For details see [24].
Definition 2.5. [6] If A is a closed linear operator with domain D(A) defined on
a Banach space E and α > 0, then we say that A is the generator of an α-resolvent
family if there exists ω ≥ 0 and a strongly continuous function Sα : R+ →L(E)
such that {λα : Re(λ) > ω} ⊂ ρ(A)) (ρ(A) being the resolvent set of A) and
Z ∞
(λα I − A)−1 x =
e−λt Sα (t)xdt, Re λ > ω, x ∈ E.
0
In this case, Sα (t) is called the α-resolvent family generated by A.
Definition 2.6 ([2, Def. 2.1]). if A is a closed linear operator with domain D(A)
defined on a Banach space E and α > 0, then we say that A is the generator
4
K. AISSANI, M. BENCHOHRA
EJDE-2013/265
of a solution operator if there exist ω ≥ 0 and a strongly continuous function
Sα : R+ →L(E) such that {λα : Re(λ) > ω} ⊂ ρ(A) and
Z ∞
λα−1 (λα I − A)−1 x =
e−λt Sα (t)xdt, Re λ > ω, x ∈ E,
0
in this case, Sα (t) is called the solution operator generated by A. For more details
see [31, 34].
In this article, we will employ an axiomatic definition for the phase space B
which is similar to those introduced by Hale and Kato [25]. Specifically, B will be a
linear space of functions mapping (−∞, 0] into E endowed with a seminorm k · kB ,
and satisfies the following axioms:
(A1) If x : (−∞, T ] → E is such that x0 ∈ B, then for every t ∈ J, xt ∈ B and
kx(t)k ≤ Ckxt kB ,
(2.1)
where C > 0 is a constant.
(A2) There exist a continuous function C1 (t) > 0 and a locally bounded function
C2 (t) ≥ 0 in t ≥ 0 such that
kxt kB ≤ C1 (t) sup kx(s)k + C2 (t)kx0 kB ,
(2.2)
s∈[0,t]
for t ∈ [0, T ] and x as in (A1).
(A3) The space B is complete.
Now we state the following lemmas which are necessary to establish our main result.
Let SF,x be a set defined by
SF,x = {v ∈ L1 (J, E) : v(t) ∈ F (t, xt , x(t)) a.e. t ∈ J}.
Lemma 2.7 ([30]). Let E be a Banach space. Let F : J × B × E → Pcp,c (E) be an
L1 -Carathéodory multivalued map and let Ψ be a linear continuous mapping from
L1 (J, E) to C(J, E), then the operator
Ψ ◦ SF : C(J, E) → Pcp,c (C(J, E)),
x
7→ (Ψ ◦ SF )(x) := Ψ(SF,x )
is a closed graph operator in C(J, E) × C(J, E).
The next result is known as the Bohnenblust-Karlin’s fixed point theorem.
Lemma 2.8 ([16]). Let E be a Banach space and D ∈ Pcl,c (E). Suppose that the
operator G : D → Pcl,c (D) is upper semicontinuous and the set G(D) is relatively
compact in E. Then G has a fixed point in D.
3. Main results
In this section we shall present and prove our main result. Before going further
we need the following lemma [37].
Lemma 3.1. Consider the Cauchy problem
Dtα x(t) = Ax(t) + F (t),
x(0) = x0 ,
0 < α < 1,
(3.1)
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IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS
5
if f satisfies the uniform Holder condition with exponent β ∈ (0, 1] and A is a
sectorial operator, then the unique solution of the Cauchy problem (3.1) is
Z t
Sα (t − s)F (s)ds,
x(t) = Tα (t)x0 +
0
where
Tα (t) =
1
2πi
Z
eλt
Bˆr
λα−1
dλ,
λα − A
Sα (t) =
1
2πi
Z
eλt
Bˆr
λα
1
dλ,
−A
B̂r denotes the Bromwich path. Sα (t) is called the α-resolvent family and Tα (t) is
the solution operator, generated by A.
Theorem 3.2 ([9, 37]). If α ∈ (0, 1) and A ∈ Aα (θ0 , ω0 ), then for any x ∈ E and
t > 0, we have
kTα (t)kL(E) ≤ M eωt ,
kSα (t)kL(E) ≤ Ceωt (1 + tα−1 ),
t > 0, ω > ω0 .
Let
fT = sup kTα (t)kL(E) ,
M
0≤t≤T
fs = sup Ceωt (1 + tα−1 ),
M
0≤t≤T
so we have
fT ,
kTα (t)kL(E) ≤ M
fs .
kSα (t)kL(E) ≤ tα−1 M
Let us consider the set of functions
n
B1 = x : (−∞, T ] → E such that x|Jk ∈ C(Jk , E) and there exist
o
−
−
x(t+
)
and
x(t
)
with
x(t
)
=
x(t
),
x
=
φ,
k
=
1,
2,
.
.
.
,
m
.
k
0
k
k
k
Endowed with the seminorm
kxkB1 = sup{|x(s)| : s ∈ [0, T ]} + kφkB , x ∈ B1 ,
where x|Jk is the restriction of x to Jk = (tk , tk+1 ], k = 1, 2, . . . , m.
From Lemma 3.1, we can define the mild solution of system (1.1) as follows.
Definition 3.3. A function x : (−∞, T ] → E is called a mild solution of (1.1)-(1.3)
if the following holds: x0 = φ ∈ B on (−∞, 0], ∆x|t=tk = Ik (x(t−
k )), k = 1, 2, . . . , m,
the restriction of x(·) to the interval Jk , (k = 0, 1, . . . , m) is continuous and there
exists v(·) ∈ L1 (Jk , E), such that v(t) ∈ F (t, xt , x(t)) a.e. t ∈ [0, T ], and x satisfies
the integral equation


