T. Jabeen, R. P. Agarwal, Donal O'Regan, and V. Lupulescu

14 Pages·2017·0.16 MB·English

Dynamic Systems and Applications 26 (2017) 411-424
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH
CAUSAL OPERATORS
TAHIRA JABEEN, RAVI P. AGARWAL, DONAL O’REGAN, AND VASILE LUPULESCU
Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
Department of Mathematics, Texas A&M University-Kingville,
Kingsville, TX, USA
School of Mathematics, Statistics and Applied Mathematics,
National University of Ireland, Galway, Ireland
Constantin Brancusi University, Republicii 1, 210152 Targu-Jiu, Romania
ABSTRACT. In this paper we present an existence result for a class of impulsive differential
equationswith causaloperatorsandprovethatthe solutionsetis compactin the space ofregulated
functions. The results are obtained under conditions with respect to the Hausdorff measure of
noncompactness. An application from optimal control is given to illustrate our main result.
KeyWordsandPhrases: Impulsivedifferentialequations,Causaloperators,Regulatedfunctions,
Measure of noncompactness, Optimal control
AMS (MOS) Subject Classification. 34A37; 34K05
1. Introduction and preliminaries
The study of functional equations with causal operators has recently been devel-
oped and some results on existence, stability and control are found in the monographs
[7, 14, 23]. The term causal operators or Volterra abstract operator was introduced
by Tonelli [39] (see also Tikhonov [38], [40]). The theory of these operators has the
advantage of unifying some classes of differential equations as: ordinary differential
equations, integrodifferential equations, differential equations with finite or infinite
delay, Volterra integral equations, and neutral functional equations, and so on. Many
papers in the literature address various aspects of the theory of causal operators.
Control problems involving causal operators were studied in [4, 8, 18, 36]. A new
class of abstract integral equations has been introduced in [17]. We note that differ-
ential equations with causal operators were studied by several authors, see [1], [3],
[9]–[12], [19], [25]–[33] and the references therein. The properties of the solutions of
the differential equations with causal operators were studied in [2, 21, 34, 35, 40].
The existence of solutions for impulsive differential equations with causal operators
in finite dimensional spaces were studied in [19, 26].
Received May 22, 2017 1056-2176 $15.00 (cid:13)cDynamic Publishers, Inc.
412 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU
Let E be a real separable Banach space endowed with the norm k·k. For x ∈ E
and r > 0 let B (x) := {y ∈ E;ky − xk < r} be the open ball centered at x with
r
radius r, and let B [x] be its closure. The space of all (classes of) strongly measurable
r
functions u(·) : [0,b] → E such that
b 1/p
ku(·)k := ku(t)kp < ∞
p (cid:18)Z (cid:19)
0
for1 ≤ p < ∞andku(·)k := esssup ku(t)k < ∞,willbedenotedbyLp([0,b],E).
∞ t∈[0,b]
This is a Banach space with respect to the norm ku(·)k . We denote by PC([0,b],E)
p
the set of all functions u : [0,b] → E such that u is continuous at t 6= t , left continu-
k
ous at t = t and the right limit u(t+) exists for k = 1,2,...,m. Then PC([0,b],E)
k k
is a Banach space with respect to the norm ku(·)k = sup ku(t)k.
0≤t≤b
The following definition of causal operator was given by Tonelli [39].
An operator Q : PC([0,b],E) → Lp ([0,b],E) is a causal operator or a Volterra
loc
operator if, for each τ ∈ [0,b) and for all u(·),v(·) ∈ PC([0,b],E) with u(t) = v(t)
for every t ∈ [0,τ], we have Qu(t) = Qv(t) for a.e. t ∈ [0,τ].
In this paper we study the following functional impulsive differential equation:
u′(t) = (Qu(t)), a.e. t ∈ [0,b]r{t ,t ,...,t },
1 2 m

