Nonconvex valued impulsive functional differential inclusions with variable times
In this paper, a fixed point theorem due to Schaefer combined with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued operators with nonempty closed and decomposable values is used to investigate the existence of solutions for first and second order impulsive funct...
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irk-123456789-1772092021-02-12T01:25:48Z Nonconvex valued impulsive functional differential inclusions with variable times Belarbi, A. Benchohra, M. Ouahab, A. In this paper, a fixed point theorem due to Schaefer combined with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued operators with nonempty closed and decomposable values is used to investigate the existence of solutions for first and second order impulsive functional differential inclusions with variable times. З допомогою теореми Шефера про нерухому точку, а також теореми Брессана i Коломбо про вибiр для нижньо напiвнеперервних багатозначних операторiв iз непорожнiми замкненими розкладними значеннями вивчено питання iснування розв’язкiв функцiонально-диференцiальних включень першого та другого порядкiв зi змiнним часом. 2007 Article Nonconvex valued impulsive functional differential inclusions with variable times / A. Belarbi, M. Benchohra, A. Ouahab // Нелінійні коливання. — 2007. — Т. 10, № 4. — С. 443-463. — Бібліогр.: 26 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177209 517.9 en Нелінійні коливання Інститут математики НАН України |
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In this paper, a fixed point theorem due to Schaefer combined with a selection theorem due to Bressan
and Colombo for lower semicontinuous multivalued operators with nonempty closed and decomposable
values is used to investigate the existence of solutions for first and second order impulsive functional differential inclusions with variable times. |
| format |
Article |
| author |
Belarbi, A. Benchohra, M. Ouahab, A. |
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Belarbi, A. Benchohra, M. Ouahab, A. Nonconvex valued impulsive functional differential inclusions with variable times Нелінійні коливання |
| author_facet |
Belarbi, A. Benchohra, M. Ouahab, A. |
| author_sort |
Belarbi, A. |
| title |
Nonconvex valued impulsive functional differential inclusions with variable times |
| title_short |
Nonconvex valued impulsive functional differential inclusions with variable times |
| title_full |
Nonconvex valued impulsive functional differential inclusions with variable times |
| title_fullStr |
Nonconvex valued impulsive functional differential inclusions with variable times |
| title_full_unstemmed |
Nonconvex valued impulsive functional differential inclusions with variable times |
| title_sort |
nonconvex valued impulsive functional differential inclusions with variable times |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/177209 |
| citation_txt |
Nonconvex valued impulsive functional differential inclusions with variable times / A. Belarbi, M. Benchohra, A. Ouahab // Нелінійні коливання. — 2007. — Т. 10, № 4. — С. 443-463. — Бібліогр.: 26 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT belarbia nonconvexvaluedimpulsivefunctionaldifferentialinclusionswithvariabletimes AT benchohram nonconvexvaluedimpulsivefunctionaldifferentialinclusionswithvariabletimes AT ouahaba nonconvexvaluedimpulsivefunctionaldifferentialinclusionswithvariabletimes |
| first_indexed |
2025-07-15T15:14:41Z |
| last_indexed |
2025-07-15T15:14:41Z |
| _version_ |
1837726414265122816 |
| fulltext |
UDC 517 . 9
NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL
INCLUSIONS WITH VARIABLE TIMES
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНI НЕОПУКЛI ВКЛЮЧЕННЯ
З IМПУЛЬСНОЮ ДIЄЮ
A. Belarbi, M. Benchohra, A. Ouahab
Univ. Sidi Bel Abbès
B.P. 89, 22000, Sidi Bel Abbès, Algérie
e-mail: aek_belarbi@yahoo.fr
benchohra@univ-sba.dz
agh_ouahab@yahoo.fr
In this paper, a fixed point theorem due to Schaefer combined with a selection theorem due to Bressan
and Colombo for lower semicontinuous multivalued operators with nonempty closed and decomposable
values is used to investigate the existence of solutions for first and second order impulsive functional di-
fferential inclusions with variable times.
З допомогою теореми Шефера про нерухому точку, а також теореми Брессана i Коломбо про
вибiр для нижньо напiвнеперервних багатозначних операторiв iз непорожнiми замкненими роз-
кладними значеннями вивчено питання iснування розв’язкiв функцiонально-диференцiальних
включень першого та другого порядкiв зi змiнним часом.
1. Introduction. In this paper, we are concerned with the existence of solutions to some classes
of initial value problems for first and second order impulsive functional and neutral functional
differential inclusions. Initially, in Section 3, we will consider the first order impulsive functional
differential inclusion
y′(t) ∈ F (t, yt), a.e. t ∈ J := [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m, (1.1)
y(t) = φ(t), t ∈ [−r, 0],
where F : J × D → P(IR) is a multivalued map with nonempty compact values, D = {ψ :
[−r, 0] → IR;ψ is continuous everywhere except for a finite number of points t̄ at which ψ(t̄−)
and ψ(t̄+) exist and ψ(t̄−) = ψ(t̄)}, φ ∈ D, 0 < r < ∞, τk : IR → IR, Ik : IR → IR, k =
= 1, 2, . . . ,m, are given functions satisfying some assumptions that will be specified later.