φ(t),
t ∈ (−∞, 0];


Rt



t ∈ [0, t1 ];
 0 Sα (t − s)v(s)ds,
Rt
−
x(t) = Tα (t − t1 )(x(t−
)
+
I
(x(t
)))
+
S
(t
−
s)v(s)ds,
t ∈ (t1 , t2 ];
1
1
1
t1 α



...,



T (t − t )(x(t− ) + I (x(t− ))) + R t S (t − s)v(s)ds, t ∈ (t , T ].
α
m
m
m
m
m
tm α
(3.2)
We shall introduce the following hypotheses:
(H1) The semigroup Sα (t) is compact for t > 0.
(H2) The multivalued map F : J × B × E → E is Carathéodory, with compact
convex values.
6
K. AISSANI, M. BENCHOHRA
EJDE-2013/265
(H3) There exists a function µ ∈ L1 (J, R+ ) and a continuous nondecreasing
function ψ : R+ → (0, +∞) such that
kF (t, v, w)k ≤ µ(t)ψ (kvkB + kwkE ) ,
(t, v, w) ∈ J × B × E.
(H4) Ik : E → E is continuous, and there exists Ω > 0 such that
Ω = max {kIk (x)k, x ∈ Dr }.
1≤k≤m
Theorem 3.4. Assume that (H1)–(H4) hold. Then problem (1.1)-(1.3) has a mild
solution on (−∞, T ].
Proof. We transform problem (1.1) into a fixed-point problem. Consider the multivalued operator N : B1 → P(B1 ) defined by N (h) = {h ∈ B1 } with

φ(t),
t ∈ (−∞, 0];


Rt



t ∈ [0, t1 ];

 0 Sα (t − s)v(s)ds,

T (t − t )(x(t− ) + I (x(t− ))) + R t S (t − s)v(s)ds, t ∈ (t , t ];
α
1
1
1 2
1
1
t1 α
h(t) =
. . . ,



−


Tα (t − tm )(x(t−
m ) + Im (x(tm )))