(1.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,
 k k k k

u(0) = ξ,




where Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator,
ξ ∈ E, m ∈ N, 0 = t < t < t < ··· < t < t = bandI : E → E isacontinuous
0 1 2 m m+1 k
operator for each k = 1,2,...,m. Now we provide some examples of impulsive
differential equations that can be included in impulsive differential equations with
causal operators of the form (1.1). The impulsive differential equation
u′(t) = F(t,u(t), a.e. t ∈ [0,b]r{t ,t ,...,t },
1 2 m

u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,
 k k k k

u(0) = ξ,




can be considered as a causal impulsive differential equations by identifying F(t,u(t))
with (Qu)(t). Another example is the general integro-differential equation
u′(t) = F(t,u(t), tK(t,s,u(s))ds), a.e. t ∈ [0,b]r{t ,t ,...,t },
 0 1 2 m
(1.2) u(t+) = u(t−)+IR(u(t−)), k = 1,2,...,m,
 k k k k

u(0) = ξ.




IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 413
Also, the differential equation with “maxima”:
u′(t) = F t,u(t), maxu(s) , a.e. t ∈ [0,b]r{t ,t ,...,t },
 (cid:18) 0≤s≤t (cid:19) 1 2 m

u(t+k) = u(t−k)+Ik(u(t−k)), k = 1,2,...,m,
u(0) = ξ,




is anotherexample of a causal impulsive differential equation. Finally, we remark
that the Fredholm operator, given by
a
(Qu)(t) = K(t,s,u(s))ds,
Z
0
is a causal operator if and only if K(t,s,u) ≡ 0 for t < s < a.
We denote by χ(A) the Hausdorff measure of non-compactness of a nonempty
bounded set A ⊂ E, and it is defined by ([13], [20]):
χ(A) = inf{ε > 0;A admits a finite cover by balls of radius ≤ ε}.
This is equivalent to the measure of non-compactness introduced by Kuratowski (see
[13], [20]).
If dim(A) = sup{kx−yk;x,y ∈ A} is the diameter of the bounded set A, then
we have that χ(A) ≤ dim(A) and χ(A) ≤ 2d if sup kxk ≤ d. We recall some
x∈A
properties of χ (see [13], [20]). If A,B are bounded subsets of E and A denotes the
closure of A, then
(1) χ(A) = 0 if and only if A is compact;
(2) χ(A) = χ(A) = χ(co(A));
(3) χ(λA) = |λ|χ(A) for every λ ∈ R;
(4) χ(A) ≤ χ(B) if A ⊂ B;
(5) χ(A+B) = χ(A)+χ(B);
(6) If T : E → E is a bounded linear operator, then γ(TA) ≤ kTkγ(A).
If V ⊂ PC([0,b],E) is equicontinuous, then
χ (V) := sup χ(V(t)),
PC
t∈[0,b]
where V(t) := {u(t) : u(·) ∈ V}, is the Hausdorff measure of non-compactness in the
space PC([0,b],E) (see [13]).
We recall the following lemma due to Kisielewicz [22, Lemma 2.2].
Lemma 1.1. Let {u (·);n ≥ 1} be a subset in L1([0,b],E) for which there exists
n
m(·) ∈ L1([0,b],R ) such that ku (t)k ≤ m(t) for each n ≥ 1 and for a.e. t ∈ [0,b].
+ n
Then the function t 7→ χ(t) := χ({u (t);n ≥ 1}) is integrable on [0,b] and, for each
n
t ∈ [0,b], we have
t t
χ u (t)dt;n ≥ 1 ≤ χ(t)dt.
n
(cid:18)(cid:26)Z (cid:27)(cid:19) Z
0 0
414 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU
2. An existence result
Consider the following functional impulsive differential equation:
u′(t) = (Qu(t)), a.e. t ∈ [0,b]r{t ,t ,...,t },
1 2 m