For any function y defined on [−r, T ] and any t ∈ J, we denote by yt the element of D
defined by
yt(θ) = y(t+ θ), θ ∈ [−r, 0].
Here yt(·) represents the history of the state from time t − r up to the present time t. Later, in
c© A. Belarbi, M. Benchohra, A. Ouahab, 2007
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4 443
444 A. BELARBI, M. BENCHOHRA, A. OUAHAB
Section 4, we study the second order impulsive functional differential inclusion of the form
y′′ ∈ F (t, yt), a.e. t ∈ J := [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t−)), t = τk(y(t)),
(1.2)
y′(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m,
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η,
where F, Ik, and φ are as in problem (1.1), Ik ∈ C(IR, IR) and η ∈ IR. Sections 5 and 6
are devoted to the existence of solutions for initial value problems for first and second order
impulsive neutral functional differential inclusions. More precisely, in these last sections, we
consider the IVPs
d
dt
[y(t)− g(t, yt)] ∈ F (t, yt), a.e. t ∈ J = [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m, (1.3)
y(t) = φ(t), t ∈ [−r, 0],
where F, Ik are as in problem (1.1), g : J ×D → IR is a given function, and
d
dt
[y′(t)− g(t, yt)] ∈ F (t, yt), a.e. t ∈ J = [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t−)), t = τk(y(t)),
(1.4)
y′(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m,
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η,
where F, Ik, φ, g, η and Ik are as in the above cited problems.
The theory of impulsive differential equations have become more important in recent years
in some mathematical models of real processes and phenomena studied in physics, chemical
technology, population dynamics, biotechnology and economics. There has been a significant
development in impulse theory, in recent years, especially in the area of impulsive differential
equations with fixed moments; see the monographs of Bainov and Simeonov [1], Lakshmi-
kantham et al. [2], and Samoilenko and Perestyuk [3] and the references therein. The theory
of impulsive differential equations with variable times is relatively less developed due to the
difficulties created by the state-dependent impulses. Recently, some interesting extensions to
impulsive differential equations with variable times have been done by Bajo and Liz [4], Fri-
gon and O’Regan [5 – 7], Kaul et al. [8], Kaul and Liu [9, 10] Lakshmikantham et al. [11, 12],
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 445
Liu and by Ballinger [13] and Vatsala and Vasundara Devi [14, 15]. Very recently, by mean
of Schaefer’s theorem and the concept of upper and lower solutions, Benchohra et al. [16 –
18] have considered different classes of impulsive functional differential equations. The same
tools have been applied to a variety of impulsive functional differential inclusions with convex
valued right-hand side by the same authors [19, 20]. The main theorems of this paper extend
those considered by Benchohra et al. [19, 20]. Our approach here is based on the Schaefer’s
fixed point theorem [21, p. 29] combined with a selection theorem due to Bressan and Colombo
[22] for lower semicontinuous multivalued operators with nonempty closed and decomposable
values.
2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts
from multivalued analysis which are used throughout this paper.
By C(J, IR) we denote the Banach space of all continuous functions from J into IR with the
norm
‖y‖∞ := sup{|y(t)| : t ∈ J}.
For φ ∈ D the norm of φ is defined by
‖φ‖D = sup{|φ(θ)| : θ ∈ [−r, 0]}.
L1([0, T ], IR) denotes the Banach space of measurable functions y : J −→ IR which are
Lebesgue integrable normed by
‖y‖L1 :=
T∫
0
|y(t)|dt for all y ∈ L1(J, IR).
ACi([0, T ], IR) is the space of i-times differentiable functions y : [0, T ] → IR, whose ith deri-
vative, y(i), is absolutely continuous.
LetA be a subset of [0, T ]×D. A isL⊗B measurable ifA belongs to the σ-algebra generated
by all sets of the form J ×D, where J is Lebesgue measurable in J and D is Borel measurable
inD.A subsetA of L1([0, T ], IR) is decomposable if for all u, v ∈ A and J ⊂ [0, T ] measurable,
the function uχJ + vχJ−J ∈ A, where χJ stands for the characteristic function of J. Let E be
a Banach space, X a nonempty closed subset of E and G : X → P(E) a multivalued operator
with nonempty closed values.G is lower semi-continuous (l.s.c.) if the set {x ∈ X : G(x)∩B 6=
6= ∅} is open for any open set B in E. G has a fixed point if there is x ∈ X such that x ∈ G(x).
For more details on multivalued maps we refer to the books of Deimling [23], Górniewicz [24]
and Hu and Papageorgiou [25].
Definition 2.1. Let Y be a separable metric space and let N : Y → P(L1([0, T ], IR)) be a
multivalued operator. We say N has property (BC) if
1) N is lower semi-continuous (l.s.c.);
2) N has nonempty closed and decomposable values.
Let F : J × D → P(IR) be a multivalued map with nonempty compact values. Assign to
F the multivalued operator
F : C([−r, T ], IR) → P(L1([0, T ], IR))
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
446 A. BELARBI, M. BENCHOHRA, A. OUAHAB
by letting
F(y) = {w ∈ L1([0, T ], IR) : w(t) ∈ F (t, yt) for a.e. t ∈ [0, T ]}.
The operator F is called the Niemytzki operator associated with F.
