Ann. Funct. Anal. 3 (2012), no. 2, 89–106
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
EXISTENCE RESULTS ON RANDOM IMPULSIVE
SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS
WITH DELAYS
A. VINODKUMAR
Communicated by C. Cuevas
Abstract. This article presents the result on existence of mild solutions for
random impulsive semilinear functional differential inclusions under sufficient
conditions. The results are obtained by using the Martelli fixed point theorem
and the fixed point theorem due to Covitz and Nadler.
1. Introduction
Impulsive differential inclusions are suitable mathematical model to simulate
the evolution of large classes of real processes. These processes are subjected to
short temporary perturbations. The duration of these perturbations is negligible
compared to the duration of whole process. These perturbations occurs in the
form of impulses (see [2, 4, 6, 1, 3, 5] and the references therein).
When the impulses are exists at random, the solutions of the equation behave
as a stochastic process. It is quite different from deterministic impulsive differential equations and stochastic differential equations (SDEs). Iwankievicz et al
[7], investigated dynamic response of non-linear systems to poisson distributed
random impulses. Tatsuyuki et al [8] presented a mathematical model of random
impulse to depict drift motion of granules in chara cells due to myosin-actin interaction. In [9], Sanz-Serna et al first brought dissipative differential equations
with random impulses and used Markov chains to simulate such systems. Wu
and Meng [10] first gave the general random impulsive ordinary differential equations and investigated boundedness of solutions to these models by Liapunov’s
Date: Received: 5 January 2012; Revised: 6 March 2012; Accepted: 23 March 2012.
2010 Mathematics Subject Classification. Primary 34K45; Secondary 47N20, 49K24, 35R60,
60H15.
Key words and phrases. Multivalued map, existence, random impulsive differential inclusions, fixed point theorem.
89
90
A. VINODKUMAR
direct method. Shujin Wu et al [11, 12, 13], studied some qualitative properties of
random impulses. In [14], the author studied the existence and exponential stability for a random impulsive semilinear functional differential equations through
the fixed point technique under non-uniqueness. The existence, uniqueness and
stability results were discussed in [15] through Banach fixed point method for
the system of differential equations with random impulsive effect. The author
[16], studied the existence results for the random impulsive neutral functional
differential equations with delays. In [17], the author studied existence results of
random impulsive neutral non-autonomous differential inclusions with delays via
Dhage’s fixed point theorem.
Motivated by the above mentioned works, the main purpose of this paper is to
study of random impulsive semilinear functional differential inclusions (RIFDIns).
We utilize the technique from [18, 19, 20].
The paper is organized as follows. In section 2, we recall briefly the notations,
definitions, lemmas and preliminaries which are used throughout this paper. In
section 3, we study the existence of RIFDIns in the convex case of the multivalued
function using a fixed point theorem for condensing map due to Martelli. In
the end of the section, we study the existence results of the problem by using
Wintner growth condition. We study the existence of RIFDIns in the non-convex
case of the multivalued function using the fixed point theorem due to Covitz and
Nadler in section 4. Finally in section 5, we present an example to illustrate the
application for the results in section 3.
2. Preliminaries
Let X be a real separable Hilbert space and Ω a nonempty set. Assume that
def.
τk is a random variable defined from Ω to Dk = (0, dk ) for k = 1, 2, · · · , where
0 < dk < +∞. Furthermore, assume that τi and τj are independent with each
other as i 6= j for i, j = 1, 2, · · · . For the sake of simplicity, we denote
<+ = [0, +∞); <τ = [τ, +∞).
We consider the semilinear functional differential inclusions with random impulses of the form

 x0 (t) ∈ Ax(t) + F (t, xt ), t 6= ξk , τ ≤ t ≤ T,
x(ξk ) = bk (τk )x(ξk− ), k = 1, 2, · · · ,
(2.1)