(2.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,
 k k k k

u(0) = ξ,



where Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator,
ξ ∈ E, m ∈ N, 0 = t < t < t < ··· < t < t = b, I : E → E is continuous for
0 1 2 m m+1 k
each k = 1,2,...,m.
We consider the following assumptions:
(H1) Q : PC([0,b],E) → Lp([0,b],E), 1 ≤ p ≤ ∞, is a continuous causal operator,
and I : E → E is continuous for each k = 1,2,...,m.
k
(H2) For each r > 0 there exist ψ,η ∈ Lp([0,b],R+) such that, for each u(·) ∈
PC([0,b],E) with sup ku(t)k ≤ r, we have
0≤t≤b
k(Qu)(t)k ≤ ψ(t) for a.e. t ∈ [0,b],
t
I (u(t−)) ≤ η(τ)dτ for s,t ∈ [0,b] with s < t.
k k Z
s<Xtk<t(cid:13) (cid:13) s
(cid:13) (cid:13)
(H3) For each bounded subsets A ⊂ PC([0,b],E) and B ⊂ E there exist constants
γ , δk > 0 (k = 1,2,...,m) such that
A B
(2.2) χ((QA)(t)) ≤ γ χ(A(t)),
A
and
χ(I (B)) ≤ δkχ(B), k = 1,2,...,m,
k B
for all t ∈ [0,b], where (QA)(t) := {(Qu)(t) : u(·) ∈ A}.
By solution of (2.1) we mean a function u(·) : [0,b] → E such that u(0) = ξ,
u(·) is continuous on (t ,t ) for k = 1,2,...,m, u′(t) = (Qu)(t) for a.e. t ∈ [0,b]r
k k+1
{t ,t ,...,t }, and u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m.
1 2 m k k k k
It is easy to show that (see [13]) a function u(·) ∈ PC([0,b],E) is a solution for
(2.1) on [0,b], if and only if
t
(2.3) u(t) = u(0)+ (Qu)(s)ds+ I (u(t−)) for t ∈ [0,b].
Z k k
0 0<Xtk<t
For a fixed ξ ∈ E, by S (ξ) we denote the set of solutions u(·) of Cauchy problem
T
(2.1) on an interval [0,T] with T ∈ (0,b].
Theorem 2.1. Let Q : PC([0,b],E) → Lp([0,b],E) be a causal operator such that
conditions (H )–(H ) hold. Then, for every ξ ∈ E, there exists T ∈ (0,b] such that
1 3
the set S (ξ) is nonempty and compact set in PC([0,T],E).
T
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 415
Proof. First we shall show that there exists T ∈ (0,b] such that the set S (ξ) is
T
nonempty. Let δ > 0 be any number and let r := kξk+δ. We choose T ∈ (0,b] such
that
T T
ψ(s)ds ≤ δ/4 and η(s)ds ≤ δ/4
Z Z
0 0
and we consider the set B defined as follows
B = {u ∈ PC([0,T],E); ku(t)−ξk ≤ δ}.
Further on, we consider the integral operator Λ : B → PC([0,T],E) given by
t
(Λu)(t) = ξ + (Cu)(s)ds+ I (u(t−)), for t ∈ [0,T]
Z k k
0 0<Xtk<t
and we prove that this is a continuous operator from B into B. First, we observe