Definition 2.2. Let F : J × D → P(IR) be a multivalued function with nonempty compact
values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Niemytzki operator
F is lower semi-continuous and has nonempty closed and decomposable values.
Next, we state a selection theorem due to Bressan and Colombo in [22].
Theorem 2.1. Let Y be a separable metric space and let N : Y → P(L1([0, T ], IR)) be
a multivalued operator which has property (BC). Then N has a continuous selection; i.e., there
exists a continuous function (single-valued) g : Y → L1(J, IR) such that g(y) ∈ N(y) for every
y ∈ Y.
Let us introduce the following hypotheses which are assumed hereafter:
(H1) F : [0, T ]×D −→ P(IR) is a nonempty, compact-valued, multivalued map such that:
a) (t, u) 7→ F (t, u) is L ⊗ B measurable;
b) u 7→ F (t, u) is lower semi-continuous for a.e. t ∈ [0, T ];
(H2) for each r > 0, there exists a function hr ∈ L1([0, T ], IR+) such that
‖F (t, u)‖P := sup{|v| : v ∈ F (t, u)} ≤ hr(t) for a.e. t ∈ [0, T ];
and for u ∈ D with ‖u‖D ≤ r.
The following lemma is crucial in the proof of our main theorem:
Lemma 2.1 [26]. Let F : [0, T ]×D → P(IR) be a multivalued map with nonempty, compact
values. Assume (H1) and (H2) hold. Then F is of l.s.c. type.
3. First order impulsive FDIs. The main result of this section concerns the IVP (1.1). Before
stating and proving this one, we give the definition of a solution of the IVP (1.1). We shall
consider the space
Ω = {y : [0, T ] −→ IR : there exist 0 < t1 < . . . < tm < T such that tk = τk(y(tk)),
yk ∈ C(Jk, IR), k = 0, . . . ,m, and there exist y(t−k ) and y(t+k ), k = 1, . . . ,m,
with y(t−k ) = y(tk)}.
Here yk := y/Jk
, k = 0, . . . ,m, Jk = (tk, tk+1], t0 = 0 and tm+1 = T.
PC = {y : [−r, T ] → IR : y ∈ D ∩ Ω}.
Definition 3.1. A function y ∈ PC ∩ ∪m
k=0AC((tk, tk+1), IR) is said to be a solution of (1.1)
if there exists v(t) ∈ F (t, yt) a.e. t ∈ [0, T ] such that y′(t) = v(t) a.e. t ∈ [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t)), t = τk(y(t)), k = 1, . . . ,m and y(t) = φ(t), t ∈ [−r, 0].
We are now in a position to state and prove our existence result for the problem (1.1). We
first list the following additional hypotheses.
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 447
(H3) The functions τk ∈ C1(IR, IR) for k = 1, . . . ,m. Moreover,
0 < τ1(x) < . . . < τm(x) < T for all x ∈ IR.
(H4) There exist constants ck > 0 such that
|Ik(x)| ≤ ck for each x ∈ IR, k = 1, . . . ,m.
(H5) There exists a continuous nondecreasing function ψ : [0,+∞) −→ (0,+∞), and
p ∈ L1([0, T ], IR+) such that
‖F (t, u)‖P ≤ p(t)ψ(‖u‖D) for a.e. u ∈ [0, T ] and each u ∈ D
with
∞∫
1
dγ
ψ(γ)
= +∞.
(H6) For all (t, x) ∈ [0, T ]× IR and for all yt ∈ D we have
τ ′k(x)v(t) 6= 1 for k = 1, . . . ,m, for all v ∈ F(y).
(H7) For all x ∈ IR
τk(Ik(x)) ≤ τk(x) < τk+1(Ik(x)) for k = 1, . . . ,m− 1.
Theorem 3.1. Suppose that hypotheses (H1 ) – (H7 ), are satisfied. Then the impulsive initial
value problem (1.1) has at least one solution.
Proof. (H1) and (H2) imply, by Lemma 2.1, that F is of lower semi-continuous type. Then
from Theorem 2.1 there exists a continuous function f : C([−r, T ], IR) → L1([0, T ], IR) such
that f(y) ∈ F(y) for all y ∈ PC.
Step 1. Consider the problem,
y′(t) = f(yt), t ∈ [0, T ],
(3.1)
y(t) = φ(t), t ∈ [−r, 0].
It is obvious that if y ∈ C([−r, T ], IR) is a solution of the problem (3.1), then y is a solution
to the problem (1.1). Transform the problem into a fixed point problem. Consider the operator
N : C([−r, T ], IR) → C([−r, T ], IR) defined by:
N(y)(t) :=
φ(t) if t ∈ [−r, 0],
φ(0) +
t∫
0
f(ys)ds if t ∈ [0, T ].
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
448 A. BELARBI, M. BENCHOHRA, A. OUAHAB
We shall show that N is a continuous and completely continuous operator.
Claim 1. N is continuous.
Let {yn} be a sequence such that yn −→ y in C([−r, T ], IR). Then
|N(yn(t))−N(y(t))| ≤
t∫
0
|f(yns)− f(ys)|ds ≤
T∫
0
|f(yns)− f(ys)|ds.
Since the function f is continuous, then
‖N(yn)−N(y)‖∞ ≤ ‖f(yn)− f(y)‖L1 → 0 as n → ∞.