xt0
= ϕ,
where A is the infinitesimal generator of strongly continuous semigroup of bounded
linear operators S(t) = {S(t), t ≥ 0} with D(A) ⊂ X.
Now we make the system (2.1) precise: The functional F : <τ × C → P(X),
P(X) is the family of all nonempty measurable subsets of X; C = C([−r, 0], X)
is the set of piecewise continuous functions mapping [−r, 0] into X with some
given r > 0; xt is a function when t is fixed, defined by xt (s) = x(t + s) for all
s ∈ [−r, 0]; ξ0 = t0 and ξk = ξk−1 + τk for k = 1, 2, . . ., here t0 ∈ <τ is arbitrary
given real number. Obviously, t0 = ξ0 < ξ1 < ξ2 < · · · < ξk < · · · ; bk : Dk → <
for each k = 1, 2, · · · ; x(ξk− ) = lim x(t) according to their paths with the norm
t↑ξk
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
kxkt =
91
sup |x(s)| for each t satisfying τ ≤ t ≤ T and τ, T ∈ <+ are given
t−r≤s≤t
numbers, k · k is any given norm in X; ϕ is a function defined from [−r, 0] to X.
Denote {Bt , t ≥ 0} the simple counting process generated by {ξn }, that is,
{Bt ≥ n} = {ξn ≤ t}, and denote Ft the σ-algebra generated by {Bt , t ≥ 0}.
Then (Ω, P, {Ft }) is a probability space. Let Lp = Lp (Ω, P, {Ft }) be the space
of all pth integrable random variables with values in X, that are measurable
with respect to {F
t , t ≥ t0 }. For the simplification, denote the Banach space
BT [t0 − r, T ], Lp , the family of all Ft -measurable, C-valued random variables ψ
with the norm
1
kψkBT =
sup Ekψkpt
p
.
t∈[t0 ,T ]
Let L0p (Ω, BT ) denote the family of all F0 - measurable, BT - valued random
variable ϕ.
We use the following notations: Pcl (X) = {Y ∈ P(X) : Y closed}, Pbd (X) =
{Y ∈ P(X) : Y bounded}, Pcv (X) = {Y ∈ P(X) : Y convex}, Pcp (X) = {Y ∈
P(X) : Y compact}. In a Hilbert space X, a multivalued map G : X → P(X) is
a convex (closed) valued, if G(x) is convex (closed) for all x ∈ X. G is bounded
on bounded sets if G(V ) = ∪x∈V G(x) is bounded in X, for all V ∈ Pbd (X) that
is,
sup{sup{kyk : y ∈ G(x)}} < ∞.
x∈V
G is called upper semi continuous (u.s.c.) on X, if for each x0 ∈ X, the set
G(x0 ) is non-empty, closed subset of X, and if for each open set V of X containing
G(x0 ) there exists an open neighborhood N of x0 such that G(N ) ⊆ V .
G is said to be completely continuous if G(V ) is relatively compact, for every
V ∈ Pbd (X).
If the multivalued map G is completely continuous with nonempty compact
value, then G is u.s.c. if and onlyif G is closed graph, ie., x(n) → x∗ , y (n) →
y ∗ , y (n) ∈ G(x(n) ) imply y ∗ ∈ G(x∗ ) .
An upper semicontinuous map G : X → X is said to be condensing if for any
bounded subset V ⊆ X with α(G(V )) < α(V ), where α denotes the Kuratowski
measure of noncompactness [21].
Remark 2.1. ([22]). A completely continuous multivalued map is the easiest example of a condensing map.
G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set
of the multivalued operator G will be denoted by F ix G.
Define the function H : Pbd,cl (X) × Pbd,cl (X) → <+ by
H(A, B) = max sup d(a, B), sup d(A, b) ,
a∈A
b∈B
where d(A, b) = inf ka − bkp , a ∈ A , d(a, B) = inf ka − bkp , b ∈ B . The
function H is called a Hausdorff metric on Pbd,cl (X).
The multivalued map Z : [τ, T ] → Pbd,cl (X) is said to be measurable if for each
x ∈ X the function Y : [τ, T ] → <+ defined by
Y (t) = d(x, Z(t)) = inf{kx − zkp : z ∈ Z(t)} is measurable.
92
A. VINODKUMAR
Definition 2.2. A multivalued operator Z : [τ, T ] → Pcl (X) is called
(a) Contraction if and only if there exists η > 0 such that
H(Z(x), Z(y)) ≤ η kx − ykp , for each x, y ∈ X, with η < 1,
(b) Z has a point if there exists x ∈ X such that x ∈ Z(x).
For more details on multivalued maps see [22, 23, 24]. Our existence results
are based on the following fixed point theorems of Martelli [25] and Covitz and
Nadler [26].
Theorem 2.3. ([25]). Let X be a Hilbert space and Z : X → Pbd,cl,cv (X) a u.s.c.
and condensing map. If the set
U = u ∈ X | λu ∈ Z(x) for some λ > 1
is bounded, then Z has a fixed point.
Theorem 2.4. ([26]). Let X be a Banach space, If Z : X → Pcl (X) is a contraction then F ix Z 6= φ.
Definition 2.5. The multivalued map F : [τ, T ] × C → P(X) is said to be Lp Carathèodory if:
(i) t 7−→ F (t, u) is measurable for each u ∈ C;
(ii) u 7−→ F (t, u) is upper semicontinuous for almost all t ∈ [τ, T ];
(iii) for each a > 0, there exists ha ∈ Lp ([τ, T ], <+ ) such that
n
o
kF (t, u)kp := sup Ekf kp : f ∈ F (t, u) ≤ ha (t),
for all kukpBT ≤ a and for a.e. t ∈ [τ, T ]. For each x ∈ Lp (X) define the set of
selections of F by
SF,x = f ∈ Lp (X) : f (t) ∈ F (t, xt ) for a.e, t ∈ [τ, T ] .
Lemma 2.6. ([27]). Let I be a compact interval and X be a Hilbert space. Let
F be an Lp - Carathèodory multivalued map with SF,x 6= φ and Γ be a linear
continuous mapping from Lp (I, X) → C(I, X). Then the operator
Γ ◦ SF : C(I, X) → Pbd,cl,cv (C(I, X)), x 7−→ (Γ ◦ SF )(x) = Γ(SF,x ),
is a closed graph operator in C(I, X) × C(I, X).
Definition 2.7. A semigroup {S(t), t ≥ 0} is said to be exponentially bounded
if there exist constants M ≥ 1 and γ ∈ < such that
kS(t)k ≤ M eγt , for t ≥ 0.
Definition 2.8. A stochastic process {x(t) ∈ BT , t0 − r ≤ t ≤ T } is called a
mild solution to system (2.1) in (Ω, P, {Ft }), if
(i) x(t) ∈ X is Ft −adapted;
(ii) x(t0 + s) = ϕ(s) ∈ L0p (Ω, BT ) when s ∈ [−r, 0], and
" k
Z ξi
+∞ Y
k Y
k
X
X
x(t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
S(t − s)f (s)ds
ξ
i−1
i=1 j=i
k=0 i=1
Z t
+
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t), f ∈ SF,x , a.s t ∈ [t0 , T ],
ξk
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
where
n
Q
(·) = 1 as m > n,
j=m
k
Q
93
bj (τj ) = bk (τk )bk−1 (τk−1 ) · · · bi (τi ), and IA (·) is the
j=i
index function, i.e.,
IA (t) =
1,
0,
if t ∈ A,
if t ∈
/ A.
3. Existence result: Convex case
In this section, we discuss the existence of mild solutions of the system (2.1).
We assume that the multivalued F has compact and convex values. We need the
following hypotheses.
Hypotheses:
(H1 ) : A : D(A) ⊂ X → X is the infinitesimal generator of strongly continuous
semigroup S(t) in X.
(
)
k
Q
(H2 ) : The condition max
kbj (τj )k is uniformly bounded, that is, there is
i,k
j=i
C > 0 such that
max
i,k
( k
Y
)
kbj (τj )k
≤ C for all τj ∈ Dj , j = 1, 2, · · · .
j=i
(H3 ) : F : [τ, T ] × C → P(X) is a compact convex Lp - Carathèodory multivalued
function.
(H4 ) : There exists a continuous nondecreasing function ψ : <+ → (0, ∞), r ∈
L1 ([τ, T ], <+ ) such that
n
o
kF (t, xt )kp = sup kf kp : f ∈ F (t, xt ) ≤ r(t)ψ(kxkpt ), a.e t ∈ [τ, T ] for all x ∈ C.
Theorem 3.1. Assume that (H1 )−(H4 ) hold. Then the problem (2.1) has atleast
one mild solution on [−r, T ], provided
Z T
Z ∞
du
−γs
Q2
e r(s)ds <
,
(3.1)
t0
Q1 ψ(u)
where Q1 = 2p−1 M p epγ(T −t0 ) C p Ekϕkp , Q2 = 2p−1 M p epγT max 1, C p (T − t0 )p−1
1
and M p epγ(T −t0 ) C p ≥ 2p−1
.
Proof. Transform the problem (2.1) into a fixed point problem. Consider the
multivalued operator Z : BT → P(BT ) defined by
Z (x) = h ∈ BT : h(t)