that if u(·) ∈ B, then sup ku(t)k < r, and so k(Cu)(t)k ≤ ψ(t) for a.e. t ∈ [0,T].
0≤t≤b
Hence, for each u(·) ∈ B, we have
t
k(Λu)(t)−ξk ≤ k(Cu)(s)kds+ I (u(t−))
Z k k
0 0<Xtk<t(cid:13) (cid:13)
(cid:13) (cid:13)
T
≤ k(Cu)(s)kds+ I (u(t−))
Z k k
0 0<Xtk<T (cid:13) (cid:13)
(cid:13) (cid:13)
T
≤ [ψ(t)+η(t)]dt ≤ δ
Z
0
and thus, Λ(B) ⊂ B. Further on, let u (·) → u(·) in B. Then we have
n
t
k(Λu )(t)−(Λu)(t)k ≤ k(Cu )(s)−(Cu)(s)kds
n n
Z
0
+ I (u (t−))−I (u(t−))
k n k k k
0<Xtk<t(cid:13) (cid:13)
(cid:13) (cid:13)
T
≤ k(Cu )(s)−(Cu)(s)kds
n
Z
0
+ I (u (t−))−I (u(t−))
k n k k k
0<Xtk<T (cid:13) (cid:13)
(cid:13) (cid:13)
T 1/p
≤ T1/q k(Cu )(s)−(Cu)(s)kpds
n
(cid:18)Z (cid:19)
0
+ I (u (t−))−I (u(t−))
k n k k k
0<Xtk<T (cid:13) (cid:13)
(cid:13) (cid:13)
if 1 ≤ p < ∞ and 1/p+1/q = 1, and
sup k(Pu )(t)−(Pu)(t)k ≤ T esssupk(Cu )(s)−(Cu)(s)k
n n
0≤t≤T 0≤t≤T
+ I (u (t−))−I (u(t−))
k n k k k
0<Xtk<T (cid:13) (cid:13)
(cid:13) (cid:13)
416 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU
if p = ∞. Using Lemma 1.15 from [30], by (H ) and (H ) it follows that
1 2
sup k(Λu )(t)−(Λu)(t)k → 0 as m → ∞,
m
0≤t≤T
so that Λ : B → B is a continuous operator. Moreover, it follows that Λ(B) is
bounded. Further on, we show that Λ(B) is equicontinuous on [0,T]. Let ε > 0. On
t t
the closed set [0,T], the functions t 7→ ψ(s)ds and t 7→ η(s)ds, are uniformly
0 0
continuous, and so there exist η > 0 suchRthat R
t t
ψ(τ)dτ ≤ ε/4 and η(τ)dτ ≤ ε/4
(cid:12)Z (cid:12) (cid:12)Z (cid:12)
(cid:12) s (cid:12) (cid:12) s (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
for every t,s ∈ [0,T] (cid:12)with |t−s|(cid:12)< η. Let t,s(cid:12)∈ [0,T] are(cid:12) such that |t−s| ≤ η. If we
suppose that 0 ≤ s ≤ t ≤ T then, for each u(·) ∈ B, we have
k(Λu)(t)−(Λu)(s)k
t s
= (Cu)(τ)dτ + I (u(t−))− (Cu)(τ)dτ − I (u(t−))
(cid:13)(cid:13)(cid:13)Z0 0<Xtk<t k k Z0 0<Xtk<s k k (cid:13)(cid:13)(cid:13)
(cid:13) (cid:13)
t t
(cid:13) (cid:13)
≤ k(Cu)(τ)kdτ + I (u(t−)) ≤ [ψ(τ)+η(τ)]dτ ≤ ε.
Z k k Z
s s<Xtk<t(cid:13) (cid:13) s
(cid:13) (cid:13)
Therefore, we conclude that Λ(B) is uniformly equicontinuous on [0,T]. Next, we
construct a sequence {u (·)} of continuous functions u (·) : [0,T] → E as follows.
n n≥1 n
Let n ∈ N. For i = 1,2,...,n, we define u1(t) = ξ, t ∈ [0,T] and
n
ui−1(t), if t ∈ [0,(i−1)T/n]
n