Claim 2. N maps bounded sets into bounded sets in C([−r, T ], IR).
Indeed, it is enough to show that for any q > 0 there exists a positive constant ` such that,
for each y ∈ Bq := {y ∈ C([−r, T ], IR) : ‖y‖∞ ≤ q}, we have ‖N(y)‖∞ ≤ `. From (H2) and
(H3) we have
|N(y)(t)| ≤ ‖φ‖D +
t∫
0
|f(ys)|ds ≤ ‖φ‖D + ‖hq‖L1 = `.
Claim 3. N maps bounded sets into equicontinuous sets of C([−r, T ], IR).
Let τ1, τ2 ∈ [0, T ], τ1 < τ2, and Bq be a bounded set of C([−r, T ], IR). Let y ∈ Bq. Then
|N(y)(τ2)−N(y)(τ1)| ≤
τ2∫
τ1
hq(s)ds.
As τ2 −→ τ1 the right-hand side of the above inequality tends to zero. The equicontinuity
for the cases τ1 < τ2 ≤ 0 and τ1 ≤ 0 ≤ τ2 is obvious.
As a consequence of Claims 1 to 3, together with the Arzela – Ascoli theorem, we conclude
that N := C([−r, T ], IR) −→ C([−r, T ], IR) is continuous and completely continuous.
Claim 4. Now it remains to show that the set
E(N) := {y ∈ C([−r, T ], IR) : y = λN(y) for some 0 < λ < 1}
is bounded.
Let y ∈ E(N). Then y = λN(y) for some 0 < λ < 1. Thus
y(t) = λ
φ(0) +
t∫
0
f(ys) ds
, t ∈ [0, T ].
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 449
This implies by (H5) that for each t ∈ [0, T ] we have
|y(t)| ≤ ‖φ‖D +
t∫
0
p(s)ψ(‖ys‖D)ds. (3.2)
We consider the function µ defined by
µ(t) := sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ T.
Let t∗ ∈ [−r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ J, by inequality (3.2) we have for t ∈ J
µ(t) ≤ ‖φ‖D +
t∫
0
p(s)ψ(µ(s))ds. (3.3)
If t∗ ∈ [−r, 0] then µ(t) = ‖φ‖D and the inequality (3.3) holds. Let us take the right-hand side
of the inequality (3.3) as v(t). Then we have
c = v(0) = ‖φ‖D, µ(t) ≤ v(t), t ∈ J,
and
v′(t) = p(t)ψ(µ(t)), t ∈ [0, T ].
Using the nondecreasing character of ψ we get
v′(t) ≤ p(t)ψ(v(t)), t ∈ [0, T ].
By using (H5) this implies for each t ∈ [0, T ] that
v(t)∫
v(0)
dτ
ψ(τ)
≤
T∫
0
p(s)ds < +∞.
This inequality implies that there exists a constant K such that v(t) ≤ K, t ∈ J, and hence
µ(t) ≤ K, t ∈ [0, T ]. Since for every t ∈ [0, T ], ‖yt‖D ≤ µ(t), we have
‖y‖∞ ≤ K ′ := max{‖φ‖D,K},
where K ′ depends only on T and on the functions p and ψ. This shows that E(N) is bounded.
Set X := C([−r, T ], IR). As a consequence of Schaefer’s theorem (see [21, p. 29]), we
deduce that N has a fixed point y which is a solution to problem (3.1). Denote this solution by
y1.
Define the function
rk,1(t) = τk(y1(t))− t for t ≥ 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
450 A. BELARBI, M. BENCHOHRA, A. OUAHAB
(H3) implies that
rk,1(0) 6= 0 for k = 1, . . . ,m.
If
rk,1(t) 6= 0 on [0, T ] for k = 1, . . . ,m,
i.e.,
t 6= τk(y1(t)) on [0, T ] and for k = 1, . . . ,m,
then y1 is a solution of the problem (1.1).
It remains to consider the case when
r1,1(t) = 0 for some t ∈ [0, T ].
Now since
r1,1(0) 6= 0
and r1,1 is continuous, there exists t1 > 0 such that
r1,1(t1) = 0, and r1,1(t) 6= 0 for all t ∈ [0, t1).
Thus by (H3) we have
rk,1(t) 6= 0 for all t ∈ [0, t1), and k = 1, . . . ,m.
Step 2. Consider now the following problem:
y′(t) = f(yt), a.e. t ∈ [t1, T ], (3.4)
y(t) = y1(t), t ∈ [t1 − r, t1], y(t+1 ) = I1(y1(t−1 )).
Set
C∗ = C([t1 − r, t1], IR) ∩ C1,
where
C1 =
{
y ∈ C((t1, T ], IR) : y(t+1 ) exists
}
.
Transform the problem (3.4) into a fixed point problem. Consider the operator N1 : C∗ → C∗
defined by
N1(y)(t) :=
y1(t) if t ∈ [t1 − r, t1],
I1(y1(t1)) +
t∫
t1
f(ys) ds if t ∈ (t1, T ].
As in Step 1 we can show that N1 is continuous and completely continuous. Now we prove only
that the set
E(N1) := {y ∈ C∗ : y = λN1(y) for some 0 < λ < 1}
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 451
is bounded.