ϕ(t − t0 ),
t ∈ [t0 − r, t0 ],






Z ξi

+∞ Y
k
k Y
k
 X
X
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
S(t − s)f (s)ds
=

ξ
i−1

i=1 j=i
k=0

Z t i=1




 +
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t),
f ∈ SF,x , a.s
t ∈ [t0 , T ].
ξk
94
A. VINODKUMAR
We shall show that the operator Z satisfies all the conditions of Theorem 2.1.
We give the proof in the following steps.
Step (1): Z(x) is convex for each y ∈ BT . Since F has convex values it follows
that SF,x is convex; so that f1 , f2 ∈ SF,x then κf1 +(1−κ)f2 ∈ SF,x , which implies
clearly that Z(x) is convex.
Step (2): Z is bounded on bounded sets of BT . Let Ba = {x ∈ BT : kxkp ≤ a},
a > 0, be a bounded subset in BT . We show that Z(Ba ) is a bounded subset of
BT . For each x ∈ Ba let h ∈ Z(x). Then there exists f ∈ SF,x such that for each
t ∈ [t0 , T ], we have
h(t) =
" k
+∞ Y
X
bi (τi )S(t − t0 )ϕ(0) +
i=1
k=0
t
Z
+
k Y
k
X
Z
ξi
S(t − s)f (s)ds
bj (τj )
ξi−1
i=1 j=i
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t)
ξk
kh(t)kp ≤
+∞
k
hX
Y
k
bi (τi )kkS(t − t0 )kkϕ(0)k
i=1
k=0
+
k
X
k
i=1
Z
t
+
k
Y
ξi
Z
kS(t − s)f (s)kds
bj (τj )k
ξi−1
j=i
ip
kS(t − s)f (s)kds I[ξk ,ξk+1 ) (t)
ξk
p−1
≤ 2
M e
p−1
+2
p pγ(t−t0 )
M
p
max
k
nY
k
h
o
kbi (τi )kp kϕ(0)kp
i=1
k
n Y
oip Z t
p
kbj (τj )k
·
max 1,
eγ(t−s) kf (s)k ds
i,k
t0
j=i
≤ 2p−1 M p epγ(T −t0 ) C p kϕ(0)kp
Z t
p−1
p
p
p−1
+2 M max 1, C (T − t0 )
epγ(t−s) kf (s)kp ds.
t0
Thus,
Ekhkpt ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕ(0)kp
Z t
p−1
p
p
p−1 pγ(T −t0 )
+ 2 M max 1, C (T − t0 ) e
e−pγ(s−t0 ) r(s)ψ(Ekxkps )ds
t0
p−1
p pγ(T −t0 ) p
p
p−1
p
≤2 M e
C Ekϕ(0)k + 2 M max 1, C p (T − t0 )p−1 epγ(T −t0 )
Z t
−pγ(s−t0 )
2
× sup ψ(Ekxkt )
e
r(s)ds .
t∈[t0 ,T ]
t0
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
95
Hence for each h ∈ Z(Ba ), we get
khkpBT
≤ 2p−1 M p epγ(T −t0 ) C p Ekϕ(0)kp
+2p−1 M p max 1, C p (T − t0 )p−1 epγ(T −t0 )
Z t
2
¯
e−pγ(s−t0 ) r(s)ds = `.
× max sup ψ(Ekxkt ) sup
x∈Sa t∈[t0 ,T ]
t∈[t0 ,T ]
t0
¯
Then, for each h ∈ Z(x), we have khkpBT ≤ `.
Step (3): Z sends bounded sets into equi-continuous sets of BT . Let t1 , t2 ∈
[t0 , T ], t0 < t1 < t2 ≤ T and Ba = {x ∈ BT : kxkp ≤ a} be a bounded set in BT .
Now for each x ∈ Ba , h ∈ Z(x), there exists a function f ∈ SF,x such that for
each t ∈ [t0 , T ], we have
h(t1 ) − h(t2 )
" k
Z
+∞ Y
k Y
k
X
X
=
bi (τi )S(t1 − t0 )ϕ(0) +
bj (τj )
i=1
k=0
Z
S(t1 − s)f (s)ds I[ξk ,ξk+1 ) (t1 ) −
+
ξk
+
Z
+∞ h Y
k
X
ξi
Z
ξi−1
Z
bi (τi )S(t1 − t0 )ϕ(0) +
S(t2 − s)f (s)ds I[ξk ,ξk+1 ) (t2 )
k Y
k
X
Z
S(t1 − s)f (s)ds
i
ξi
S(t1 − s)f (s)ds
bj (τj )
ξi−1
i=1 j=i
t1
+
t2
ξk
i=1
k=0
bi (τi )S(t2 − t0 )ϕ(0)
i=1
S(t2 − s)f (s)ds +
bj (τj )
i=1 j=i
=
+∞ Y
X
k
k=0
k Y
k
X
S(t1 − s)f (s)ds
ξi−1
i=1 j=i
t1
ξi
I[ξk ,ξk+1 ) (t1 ) − I[ξk ,ξk+1 ) (t2 )
ξk
+
+∞ h Y
k
X
k=0
+
i=1
k Y
k
X
i=1 j=i
Z
t1
+
ξk
bi (τi ) S(t1 − t0 ) − S(t2 − t0 ) ϕ(0)
Z
ξi
bj (τj )
S(t1 − s) − S(t2 − s) f (s)ds
ξi−1
Z
S(t1 − s) − S(t2 − s) f (s)ds +
t2
i
S(t2 − s)f (s)ds I[ξk ,ξk+1 ) (t2 ).
t1
Then,
Ekh(t1 ) − h(t2 )kp ≤ 2p−1 EkI1 kp + 2p−1 EkI2 kp ,
(3.2)
96
A. VINODKUMAR
where
I1 =
+∞ h Y
k
X
bi (τi )S(t1 − t0 )ϕ(0) +
i=1