uin(t) = ξ + 0t−T/n(Cuin−1)(s)ds
 R+ I (ui−1(t−)), if t ∈ [(i−1)T/n,iT/n],
k n k
 0<tk<Pt−T/n


for i > 1. It is easy to see that if i ∈ {1,2,...,n − 1} and kui(t)k ≤ r for t ∈
n
[0,iT/n], then kui+1(t)k ≤ r for t ∈ [0,iT/n] and, by (H ), k(Cui)(t)k ≤ ψ(t) for a.e.
n 2 n
t ∈ [0,iT/n] and
t−T/n
kI (ui−1(t−))k ≤ η(s)ds
k n k Z
0<tkX<t−T/n 0
for t ∈ [0,iT/n]. It follows that
t−T/n
kui+1(t)−ξk = (cid:13) (Cui)(s)ds+ I (ui(t−))(cid:13)
n (cid:13)Z n k n k (cid:13)
(cid:13) 0 0<tkX<t−T/n (cid:13)
(cid:13) (cid:13)
(cid:13) t−T/n (cid:13)
(cid:13) (cid:13)
≤ k(Cui)(s)kds+ I (ui(t−))
Z n k n k
0 0<tkX<t−T/n(cid:13) (cid:13)
(cid:13) (cid:13)
t−T/n
≤ [ψ(τ)+η(τ)]dτ < δ,
Z
0
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 417
for all t ∈ [0,(i + 1)T/n]. Since ku1(t)k ≤ r for t ∈ [0,T/n], then by induction on
n
k we have that kui(t)k ≤ r for all k = 1,2,...,n, t ∈ [0,iT/n]. In the following,
n
to simplify the notation, we put u (·) = un(·), n ∈ N. Since un(s) = un−1(s) for all
n n n n
s ∈ [0,(n−1)T/n] and C is a causal operator, then
(Cun)(s) = (Cun−1)(s) for all s ∈ [0,(n−1)T/n].
n n
Moreover, we have that un(t−) = un−1(t−) and so I un(t−) = I un−1(t−) for
n k n k k n k k n k
0 < t < t − T/n with t ∈ [T/n,(n − 1)T/n]. Next,(cid:0)if t ∈ (cid:1)[(n − 1)(cid:0)T/n,T], (cid:1)then
k
t−T/n ≤ (n−1)T/n and consequently
t−T/n t−T/n
(Cun)(s)ds = (Cun−1)(s)ds
Z n Z n
0 0
for t ∈ [(n−1)T/n,T]. It follows that the sequence {u (·)} can be written as
n n≥1
ξ for t ∈ [0,T/n]
u (t) = 
n ξ + t−T/n(Cu )(s)ds+ I (u (t−)) for t ∈ [T/n,T],
 0 n k n k
R 0<tk<Pt−T/n

for every n ∈N. Moreover, it is easy to see that u (·) ∈ PC([0,T],E) for all n ≥ 1.
n
Further, if 0 ≤ t ≤ T/n, then we have
t
k(Λu )(t)−u (t)k = (Cu )(s)ds
n n n
(cid:13)Z (cid:13)
(cid:13) 0 (cid:13)
(cid:13) (cid:13)
T/n
(cid:13) (cid:13)
≤ kS(t−s)(Cu )(s)kds
n
Z
0
T/n
≤ ψ(s)ds.
Z
0
If T/n ≤ t ≤ T, then we have
t
k(Λu )(t)−u (t)k = (Cu )(s)ds+ I (u (t−))
n n (cid:13)(cid:13)(cid:13)Z0 n 0<Xtk<t k n k
(cid:13)
(cid:13) t−T/n
− (Cu )(s)ds− I (u (t−))(cid:13)
Z n k n k (cid:13)
0 0<tkX<t−T/n (cid:13)
(cid:13)
t (cid:13)
≤ k(Cu )(s)kds+ I (u (t−))(cid:13)
Z n k n k
t−T/n t−T/Xn<tk<t(cid:13) (cid:13)
(cid:13) (cid:13)
t
≤ [ψ(τ)+η(τ)]dτ.
Z
t−T/n
Therefore, it follows that
(2.4) sup k(Λu )(t)−u (t)k → 0 as m → ∞.
n n
0≤t≤T
Let A = {u (·);n ≥ 1}. Denote by I the identity mapping onB. From (2.4) it follows
n
that (I −Λ)(A) is a equicontinuous subset of B. Since A ⊂ (I −Λ)(A)+Λ(A) and
418 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU
the set Λ(A) is equicontinuous, then we infer that the set A is also equicontinuous
on [0,T]. Set A(t) = {u (t);n ≥ 1} for t ∈ [0,T]. Then, by Lemma 1.1 and the
n
properties of the measure of non-compactness we have
t t
χ(A(t)) ≤ χ (CA)(s)ds +χ (CA)(s)ds
(cid:18)Z (cid:19) (cid:18)Z (cid:19)
0 t−T/n
+χ I (A(t−)).
k k
X
0<tk<t−T/n
 