Let y ∈ E(N1). Then y = λN1(y) for some 0 < λ < 1. Thus
y(t) = λ
I1(y(t−1 )) +
t∫
t1
f(ys) ds
, t ∈ [t1, T ].
This implies by (H4) and (H5) that for each t ∈ [t1, T ] we have
|y(t)| ≤ c1 +
t∫
t1
p(s)ψ(‖ys‖D)ds. (3.5)
We consider the function µ defined by
µ(t) := sup{|y(s)| : t1 − r ≤ s ≤ t}, t1 ≤ t ≤ T.
Let t∗ ∈ [r− t1, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ [t1 − r, T ], by inequality (3.5) we have for
t ∈ [t1 − r, T ]
µ(t) ≤ c1 +
t∫
t1
p(s)ψ(µ(s))ds. (3.6)
If t∗ ∈ [t1 − r, t1] then µ(t) = ‖y1‖∞ and the inequality (3.6) holds. Let us take the right-hand
side of the inequality (3.6) as v(t). Then we have
c∗ = v(t1) = c1, µ(t) ≤ v(t), t ∈ [t1, T ],
and
v′(t) = p(t)ψ(µ(t)), t ∈ [t1, T ].
Using the nondecreasing character of ψ we get
v′(t) ≤ p(t)ψ(v(t)), t ∈ [t1, T ].
By using (H5) this implies for each t ∈ [t1, T ] that
v(t)∫
v(t1)
dτ
ψ(τ)
≤
T∫
0
p(s)ds < +∞.
This inequality implies that there exists a constant K such that v(t) ≤ K1, t ∈ [t1, T ], and
hence µ(t) ≤ K1, t ∈ [t1, T ]. Since for every t ∈ [t1, T ], ‖yt‖D ≤ µ(t), we have
‖y‖∞ ≤ K2 := max{‖y1‖∞,K1},
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452 A. BELARBI, M. BENCHOHRA, A. OUAHAB
where K2 depends only on T and on the functions p, ψ1 and ψ. This shows that E(N1) is
bounded.
Set X := C([t1− r, T ], IR). As a consequence of Schaefer’s theorem we deduce that N1 has
a fixed point y which is a solution to problem (3.4). Denote this solution by y2. Define
rk,2(t) = τk(y2(t))− t for t ≥ t1.
If
rk,2(t) 6= 0 on (t1, T ] and for all k = 1, . . . ,m
then
y(t) =
{
y1(t) if t ∈ [0, t1],
y2(t) if t ∈ (t1, T ],
is a solution of the problem (1.1).
It remains to consider the case when
r2,2(t) = 0 for some t ∈ (t1, T ].
By (H7) we have
r2,2(t+1 ) = τ2(y2(t+1 ))− t1 = τ2(I1(y1(t1)))− t1 > τ1(y1(t1))− t1 = r1,1(t1) = 0.
Since r2,2 is continuous, there exists t2 > t1 such that
r2,2(t2) = 0,
and
r2,2(t) 6= 0 for all t ∈ (t1, t2).
It is clear by (H3) that
rk,2(t) 6= 0 for all t ∈ (t1, t2), k = 2, . . . ,m.
Suppose now that there is s̄ ∈ (t1, t2] such that
r1,2(s̄) = 0.
From (H7) it follows that
r1,2(t+1 ) = τ1(y2(t+1 ))− t1 = τ1(I1(y1(t1)))− t1 ≤ τ1(y1(t1))− t1 = r1,1(t1) = 0.
Thus the function r1,2 attains a nonnegative maximum at some point s1 ∈ (t1, T ]. Since
y′2(t) = f(y2t),
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 453
we have
r′1,2(s1) = τ ′1(y2(s1))y′2(s)− 1 = 0.
Therefore
τ ′1(y2(s1)).f(y2s1
) = 1,
which contradicts (H6).
Step 3. We continue this process and taking into account that ym+1 := y
∣∣∣
[tm,T ]
is a solution
to the problem
y′(t) = f(yt), a.e. t ∈ (tm, T ),
y(t) = ym(t), t ∈ [tm − r, tm], y(t+m) = Im(ym(t−m)).
The solution y of the problem (1.1) is then defined by
y(t) =
y1(t) if t ∈ [−r, t1],
y2(t) if t ∈ (t1, t2],
. . . . . . . . . . . . . . . . . .
ym+1(t) if t ∈ (tm, T ].
4. Second order impulsive FDIs. In this section we give an existence result for the IVP (1.2).
Let us start by defining what we mean by a solution of problem (1.2).
Definition 4.1. A function y ∈ PC ∩ ∪m
k=0AC
1((tk, tk+1), IR) is said to be a solution of (1.2)
if there exists v(t) ∈ F (t, yt) a.e. t ∈ [0, T ] such that y′′(t) = v(t) a.e. on [0, T ], t 6= τk(y(t)),
y(t+) = Ik(y(t−)), t = τk(y(t)), y′(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m, y(t) = φ(t),
t ∈ [−r, 0] and y′(0) = η.
Theorem 4.1. Assume (H1) – (H7) and the condition
(H8) there exist constants dk > 0 such that
|Ik(y)| ≤ dk for each y ∈ IR, k = 1, . . . ,m,
are satisfied. Then the IVP (1.2) has at least one solution.