k=0
Z
Z
S(t1 − s)f (s)ds
i
ξi
S(t1 − s)f (s)ds
bj (τj )
ξi−1
i=1 j=i
t1
+
k Y
k
X
I[ξk ,ξk+1 ) (t1 ) − I[ξk ,ξk+1 ) (t2 ) ,
ξk
and
I2 =
+∞ h Y
k
X
i=1
k=0
+
bi (τi ) S(t1 − t0 ) − S(t2 − t0 ) ϕ(0)
k Y
k
X
Z
t1
+
bj (τj )
S(t1 − s) − S(t2 − s) f (s)ds
ξi−1
i=1 j=i
Z
ξi
Z
t2
i
S(t2 − s)f (s)ds I[ξk ,ξk+1 ) (t2 ).
S(t1 − s) − S(t2 − s) f (s)ds +
t1
ξk
Furthermore,
EkI1 k
p
≤ 2 M e
C Ekϕ(0)k E I[ξk ,ξk+1 ) (t1 ) − I[ξk ,ξk+1 ) (t2 )
Z t1
p−1
p
p−1
+2 max {1, C } (t1 − t0 ) E
kS(t1 − s)kp kf (s)kp ds
t0
×E I[ξk ,ξk+1 ) (t1 ) − I[ξk ,ξk+1 ) (t2 ) → 0 as t2 → t1 ,(3.3)
p pγ(t1 −t0 )
p−1
p
p
and
EkI2 kp ≤ 3p−1 C p kS(t1 − t0 ) − S(t2 − t0 )kp Ekϕ(0)kp
Z t1
p−1
p
p−1
+ 3 max {1, C } (t1 − t0 )
kS(t1 − s) − S(t2 − s)kp Ekf (s)kp ds
t0
+ 3p−1 (t2 − t1 )p−1
Z
t2
kS(t2 − s)kp Ekf (s)kp ds → 0 as t2 → t1 . (3.4)
t1
From (3.3) and (3.4), it follows that the right hand side of (3.2) tends to zero
as t2 → t1 . Since the compactness of S(t − t0 ) for t > t0 implies the continuity in
the uniform operator topology.
Step (4): Z maps bounded sets into relatively compact sets in BT .
Let a real number satisfying ∈ (0, t − t0 ), for t ∈ [t0 , T ]. For x ∈ Ba we
define a function h by
" k
Z ξi
+∞ Y
k
k Y
X
X
h (t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
S(t − s)f (s)ds
i=1
k=0
Z
t−
i=1 j=i
ξi−1
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t), t ∈ (t0 , t − )
+
ξk
where f ∈ SF,x . Since S(t − t0 ) is a compact operator, the set H (t) = {h (t) :
h ∈ Z(x)} is relatively compact in BT for every ∈ (0, t − t0 ). Moreover, for
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
97
every h ∈ Z(x) we have
h(t) − h (t)
" k
Z
+∞ Y
k Y
k
X
X
=
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
i=1
k=0
Z
t
+
S(t − s)f (s)ds
ξi−1
i=1 j=i
+∞ Y
X
k
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t) −
bi (τi )S(t − t0 )ϕ(0)
ξk
+
ξi
i=1
k=0
k Y
k
X
Z
ξi
Z
S(t − s)f (s)ds +
bj (τj )
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t).
ξk
ξi−1
i=1 j=i
t−
By using (H1 ) − (H4 ), we obtain
Ekh −
h kpt
p
p
p−1
Z
t
≤ M max{1, C }(T − t0 )
epγ(t−s) ψ(a)r(s)ds
t−
Therefore, there are relatively compact sets arbitrarily close to the set {h(t) : h ∈
Z(Ba )}. Hence the set {h(t) : h ∈ Z(Ba )} is also relatively compact in BT .
As the consequence of Step 1-4, together with Ascoli-Arzela theorem, we can
conclude that Z is a compact multivalued map, and therefore, a condensing map.
Step (5): Z has a closed graph
Let x(n) → x∗ and h(n) ∈ Z(x(n) ) with h(n) → h∗ . We shall show that h∗ ∈
Z(x∗ ). There exists f (n) ∈ SF,x(n) , such that
" k
Z ξi
+∞ Y
k Y
k
X
X
(n)
S(t − s)f (n) (s)ds
h (t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
i=1
k=0
Z
+
ξi−1
i=1 j=i
t
(n)
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t).
ξk
We must prove that there exists f ∗ ∈ SF,x∗ , such that
" k
Z
+∞ Y
k
k Y
X
X
∗
h (t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
i=1
k=0
Z
S(t − s)f ∗ (s)ds
ξi−1
i=1 j=i
t
+
ξi
S(t − s)f ∗ (s)ds I[ξk ,ξk+1 ) (t).
ξk
Consider the linear continuous operator Γ : Lp (X) → BT defined by
" k k
#
Z ξi
Z t
+∞ X
X
Y
Γ(f )(t) =
bj (τj )
S(t − s)f (s)ds +