 Rt
+ tm Sα (t − s)v(s)ds, v ∈ SF,x ,
t ∈ (tm , T ].
It is clear that the fixed points of the operator N are mild solutions of problem
(1.1). Let us define y(.) : (−∞, T ] → E as
(
φ(t),
t ∈ (−∞, 0];
y(t) =
0,
t ∈ J.
Then y0 = φ. For each z ∈ C(J, E) with z(0) = 0, we denote by z the function
defined by
(
0,
t ∈ (−∞, 0];
z(t) =
z(t), t ∈ J.
Let xt = yt + z t , t ∈ (−∞, T ]. It is easy to see that x(.) satisfies
z0 = 0 and for t ∈ J, we have
R t
S (t − s)v(s)ds,


0 α


−
−
−
−

T
(t

α − t1 )[y(t1 ) + z(t1 ) + I1 (y(t1 ) + z(t1 ))]

R

+ t S (t − s)v(s)ds,
t1 α
z(t) =

...,



−
−
−


Tα (t − tm )[y(t−
m ) + z(tm ) + Im (y(tm ) + z(tm ))]


 Rt
+ tm Sα (t − s)v(s)ds,
where v(s) ∈ SF,y+z . Let
B2 = {z ∈ B1 : z0 = 0}.
For any z ∈ B2 , we have
kzkB2 = sup kz(t)k + kz0 kB = sup kz(t)k.
t∈J
t∈J
(3.2) if and only if
t ∈ [0, t1 ];
t ∈ (t1 , t2 ];
t ∈ (tm , T ],
EJDE-2013/265
IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS
Thus (B2 , k.kB2 ) is a Banach space. We define the operator P
P (z) = {h ∈ B2 } with
R t
S (t − s)v(s)ds,


 0 α

−
−
−
−


TαR(tt − t1 )[y(t1 ) + z(t1 ) + I1 (y(t1 ) + z(t1 ))]

+
S (t − s)v(s)ds,
t1 α
h(t) =
. . . ,



−
−
−


Tα (t − tm )[y(t−
m ) + z(tm ) + Im (y(tm ) + z(tm ))]


 Rt
+ tm Sα (t − s)v(s)ds,
7
: B2 → P(B2 ) by
t ∈ [0, t1 ];
t ∈ (t1 , t2 ];
t ∈ (tm , T ],
where v(s) ∈ SF,y+z . It is clear that the operator N has a fixed point if and only
if P has a fixed point. So let us prove that P has a fixed point. Let
Dr = {z ∈ B2 : z(0) = 0, kzkB2 ≤ r},
where r is any fixed finite real number which satisfies the inequality
Z T
α
fT (r + Ω) + M
fS T ψ(C2∗ kφkB + (C1∗ + 1)r)
µ(s)ds.
r>M
α
0
It is clear that Dr is a closed, convex, bounded set in B2 . We need the following
lemma.
Lemma 3.5. Set
C1∗ = sup C1 (t),
t∈J
C2∗ = sup C2 (η).
(3.3)
η∈J
Then for any z ∈ Dr we have
kyt + z t kB ≤ C2∗ kφkB + C1∗ r,
Proof. Using (2.2) and (3.3), we obtain
kyt + z t kB ≤ kyt kB + kz t kB
≤ C1 (t) sup ky(τ )k + C2 (t)ky0 kB + C1 (t) sup kz(τ )k + C2 (t)kz0 kB
0≤τ ≤t
0≤τ ≤t
≤ C2 (t)kφkB + C1 (t) sup kz(τ )k
0≤τ ≤t
≤ C2∗ kφkB + C1∗ r.
The proof is complete.
Now we shall show that P satisfies all the assumptions of Lemma 2.8. The proof
will be given in several steps.
Step 1: P (z) is convex for each z ∈ B2 . Indeed, if h1 and h2 belong to P (z), then
there exist v1 , v2 ∈ SF,y+z such that, for t ∈ J and i = 1, 2, we have
R t
S (t − s)vi (s)ds,
t ∈ [0, t1 ];