Proof. (H1) and (H2) imply, by Lemma 2.1, that F is of lower semi-continuous type. Then
from Theorem 2.1 there exists a continuous function f : C([−r, T ], IR) → L1([0, T ], IR) such
that f(y) ∈ F(y) for all y ∈ C([−r, T ], IR).
Step 1. Consider the following problem:
y′′(t) = f(yt), t ∈ [0, T ],
(4.1)
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η.
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454 A. BELARBI, M. BENCHOHRA, A. OUAHAB
Consider the operator N : C([−r, T ], IR) → C([−r, T ], IR) defined by
N(y)(t) :=
φ(t) if t ∈ [−r, 0],
φ(0) + tη +
t∫
0
(t− s)f(ys) ds if t ∈ [0, T ].
As in Theorem 3.1 we can show thatN is continuous and completely continuous. Now we prove
only that the set
E(N) := {y ∈ C([−r, T ], IR) : y = λN(y) for some 0 < λ < 1}
is bounded. Let y ∈ E(N). Then y = λN(y) for some 0 < λ < 1. Thus
y(t) = λ
φ(0) + tη +
t∫
0
(t− s)f(ys)ds
.
This implies by (H5) and (H8) that for each t ∈ J we have
|y(t)| ≤ ‖φ‖D + T |η|+
t∫
0
(T − s)p(s)ψ(‖ys‖)ds. (4.2)
We consider the function µ defined by
µ(t) := sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ T.
Let t∗ ∈ [−r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ J, by the inequality (4.2) we have for t ∈ J
|µ(t)| ≤ ‖φ‖D + T |η|+
t∫
0
(T − s)p(s)ψ(µ(s))ds. (4.3)
If t∗ ∈ [−r, 0] then µ(t) = ‖φ‖ and the inequality (4.3) holds. Let us take the right-hand side of
inequality (4.3) as v(t). Then we have
v(0) = ‖φ‖D + T |η| and v′(t) = (T − t)p(t)ψ(µ(t)), t ∈ [0, T ].
Using the nondecreasing character of ψ we get
v′(t) ≤ (T − t)p(t)ψ(v(t)), t ∈ [0, T ].
This implies together with (H5) for each t ∈ [0, T ] that
v(t)∫
v(0)
dτ
ψ(τ)
≤
T∫
0
(T − s)p(s)ds ≤ T
T∫
0
p(s)ds < +∞.
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 455
This inequality implies that there exists a constant b such that v(t) ≤ b, t ∈ [0, T ], and hence
µ(t) ≤ b, t ∈ J. Since for every t ∈ [0, T ], ‖yt‖D ≤ µ(t), we have
‖y‖∞ ≤ max{‖φ‖D, b},
where b depends only on T and on the functions p and ψ. This shows that E(N) is bounded.
Set X := C([−r, T ], IR). As a consequence of Schaefer’s theorem we deduce that N has a
fixed point y which is a solution to problem (4.1). Denote this solution by y1 and continue as in
Theorem 3.1.
Step 2. Consider the following problem:
y′′(t) = f(yt), a.e. t ∈ [t1, T ],
y(t) = y1(t), t ∈ [t1 − r, t1], y(t+1 ) = I1(y1(t−1 )), (4.4)
y′(t+1 ) = I1(y1(t−1 )).
Consider the operator, N1 : C∗ → C∗ defined by
N1(y)(t) :=
y1(t) if t ∈ [t1 − r, t1],
I1(y1(t1)) + tI1(y1(t1)) +
t∫
t1
(t− s)f(ys) ds if t ∈ (t1, T ].
As in Theorem 3.1 we can show that N1 is continuous and completely continuous. Now we
prove only that the set
E(N1) := {y ∈ C∗ : y = λN1(y) for some 0 < λ < 1}
is bounded. Let y ∈ E(N1). Then y = λN1(y) for some 0 < λ < 1. Thus
y(t) = λ
I1(y1(t−1 )) + tI1(y1(t−1 )) +
t∫
t1
(t− s)f(ys)ds
.
This implies by (H4), (H5) and (H8) that for each t ∈ J we have
|y(t)| ≤ c1 + Td1 +
t∫
t1
(T − s)p(s)ψ(‖ys‖) ds. (4.5)
We consider the function µ defined by
µ(t) := sup{|y(s)| : t1 − r ≤ s ≤ t}, t1 ≤ t ≤ T.
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456 A. BELARBI, M. BENCHOHRA, A. OUAHAB
Let t∗ ∈ [t1 − r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ [t1, T ], by the inequality (4.5) we have for
t ∈ [t1, T ]
|µ(t)| ≤ c1 + Td1 +
t∫
0
(T − s)p(s)ψ(µ(s))ds. (4.6)
If t∗ ∈ [t1 − r, t1] then µ(t) = ‖y1‖∞ and the inequality (4.6) holds. Let us take the right-hand
side of inequality (4.6) as v(t). Then we have
v(t1) = c1 + Td1, and v′(t) = (T − t)p(t)ψ(µ(t)), t ∈ [t1, T ].
Using the nondecreasing character of ψ we get
v′(t) ≤ (T − t)p(t)ψ(v(t)), t ∈ [t1, T ].