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t).
k=0
Then we have
i=1 j=i
ξi−1
ξk
98
A. VINODKUMAR
+∞ Y
X
k
(n)
k h (t) −
bi (τi )S(t − t0 )ϕ(0) I[ξk ,ξk+1 ) (t)
k=0
− h∗ (t) −
i=1
+∞
X
k
Y
k=0
i=1
bi (τi )S(t − t0 )ϕ(0) I[ξk ,ξk+1 ) (t) kp → 0, as n → ∞.
From Lemma (2.3), it follows that Γ ◦ SF is a closed graph operator and from the
definition of Γ one has
h(n) (t) −
+∞ Y
X
k
k=0
bi (τi )S(t − t0 )ϕ(0) I[ξk ,ξk+1 ) (t) ∈ Γ ◦ SF,x(n) .
i=1
As x(n) → x∗ and h(n) → h∗ , there is a f ∗ ∈ SF,x∗ such that
h∗ (t) −
+∞ Y
X
k
i=1
k=0
=
+∞
X
k=0
bi (τi )S(t − t0 )ϕ(0) I[ξk ,ξk+1 ) (t)
k Y
k
X
Z
ξi
∗
t
S(t − s)f (s)ds +
bj (τj )
ξi−1
i=1 j=i
Z
S(t − s)f ∗ (s)ds I[ξk ,ξk+1 ) (t)
ξk
for all t ∈ [t0 , T ]. Hence h∗ ∈ Z(x∗ ), which shows that the graph Z is closed.
Step (6): A priori bounds
Now it remains to show that the set
U = {x ∈ BT : λ x ∈ Z(x), for some λ > 1} is bounded.
Let x ∈ U , then for some λ > 1, λ x ∈ Z(x) and there exists f ∈ SF,x such that
x(t) = λ
" k
+∞ Y
X
−1
Z
k=0
t
+
ξk
i=1
bi (τi )S(t − t0 )ϕ(0) +
k Y
k
X
Z
ξi
S(t − s)f (s)ds
bj (τj )
i=1 j=i
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t), t ∈ [t0 , T ].
ξi−1
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
99
Thus, by (H1 ) − (H4 ), for each t ∈ [t0 , T ], we have
+∞ h Y
k
X
p
kx(t)k ≤
kbi (τi )kkS(t − t0 )kkϕ(0)k
k=0
+
i=1
k Y
k
X
kbj (τj )k
kS(t − s)kkf (s)kds
ξi−1
i=1 j=i
Z
t
+
ξi
Z
i
p
kS(t − s)kkf (s)kds I[ξk ,ξk+1 ) (t)
ξk
≤ 2p−1 M p epγ(t−t0 ) max
k
nY
k
+2p−1 M p
h
o
kbi (τi )kp kϕ(0)kp
i=1
k
p
n Y
oip Z t
γ(t−s)
kf (s)k ds
max 1,
e
kbj (τj )k
·
i,k
t0
j=i
Ekxkpt ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕ(0)kp
Z t
p−1
p
p
p−1
+2 M max 1, C (T − t0 ) E
epγ(t−s) kf (s)kp ds.
t0
Noting that the last term of the right hand side of the above inequality increases
1
, we obtain that
in t and choose M p epγ(T −t0 ) C p ≥ 2p−1
Ekxkpt ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕkp
Z t
p−1
p pγ(T −t0 )
p
p−1
+2 M e
max 1, C (T − t0 )
e−pγ(s−t0 ) r(s)ψ(Ekxkps )ds.
t0
The function µ defined by
µ(t) = sup Ekxkps , t ∈ [t0 , T ].
(3.5)
t0 ≤s≤t
Then, for any [t0 , T ], it follows that
µ(t) ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕkp
(3.6)
Z t
+2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1
e−pγ(s−t0 ) r(s)ψ(µ(s))ds.
t0
Denoting the right hand side of the above inequality (3.6) as v(t), we obtain that
µ(t) ≤ v(t), t ∈ [t0 , T ],
v(t0 ) = 2p−1 M p epγ(T −t0 ) C p Ekϕkp = Q1 ,
and
v 0 (t) = 2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1 e−pγ(t−t0 ) r(t)ψ(µ(t))
≤ 2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1 e−pγ(t−t0 ) r(t)ψ(v(t)).
Then
v 0 (t)
≤ 2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1 e−pγ(t−t0 ) r(t).
ψ(v(t))
(3.7)
100
A. VINODKUMAR
Integrating (3.7) from t0 to t and by making use of change of variables, we obtain
Z
v(t)
v(t0 )
du
≤ 2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1
ψ(u)
Z T
e−pγs r(s)ds
≤ Q2
t0
Z ∞
du
<
.
Q1 ψ(u)
Z
t
e−pγ(s−t0 ) r(s)ds
t0
Hence by (3.1) there exists a constant β1 such that µ(t) ≤ v(t) ≤ β1 , for all
t ∈ [t0 , T ].
Since for every t ∈ [t0 , T ], Ekxkp ≤ µ(t) we have
kxkpBT = sup {Ekxkpt } ≤ β1 ,