0 α


−
−
−
−

z(t
)
+
I
(y(t
)
+
z(t
))]
T
(t
−
t
)[y(t
)
+

α
1
1
1
1
1
1


+ R t S (t − s)v (s)ds,
t ∈ (t1 , t2 ];
α
i
t1
hi (t) =

...,



−
−
−


Tα (t − tm )[y(t−
m ) + z(tm ) + Im (y(tm ) + z(tm ))]


 Rt
t ∈ (tm , T ].
+ tm Sα (t − s)vi (s)ds,
8
K. AISSANI, M. BENCHOHRA
EJDE-2013/265
Let d ∈ [0, 1]. Then for each t ∈ [0, t1 ], we get
Z t
dh1 (t) + (1 − d)h2 (t) =
Sα (t − s)[dv1 (s) + (1 − d)v2 (s)]ds.
0
Similarly, for any t ∈ (ti , ti+1 ], i = 1, . . . , m, we have
−
−
−
dh1 (t) + (1 − d)h2 (t) = Tα (t − ti )[y(t−
i ) + z(ti ) + Ii (y(ti ) + z(ti ))]
Z t
+
Sα (t − s)[dv1 (s) + (1 − d)v2 (s)]ds
ti
Since F has convex values, SF,y+z is convex, we see that
dh1 + (1 − d)h2 ∈ P (z).
Step 2: P (Dr ) ⊂ Dr . Let h ∈ P (z) and z ∈ Dr , for t ∈ [0, t1 ], then by Lemma
3.5, we have
Z t
kh(t)k ≤
kSα (t − s)kkv(s)kds
0
Z t
fS
≤M
(t − s)α−1 µ(τ )ψ(kys + z s k + ky(s) + z(s)k)ds
0
Z t
α
T
∗
∗
f
≤ MS
ψ(C2 kφkB + (C1 + 1)r)
µ(s)ds < r.
α
0
Moreover, when t ∈ (ti , ti+1 ], i = 1, . . . , m, we have the estimate
Z t
−
−
kh(t)k ≤ kTα (t − ti )[z(ti ) + Ii (z(ti ))]k +
kSα (t − s)kkv(s)kds
ti
Z
t
fT (r + Ω) + M
fS
≤M
(t − s)α−1 µ(τ )ψ(kys + z s k + ky(s) + z(s)k)ds
0
α
fT (r + Ω) + M
fS
≤M
T
ψ(C2∗ kφkB + (C1∗ + 1)r)
α
Z
T
µ(s)ds < r,
ti
which proves that P (Dr ) ⊂ Dr .
Step 3: We will prove that P (Dr ) is equicontinuous. Let τ1 , τ2 ∈ [0, t1 ], with
τ1 < τ2 , we have
Z τ1
kh(τ2 ) − h(τ1 )k ≤
kSα (τ2 − s) − Sα (τ1 − s)kkv(s)kds
0
Z τ2
+
kSα (τ2 − s)kkv(s)kds
τ1
≤ Q1 + Q2 ,
where
Z
τ1
kSα (τ2 − s) − Sα (τ1 − s)kkv(s)kds
Z τ1
≤ ψ(C2∗ kφkB + (C1∗ + 1)r)
kSα (τ2 − s) − Sα (τ1 − s)kµ(s)ds.
Q1 =
0
0
fs (t1 − s)α−1 which belongs to L1 (J, R+ )
Since kSα (τ2 − s) − Sα (τ1 − s)kL(E) ≤ 2M
for s ∈ [0, t1 ], and Sα (τ2 − s) − Sα (τ1 − s) → 0 as τ1 → τ2 , Sα is strongly continuous.
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IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS
9
This implies that
lim Q1 = 0.
τ1 →τ2
Where
Z
τ2
kSα (τ2 − s)kkv(s)kds
Q2 =
τ1
Z τ2
fs (τ2 − τ1 )α
M
∗
∗
≤
ψ(C2 kφkB + (C1 + 1)r)
µ(s)ds.
α
τ1
Hence, we deduce that
lim Q2 = 0.
τ1 →τ2
Similarly, for τ1 , τ2 ∈ (ti , ti+1 ], i = 1, . . . , m, we have
kh(τ2 ) − h(τ1 )k
−
≤ kTα (τ2 − ti ) − Tα (τ1 − ti )k kz(t−
i )k + kIi (z(ti ))k + Q1 + Q2
≤ kTα (τ2 − ti ) − Tα (τ1 − ti )k(r + Ω) + Q1 + Q2 .
Since Tα is also strongly continuous, so Tα (τ2 − ti ) − Tα (τ1 − ti ) → 0 as τ1 → τ2 .