This implies together with (H5) for each t ∈ [t1, T ] that
v(t)∫
v(t1)
dτ
ψ(τ)
≤
T∫
t1
(T − s)p(s)ds ≤ T
T∫
t1
p(s)ds < +∞.
This inequality implies that there exists a constant b such that v(t) ≤ b, t ∈ [t1, T ], and hence
µ(t) ≤ b, t ∈ [t1, T ]. Since for every t ∈ [t1, T ], ‖yt‖D ≤ µ(t), we have
‖y‖∞ ≤ max{‖y1‖∞, b},
where b depends only on T and on the functions p, and ψ.
Set X := C∗. As a consequence of Schaefer’s theorem [21] we deduce that N1 has a fixed
point y which is a solution to problem (4.4). Denote this one by y2 and continue as in Step 2 of
the Theorem 3.1.
Step 3. We continue this process and taking into account that ym+1 := y
∣∣∣
[tm,T ]
is a solution
to the problem
y′′(t) = f(yt), a.e. t ∈ (tm, T ),
y(t) = ym(t), t ∈ [tm − r, tm], y(t+m) = Im(ym(t−m)),
y′(t+m) = Im(ym−1(t−m)).
The solution y of the problem (1.2) is then defined by
y(t) =
y1(t) if t ∈ [−r, t1],
y2(t) if t ∈ (t1, t2],
. . . . . . . . . . . . . . . . . .
ym+1(t) if t ∈ (tm, T ].
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 457
5. First order impulsive neutral FDIs. In this section we are concerned with the existence
of solutions for problem (1.3).
Definition 5.1. A function y ∈ Ω ∩ ∪m
k=0AC((tk, tk+1), IR) is said to be a solution of (1.3)
if there exists v(t) ∈ F (t, yt) a.e. t ∈ [0, T ] such that
d
dt
[y(t)− g(t, yt)] = v(t) a.e. t ∈ [0, T ],
t 6= τk(y(t)), y(t+) = Ik(y(t−)), t = τk(y(t)), k = 1, . . . ,m, and y(t) = φ(t), t ∈ [−r, 0].
We first list the following hypotheses:
(A1) the function g is completely continuous and for any bounded set B in C([−r, T ], IR),
the set {t → g(t, yt) : y ∈ B} is equicontinuous in C([0, T ], IR) and there exist constants
0 ≤ d1 < 1 and d2 ≥ 0 such that
|g(t, u)| ≤ d1‖u‖D + d2, t ∈ [0, T ], u ∈ D, k = 1, . . . ,m;
(A2) g is a nonnegative function;
(A3) τk is a nonincreasing function and
Ik(x) ≤ x for all x ∈ IR, k = 1, . . . ,m;
(A4) for all x ∈ IR
τk(x) < τk+1(Ik(x)) for k = 1, . . . ,m;
(A5) for all t ∈ [0, T ] and for all yt ∈ D we have
τ ′k(y(t)− g(t, yt))v(t) 6= 1 for k = 1, . . . ,m for all v ∈ F(y).
Theorem 5.1. Assume that hypotheses (H1) – (H5) and (A1) – (A4) hold, then the IVP (1.3)
has at least one solution on [−r, T ].
Proof. (H1) and (H2) imply by Lemma 2.1 that F is of lower semi-continuous type. Then
from Theorem 2.1 there exists a continuous function f : C([−r, T ], IR) → L1([0, T ], IR) such
that f(y) ∈ F(y) for all y ∈ C([−r, T ], IR).
Step 1. Consider now the following problem:
d
dt
[y(t)− g(t, yt)] = f(yt), t ∈ [0, T ],
(5.1)
y(t) = φ(t), t ∈ [−r, 0].
Consider the operator N2 : C([−r, T ], IR) → C([−r, T ], IR) defined by
N2(y)(t) :=
φ(t) if t ∈ [−r, 0],
φ(0)− g(0, φ(0)) + g(t, yt) +
t∫
0
f(ys)ds if t ∈ [0, T ].
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458 A. BELARBI, M. BENCHOHRA, A. OUAHAB
Clearly from (A1) N2 is a continuous and completely continuous operator. Now it remains to
show that the set
E(N2) := {y ∈ C([−r, T ], IR) : y = λN2(y) for some λ ∈ (0, 1)}
is bounded.
Let y ∈ E(N2). Then λN2(y) = y for some 0 < λ < 1 and
y(t) = λ
φ(0)− g(0, φ(0)) + g(t, yt) +
t∫
0
f(ys)ds
.
This implies by (H5) and (A1) that for each t ∈ [0, T ] we have
|y(t)| ≤ |φ(0)|+ |g(0, φ(0))|+ |g(t, yt)|+
t∫
0
p(s)ψ(‖ys‖)ds
or
|y(t)| ≤ (1 + d1)‖φ‖D + 2d2 + d1‖yt‖+
t∫
0
p(s)ψ(‖ys‖)ds. (5.2)
We consider the function µ defined by
µ(t) := sup{|y(s)| : −r ≤ s ≤ t}, t ∈ [0, T ].
Let t∗ ∈ [−r, t] be such that µ = |y(t∗)|. If t∗ ∈ [0, T ], by the inequality (5.2), we have for
t ∈ [0, T ]
µ(t) ≤ (1 + d1)‖φ‖D + 2d2 + d1µ(t) +
t∫
0
p(s)ψ(µ(s))ds.