t0 ≤t≤T
where β1 depends only on T and the function ψ and r. This shows that U is
bounded. As a consequence of Theorem 2.1, we deduce that Z has a fixed point
x defined on the interval [−r, T ], which is a solution of (2.1).
We now present another existence result for the problem (2.1). The multivalued
F is relaxed by using Wintner-type growth condition in the following Theorem.
Theorem 3.2. Assume that (H1 )- (H3 ) and the following condition holds
(HF ) : There exists some function ` ∈ Lp ([t0 , T ], <+ ) such that
H(F (t, xt ), F (t, yt )) ≤ `(t) kx − ykpt , for all t ∈ [t0 , T ], x, y ∈ C,
H(0, F (t, 0)) ≤ `(t), for a.e t ∈ [t0 , T ],
R T −pγs
where, t0 e
`(s)ds < ∞,
Q3 = 2p−1 M p epγ(T −t0 ) C p Ekϕkp
Z T
p−1
p pγT
p
p−1
+2 M e max 1, C (T − t0 )
e−pγs `(s)ds
t0
p−1
p pγT
p
p−1
Q4 = 2 M e max 1, C (T − t0 ) ,
then the problem (2.1) has at least one mild solution on [−r, T ].
Proof. Let the operator Z is defined as in Theorem 3.1. It can be shown, as in the
proof of Theorem 3.1 that Z is completely continuous and upper semi-continuous.
Now we prove that
U = {x ∈ BT : λ x ∈ Z(x) for some λ > 1} is bounded.
Let x ∈ U , then there exists f ∈ SF,x , for t ∈ [t0 , T ],
" k
Z
k Y
k
+∞ Y
X
X
−1
x(t) = λ
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
Z
k=0
t
+
ξk
i=1
i=1 j=i
S(t − s)f (s)ds I[ξk ,ξk+1 ) (t), t ∈ [t0 , T ].
ξi
ξi−1
S(t − s)f (s)ds
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
101
Thus, by (H1 ) − (H3 ) and (HF ), we have
Ekxkpt ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕ(0)kp
Z t
p−1
p−1
p
p
epγ(t−s) E kf (s)kp ds.
+2 M max 1, C (T − t0 )
t0
Noting that the last term of the right hand side of the above inequality increases
1
in t and choose M p epγ(T −t0 ) C p ≥ 2p−1
, we obtain that
Ekxkpt ≤ 2p−1 M p epγ(T −t0 ) C p Ekϕkp
Z t
p−1
p−1
p pγ(T −t0 )
p
e−pγ(s−t0 ) `(s)ds
+2 M e
max 1, C (T − t0 )
t0
t
+2p−1 M p epγ(T −t0 ) max 1, C p (T − t0 )p−1
Z
e−pγ(s−t0 ) `(s)Ekxkps ds.
t0
Using the function µ(t) defined by (3.5), we obtain
Z t
µ(t) ≤ Q3 + Q4
e−pγs `(s)µ(s)ds.
t0
Grownwall inequality, we get
Z
t
µ(t) ≤ Q3 exp Q4
e−γs l(s)ds , for all t ∈ [t0 , T ].
t0
Therefore, there exists β2 > 0 such that
µ(t) ≤ β2 for all t ∈ [t0 , T ],
which implies that
kxkpBT ≤ β2 .
This shows that the set U is bounded. As a consequence of Theorem 2.1, we
deduce that Z has a fixed point which is a mild solution of (2.1).
4. Existence results : non- convex case
In this section, we consider problem (2.1) with a non-convex valued right hand
side. We assume that the multivalued map F has compact values. Our result
in this section is based on the fixed point theorem for contraction multivalued
operators given by Covitz and Nadler.
We now make additional assumption:
(HF cp ) : F : [τ, T ] × C → Pcp (X) has the property that F (·, x) : [τ, T ] → Pcp (X)
is measurable for each x ∈ C.
Theorem 4.1. Assume that hypotheses (H1 ) − (H2 ), (HF ) and (HF cp ) are satisfied, then the IVP (2.1) has at least one mild solution on [−r, T ], provided
Z T
h
i
p−1
p
p
η = M max 1, C (T − t0 )
epγ(t−s) `(s)ds < 1.
(4.1)
t0
102
A. VINODKUMAR