t
Note that, given ε > 0, we can find n(ε) > 0 such that ψ(s)ds < ε/2 for
t−T/n
t ∈ [0,T] and n ≥ n(ε). Hence we have that R
t t
χ (CA)(s)ds = χ (Cu )(s)ds;n ≥ n(ε)
m
(cid:18)Z (cid:19) (cid:18)(cid:26)Z (cid:27)(cid:19)
t−T/n t−T/n
t
≤ 2 sup ψ(s)ds < ε.
Z
n≥n(ε) t−T/n
Using the last inequality, we obtain that
t
χ(A(t)) ≤ χ (CA)(s)ds + χ (I A)(t−) .
(cid:18)Z (cid:19) k k
0 0<tkX<t−T/n (cid:0) (cid:1)
Since for every t ∈ [0,T], A(t) is bounded then, by Lemma 1.1, (H ) and the and
3
properties of the measure of non-compactness we have that
t m
χ(A(t)) ≤ χ((CA)(s))ds+ δkχ(I (A(t−)))
Z r k k
0 X
k=1
t m
≤ γ χ(A(s))ds+ δkχ(A(t−))
Z r r k
0 X
k=1
for every t ∈ [0,T]. Therefore, if we put m(t) := χ(A(t)), t ∈ [0,T], then we infer
that
t m
m(t) ≤ γ m(s)ds+ δkm(t−),
Z r r k
0 X
k=1
for every t ∈ [0,T]. Then, by Gronwall’s lemma for impulsive integral inequalities
(see [13, Theorem 1.5.1]), we must have that m(t) = χ(A(t)) = 0 for every t ∈ [0,T].
Moreover, since (see [13]) χ (A) = sup χ(A(t)) we deduce that χ (A) = 0.
PC 0≤t≤T PC
Therefore, A is relatively compact subset of PC([0,T],E). Then, by Arzela-Ascoli
theorem (see [13, Theorem 1.1.5]), and extracting a subsequence if necessary, we may
assume that the sequence {u (·)} converges on [0,T] to a function u(·) ∈ B.
n m≥1
Therefore, since
sup k(Λu)(t)−u(t)k ≤ sup k(Λu)(t)−(Λu )(t)k
n
0≤t≤T 0≤t≤T
+ sup k(Λu )(t)−u (t)k+ sup ku (t)−u(t)k
n n n
0≤t≤T 0≤t≤T
IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS 419
then, by (2.4) and by the fact that Λ is a continuous operator, we obtain that
sup)0 ≤ t ≤ Tk(Λu)(t)−u(t)k = 0. It follows that
t
u(t) = (Λu)(t) = ξ + (Cu)(s))ds+ I (u(t−))
Z k k
0 0<Xtk<t
for every t ∈ [0,T], that is u(·) = Λu(·). Hence
t
u(t) = ξ + (Cu)(s)ds+ I (u(t−)), for t ∈ [0,T]
Z k k
0 0<Xtk<t
solve the Cauchy problem (2.1), that is, u(·) ∈ S (ξ) and so S (ξ) is a nonempty
T T
set. Since Λ is continuous, then S (ξ) is a closed subset in PC([0,T],E). Moreover,
T
since S (ξ) = Λ(S (ξ)) it follows that χ((S (ξ))(t)) = χ(Λ(S (ξ))(t)) for every
T T T T
t ∈ [0,T], where (S (ξ))(t) := {u(t);u ∈ S (ξ)}. Therefore, following the same
T T
argument as above, we obtain that relatively compact subset of PC([0,T],E). Since
S (ξ) is a closed subset in PC([0,T],E) it follows that S (ξ) is a compact subset in
T T
PC([0,T],E).
Remark 2.2. The conclusion of Theorem 2.1 is also true if we replace the condition
(2.2) with the condition:
(H′) For each bounded subsets A ⊂ PC([0,b],E) there exists γ > 0 such that
3 A
(2.5) χ((CA)(t)) ≤ γ sup χ(A(s)) for every t ∈ [0,b].
A
0≤s≤t
3. An optimal control problem
In the following, we shall establish necessary conditions for the existence of an
optimal solution for the control problem:
u′(t) = (Qu)(t), for a.e. t ∈ [0,T]r{t ,t ,...,t }
1 2 m