Thus, from the above inequalities, we have
lim kh(τ2 ) − h(τ1 )k = 0.
τ1 →τ2
So, P (Dr ) is equicontinuous.
As a consequence of Steps 1, 2 and 3 with the Arzelá-Ascoli theorem we conclude
that P : B2 → P(B2 ) is completely continuous.
Step 4: P has a closed graph. Suppose that zn → z∗ , hn ∈ P (zn ) with hn → h∗ .
We claim that h∗ ∈ P (z∗ ). In fact, the assumption hn ∈ P (zn ) implies that there
exists vn ∈ SF,yn +zn such that, for each t ∈ [0, t1 ],
Z t
hn (t) =
Sα (t − s)vn (s)ds.
0
We will show that there exists v∗ ∈ SF,z∗ such that, for each t ∈ [0, t1 ],
Z t
h∗ (t) =
Sα (t − s)v∗ (s)ds.
0
Consider the linear continuous operator Υ : L1 ([0, t1 ], E) → C([0, t1 ], E),
Z t
v 7→ (Υv)(t) =
Sα (t − s)v(s)ds.
0
By Lemma 2.7, we know that ΥoSF is a closed graph operator. Moreover, for every
t ∈ [0, t1 ], we obtain
hn (t) ∈ Υ(SF,yn +zn ).
Since zn → z∗ and hn → h∗ , it follows, that for every t ∈ [0, t1 ],
Z t
h∗ (t) =
Sα (t − s)v∗ (s)ds,
0
for some v∗ ∈ SF,y∗ +z∗ .
Similarly, for any t ∈ (ti , ti+1 ], i = 1, . . . , m, we have
−
−
−
hn (t) = Tα (t − ti ) yn (t−
i ) + z n (ti ) + Ii (yn (ti ) + z n (ti ))
10
K. AISSANI, M. BENCHOHRA
Z
EJDE-2013/265
t
Sα (t − s)vn (s)ds.
+
ti
We must prove that there exists v∗ ∈ SF,y∗ +z∗ such that, for each t ∈ (ti , ti+1 ],
−
−
−
h∗ (t) = Tα (t − ti ) y∗ (t−
i ) + z ∗ (ti ) + Ii (y∗ (ti ) + z ∗ (ti ))
Z t
+
Sα (t − s)v∗ (s)ds.
ti
Now, for every t ∈ (ti , ti+1 ], i = 1, . . . , m, we have
−
−
−
hn (t) − Tα (t − ti ) yn (t−
i ) + z n (ti ) + Ii (yn (ti ) + z n (ti ))
−
−
−
− h∗ (t) − Tα (t − ti ) y∗ (t−
)
+
z
(t
)
+
I
(y
(t
)
+
z
(t
))
→ 0
∗ i
i ∗ i
∗ i
i
as n → ∞.
Consider the linear continuous operator Υ : L1 ((ti , ti+1 ], E) → C((ti , ti+1 ], E),
Z t
v 7→ (Υv)(t) =
Sα (t − s)v(s)ds.
ti
From Lemma 2.7, it follows that ΥoSF is a closed graph operator. Also, from the
definition of Υ, we have that, for every t ∈ (ti , ti+1 ], i = 1, . . . , m,
−
−
−
hn (t) − Tα (t − ti ) yn (t−
∈ Υ(SF,yn +zn ).
i ) + z n (ti ) + Ii (yn (ti ) + z n (ti ))
Since zn → z∗ , for some v∗ ∈ SF,y∗ +z∗ it follows that, for every t ∈ (ti , ti+1 ], we
have
−
−
−
h∗ (t) = Tα (t − ti ) y∗ (t−
i ) + z ∗ (ti ) + Ii (y∗ (ti ) + z ∗ (ti ))
Z t
+
Sα (t − s)v∗ (s)ds.
ti
Hence the multivalued operator P is upper semi-continuous.
It follows from Lemma 2.8 that P has a fixed point z ∈ B2 . Then the operator