Proof. Transform problem (2.1) into a fixed point problem. Consider the multivalued operator Z defined in Theorem 3.1. We shall show that Z satisfies the
assumptions of Theorem 2.2.
Step (1): Z(x) ∈ Pcl (BT ) for each x ∈ BT .
Indeed, let (x(n) )n≥0 ∈ Z(x) such that x(n) → x in BT . Then x ∈ BT and there
exists f n ∈ SF,x such that, for every t ∈ [t0 , T ],
Z ξi
+∞ Y
k Y
k
X
X
k
(n)
x (t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
S(t − s)f (n) (s)ds
i=1
k=0
i=1 j=i
ξi−1
t
Z
S(t − s)f (n) (s)ds I[ξk ,ξk+1 ) (t),
+
ξk
using the fact that F has compact values and from (HF ). We may pass to a
subsequence if necessary to get that f (n) converges to f in Lp ([τ, T ], X) and
hence f ∈ SF,x . Then, for each t ∈ [t0 , T ],
x(n) (t) → x(t)
Z
k Y
k
+∞ Y
X
X
k
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
=
i=1
k=0
Z
i=1 j=i
ξi
S(t − s)f (n) (s)ds
ξi−1
t
S(t − s)f (n) (s)ds I[ξk ,ξk+1 ) (t).
+
ξk
So x ∈ Z(x).
Step (2): Contraction
Let x1 , x2 ∈ BT and h1 ∈ Z(x1 ) then there exists f 1 (t) ∈ F (t, x1t ) such that for
t ∈ [t0 , T ],
Z ξi
+∞ Y
k Y
k
X
X
k
1
h (t) =
bi (τi )S(t − t0 )ϕ(0) +
bj (τj )
S(t − s)f 1 (s)ds
i=1
k=0
Z
i=1 j=i
ξi−1
t
+
S(t − s)f 1 (s)ds I[ξk ,ξk+1 ) (t).
ξk
From (HF ) it follows that
p
H(F (t, x1t ), F (t, x2t )) ≤ `(t) x1 − x2 t , t ∈ [t0 , T ].
Hence, there is y ∈ F (t, x2t ) such that
kf 1 (t) − ykp ≤ `(t)kx1 − ykpt , t ∈ [t0 , T ].
Consider N : [t0 , T ] → Pcp (X), given by
N (t) = {y ∈ X : kf 1 (t) − ykp ≤ `(t)kx1 − ykpt }.
Since the multivalued operator V (t) = N (t) ∩ F (t, x2t ) is measurable, there exists
f 2 (t) a measurable selection for V . So f 2 (t) ∈ F (t, x2t ) and
kf 1 (t) − f 2 (t)kp ≤ `(t)kx1 − ykpt , for each t ∈ [t0 , T ].
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
103
Let us define for each t ∈ [t0 , T ],
+∞ Y
X
k
2
h (t) =
k Y
k
X
bi (τi )S(t − t0 )ϕ(0) +
i=1
k=0
Z
Z
ξi
bj (τj )
S(t − s)f 2 (s)ds
ξi−1
i=1 j=i
t
S(t − s)f 2 (s)ds I[ξk ,ξk+1 ) (t).
+
ξk
Then we have
1
2
h (t) − h (t) =
+∞ h X
k Y
k
X
Z
Z
t
+
S(t − s) f 1 (s) − f 2 (s) ds
ξi−1
i=1 j=i
k=0
ξi
bj (τj )
i
S(t − s) f 1 (s) − f 2 (s) ds I[ξk ,ξk+1 ) (t)
ξk
1
2
p
kh (t) − h (t)k ≤
+∞ h X
k Y
k
X
k=0
Z t
+
ξi
Z
kS(t − s)kkf 1 (s) − f 2 (s)kds
kbj (τj )k
ξi−1
i=1 j=i
i
p
kS(t − s)kkf 1 (s) − f 2 (s)kds I[ξk ,ξk+1 ) (t)
ξk
≤M
p
h
k
n Y
oip Z t
p
max 1,
kbj (τj )k
·
eγ(t−s) kf 1 (s) − f 2 (s)kds
i,k
t0
j=i
Ekh1 − h2 kpt ≤ M p max 1, C p (T − t0 )p−1 E
t
Z
epγ(t−s) kf 1 (s) − f 2 (s)kp ds
t0
p
≤ M max 1, C
p
p−1
Z
t
epγ(t−s) `(s)Ekx1 − x2 kps ds
(T − t0 )
t0
h
≤ M p max 1, C p (T − t0 )p−1
T
Z
i
epγ(t−s) `(s)ds Ekx1 − x2 kpt .
t0
Taking supremum over t, we get
Z
h
p
p
p−1
1
2 p
kh − h kBT ≤ M max 1, C (T − t0 )
T
t0
i
epγ(t−s) `(s)ds kx1 − x2 kpBT .
By the analogous relation, obtained by interchanging the role of x1 and x2 , it
follows that
H(M (x1 ), M (x2 )) ≤ η kx1 − x2 kpBT ,
h
i
RT
where, η = M p max 1, C p (T − t0 )p−1 t0 epγ(t−s) `(s)ds .
From (4.1), 0 < η < 1 and hence Z is a contraction, and thus by Theorem 2.2,
Z has a fixed point x, which is a mild solution of (2.1).
5. An example
Consider the following partial differential inclusion with finite delay of the form
104
A. VINODKUMAR