(3.1) u(t+) = u(t−)+I (u(t−)), k = 1,2,...,m,
 k k k k

u(0) = ξ minimize g(u(T)),




where g(·) : E → R is a given function. For this aim, it will need to establish some
preliminary results. For a fixed ξ ∈ E we denote by A (ξ) the attainable set of
T
Cauchy problem (2.1); that is, A (ξ) = {u(T);u(·) ∈ S (ξ)}.
T T
Lemma 3.1. Assume that Q : PC([0,b],E) → Lp([0,b],E) is a causal operator such
that the condition (H )–(H ) hold. Then the multifunction S : E → PC([0,T],E) is
1 3 T
upper semicontinuous.
420 T. JABEEN, R. P. AGARWAL, D. O’REGAN, AND V. LUPULESCU
Proof. Let K be a closed set in PC([0,T],E) and G = {ξ ∈ E;S (ξ)∩K =6 ∅}. We
T
must show that G is closed in E. For this, let {ξ } be a sequence in G such that
n n≥1
ξ → ξ. Further on, for each n ≥ 1, let u (·) ∈ S (ξ )∩K. Then
n n T n
t
u (t) = ξ + (Qu )(s)ds+ I (u (t−))
n n Z n k n k
0 0<Xtk<t
forevery t ∈ (0,T]. AsinproofofTheorem1.1wecanshow that{u (·)} converges
n n≥1
uniformly on [0,T] to a continuous function u(·) ∈ K. Since
t
u(t) = lim u (t) = ξ + (Qu)(s)ds+ I (u(t−))
n→∞ n Z k k
0 0<Xtk<t
for every t ∈ [0,T], we deduce that u(·) ∈ S (ξ)∩K. This prove that G is closed and
T
so ξ 7→ S (ξ) is upper semicontinuous.
T
Corollary 3.2. Assume that Q : PC([0,b],E) → Lp([0,b],E) is a causal operator
such that conditions (H )–(H ) hold. Then, for any ξ ∈ E and any t ∈ [0,T] the
1 3
attainable set A (ξ) is compact in C([0,t],E)and the multifunction (t,ξ) → A (ξ)is
t t
jointly upper semicontinuous.
Theorem 3.3. Let K be a compact set in E and let g(·) : E → R be a lower
0
semicontinuous function. If Q : PC([0,b],E) → Lp([0,b],E) is a causal operator
such that the condition (H )–(H ) hold, then the control problem (3.1) has an optimal
1 3
solution; that is, there exists ξ ∈ K and u (·) ∈ S (ξ ) such that g(u (T)) =
0 0 0 T 0 0
inf{g(u(T));u(·) ∈ S (ξ ),ξ ∈ K }.
T 0 0 0
Proof. From Corollary 3.2 we deduce that the attainable set A (ξ) is upper semi-
T
continuous. Then the set A (K ) = {u(T);u(·) ∈ S (ξ),ξ ∈ K } = ∪ A (ξ) is
T 0 T 0 ξ∈K0 T
compact in E and so, since g(·) is lower semicontinuous, there exists ξ ∈ K such
0 0
that g(u (T)) = inf{g(u(T));u(·) ∈ S (ξ ),ξ ∈ K }.
0 T 0 0 0
4. An example
Consider the following impulsive differential equation:
u′(t) = g(t)+ tK(t,s)f(s,u(s))ds, a.e. t ∈ [0,b]r{t ,t ,...,t },
0 1 2 m

(4.1) u(t+) = u(t−)R+ tk+1λ(t)dt u(t−), k = 1,2,...,m,
 k k (cid:16) tk (cid:17) k
R
u(0) = ξ,




where g(·),λ(·) ∈ Lp([0,b],E), p ≥ 1, and K : [0,b] × [0,b] → L(E) is strongly
continuous. Let M := sup kK(t,s)k. Assume that
s,t∈[0,b]