N has a fixed point which gives rise to a mild solution to problem (1.1)-(1.3). This
completes the proof.
4. An example
To apply our abstract results, we consider the impulsive fractional integrodifferential inclusion
Z 0
∂tq
∂2
v(t, ζ) − 2 v(t, ζ) ∈
H(t, v(θ, ζ))η(t, θ, ζ)dθ
∂tq
∂ζ
−∞
v(t, 0) = 0, v(t, π) = 0
(4.1)
v(t, ζ) = v0 (θ, ζ), −∞ < θ ≤ 0
Z tk
pk (tk − y)dy cos(v(tk )(ζ))
∆v(tk )(ζ) =
−∞
where 0 < q < 1, t ∈ [0, T ], ζ ∈ [0, π], γ : (−∞, 0] → R, pk : R → R, k = 1, 2, . . . , m,
and H : [0, T ]×R → P (R) is an u.s.c. multivalued map with compact convex values.
Set E = L2 ([0, π]), D(A) ⊂ E → E is the map defined by Aω = ω 00 with domain
D(A) = {ω ∈ E : ω, ω 0 are absolutely continuous, ω 00 ∈ E, ω(0) = ω(π) = 0}.
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IMPULSIVE FRACTIONAL DIFFERENTIAL INCLUSIONS
Then
Aω =
∞
X
n2 (ω, ωn )ωn ,
11
ω ∈ D(A),
n=1
q
where ωn (x) = π2 sin(nx), n ∈ N is the orthogonal set of eigenvectors of A. It is
well known that A is the infinitesimal generator of an analytic semigroup {T (t)}t≥0
in E and is given by
∞
X
2
T (t)ω =
e−n t (ω, ωn )ωn , ∀ω ∈ E, and every t > 0.
n=1
From these expressions, it follows that {T (t)}t≥0 is a uniformly bounded compact
semigroup, so that R(λ, A) = (λ − A)−1 is a compact operator for all λ ∈ ρ(A);
that is, A ∈ Aα (θ0 , ω0 ). For the phase space, we choose B = Bγ defined by
Bγ := φ ∈ C((−∞, 0], E) : lim eγθ φ(θ) exists in E
θ→−∞
endowed with the norm
kφk = sup{eγθ |φ(θ)| : θ ≤ 0}.
Clearly, we can see that Bγ is an admissible phase space which satisfies (A1)–(A3).
Set
x(t)(ζ) = v(t, ζ),
t ∈ [0, T ], ζ ∈ [0, π];
φ(θ)(ζ) = v0 (θ, ζ), θ ∈ (−∞, 0], ζ ∈ [0, π];
Z 0
F (t, ϕ, x(t))(ζ) =
H(t, ϕ(θ)(ζ))η(t, θ, ζ)dθ, t ∈ [0, T ], ζ ∈ [0, π];
Ik (x(t−
k ))(ζ) =
Z
−∞
0
pk (tk − y)dy cos(x(tk )(ζ)),
k = 1, 2, . . . , m.
−∞
Then problem (4.1) can be rewritten in the abstract form (1.1). If conditions
(H1)–(H4) are fulfilled, then from Lemma 2.8, system (4.1) has a mild solution on
(−∞, T ].
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Khalida Aissani
Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000, Sidi BelAbbès, Algeria
E-mail address: aissani k@yahoo.fr
Mouffak Benchohra
Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000, Sidi BelAbbès, Algeria
E-mail address: benchohra@univ-sba.dz