Z t
∂u(x, t)
∂ 2 u(x, t)



∈
µ(t, x, θ)G(u(t + θ, x))dθ,
+


∂t
∂x2

−r

0 < x < π, t0 ≤ t ≤ T, t 6= ξk ,
u(x, ξk ) = q(k) τk u(x, ξk− ), t = ξk ,




u(0, t)
= u(π, t) = 0,



u(x, t)
= ϕ(x, t), −r ≤ t ≤ 0, 0 ≤ x ≤ π,
Let X = Lp [0, π] and the operator A =
∂2
∂x2
(5.1)
with the domain
D(A)
n
o
∂u
∂ 2u
= u ∈ X u and
are absolutely continuous,
∈
X,
u(0)
=
u(π)
=
0
.
∂x
∂x2
Then,
Au =
∞
X
n2 (u, un ), u ∈ D(A),
n=1
q
where un (x) = n2 sin(nx), n = 1, 2 . . . is the orthogonal set of eigenvectors in A.
It is well known that A generates a strongly continuous semigroup S(t) which is
compact, analytic and self adjoint and
kS(t)k ≤ eγt , for t ≥ 0, where M = 1 and γ ∈ < .
Thus S(t) is exponentially bounded.
We assume that the following conditions hold:
(i) The function µ(·) ≥ 0 is continuous in [t0 , T ] × [0, π] × [−r, 0] with
Z 0
Z π
p1
p
µ(t, x, θ)dθ = K(t, x) < ∞, K(t) =
K (t, x)dx < ∞.
−r
0
(ii) The multifunction G(·) is an Lp - Carathèodory multivalued function with
compact and convex values and
0 ≤ kG(u(θ, x))k ≤ ψ̂ ku(θ, ·)kLp , (θ, x) ∈ [t0 , T ] × [0, π],
where ψ̂(·)
( : [0, ∞) → (0,
) ∞) is continuous and nondecreasing.
h
i
k
Q
kq(j)(τj )k
(iii) E max
≤ Ĉ < ∞.
i,k
j=i
Assuming that conditions (i) -(iii) are verified, then the problem (5.1) can be
modeled as the abstract random impulsive functional differential inclusions of
the form (2.1), with
Z t
F (t, xt ) =
µ(t, x, θ)G(u(t + θ, x))dθ and bk (τk ) = q(k)τk .
−r
The next results are consequence of Theorem 3.1.
EXISTENCE RESULTS ON RANDOM IMPULSIVE DIFFERENTIAL INCLUSIONS
105
Proposition 5.1.
Assume that the conditions (i) − (iii) hold. Then there exists at least one mild
solution u of the system (5.1) provided that
Z ∞
Z T
du
−γs
e K(s)ds <
N2
,
N1 ψ̂(u)
t0
where, N1 = 2p−1 epγ(T −t0 ) Ĉ p Ekϕkp , N2 = 2p−1 epγT max 1, Ĉ p (T − t0 )p−1
1
and epγ(T −t0 ) Ĉ p ≥ 2p−1
is satisfied.
Acknowledgement. The author sincerely thank the anonymous reviewer for
his careful reading, constructive comments and suggestions to improve the quality
of the manuscript. .
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Department of Mathematics, PSG College of Technology, Coimbatore-641
004, Tamil Nadu, India.
E-mail address: vinod026@gmail.com