Thus
µ(t) ≤ 1
1− d1
(1 + d1)‖φ‖D + 2d2 +
t∫
0
p(s)ψ(µ(s))ds
. (5.3)
If t∗ ∈ [−r, 0] then µ(t) = ‖φ‖D and the inequality (5.3) holds. Let us take the right-hand side
of the inequality (5.3) as v(t), then we have
v(0) =
1
1− d1
[(1 + d1)‖φ‖D + 2d2] and v′(t) = p(t)ψ(µ(t)).
Since ψ is nondecreasing we have
v′(t) = p(t)ψ(µ(t)) ≤ p(t)ψ(v(t)) for all t ∈ [0, T ].
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NONCONVEX VALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS WITH VARIABLE TIMES 459
From this inequality, it follows that
t∫
0
v′(s)
ψ(v(s))
ds ≤
t∫
0
p(s)ds.
By using (H5) we then have
v(t)∫
v(0)
du
ψ(u)
≤
t∫
0
p(s)ds ≤
T∫
0
p(s) ds < +∞.
This inequality implies that there exists a constant b depending only on T and on the function p
such that
|y(t)| ≤ b for each t ∈ [0, T ].
Hence
‖y‖∞ ≤ b.
This shows that E(N2) is bounded, and hence N2 has a fixed point y which is a solution to
problem (5.1).
Denote this solution by y1. Define the function
rk,1(t) = τk(y1(t))− t for t ≥ 0.
(H3) implies that
rk,1(0) 6= 0 for k = 1, . . . ,m.
If
rk,1(t) 6= 0 on [0, T ] for k = 1, . . . ,m,
i.e.,
t 6= τk(y1(t)) on [0, T ] and for k = 1, . . . ,m,
then y1 is a solution of the problem (1.3).
It remains to consider the case when
r1,1(t) = 0 for some t ∈ [0, T ].
Now since
r1,1(0) 6= 0
and r1,1 is continuous, there exists t1 > 0 such that
r1,1(t1) = 0 and r1,1(t) 6= 0 for all t ∈ [0, t1).
Thus by (H3) we have
rk,1(t) 6= 0 for all t ∈ [0, t1), k = 1, . . . ,m.
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460 A. BELARBI, M. BENCHOHRA, A. OUAHAB
Step 2. Consider now the following problem:
d
dt
[y(t)− g(t, yt)] = f(yt), a.e. t ∈ [t1, T ],
(5.4)
y(t) = y1(t), t ∈ [t1 − r, t1], y(t+1 ) = I1(y1(t−1 )).
Consider the operator N2 : C∗ → C∗ defined by
N2(y)(t) :=
y1(t) if t ∈ [t1 − r, t1],
I1(y1(t1))− g(t1, y1t1) + g(t, yt) +
t∫
t1
f(ys)ds if t ∈ [t1, T ].
As in Step 1 we can show that N2 is continuous and completely continuous, and the set
E(N2) := {y ∈ C∗ : y = λN2(y) for some 0 < λ < 1}
is bounded.
Set X := C∗. As a consequence of Schaefer’s theorem we deduce that N2 has a fixed point
y which is a solution to problem (5.4). Denote this solution by y2. Define
rk,2(t) = τk(y2(t))− t for t ≥ t1.
If
rk,2(t) 6= 0 on (t1, T ] and for all k = 1, . . . ,m
then
y(t) =
y1(t) if t ∈ [0, t1],
y2(t) if t ∈ (t1, T ],
is a solution of the problem (1.3). It remains to consider the case when
r2,2(t) = 0 for some t ∈ (t1, T ].
By (A4) we have
r2,2(t+1 ) = τ2(y2(t+1 ))− t1 = τ2(I1(y1(t1)))− t1 > τ1(y1(t1))− t1 = r1,1(t1) = 0.
Since r2,2 is continuous, there exists t2 > t1 such that
r2,2(t2) = 0,
and
r2,2(t) 6= 0 for all t ∈ (t1, t2).
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It is clear by (H3) that
rk,2(t) 6= 0 for all t ∈ (t1, t2), k = 2, . . . ,m.
Suppose now that there is s̄ ∈ (t1, t2] such that
r1,2(s̄) = 0.
Consider the function L1(t) = τ1(y2(t)− g(t, y2t))− t.
From (A2) – (A4) it follows that
L1(s̄) = τ1(y2(s̄)− g(s̄, y2s̄))− s̄ ≥ τ1(y2(s̄))− s̄ = r1,2(s̄) = 0.
Thus the function L1 attains a nonnegative maximum at some point s1 ∈ (t1, T ]. Since
d
dt
[y2(t)− g(t, y2t)] = f(y2t),
it follows that
L′1(s1) = τ ′1(y2(s1)− g(s1, y2s1
))
d
dt
[y2(s1)− g(s1, y2s1
)]− 1 = 0.
Therefore
[τ ′1(y2(s1)− g(s1, y2s1
))]f(y2s1
) = 1,
which contradicts (A5).
Step 3. We continue this process and taking into account that ym+1 := y
∣∣∣
[tm,T ]
is a solution
to the problem
d
dt
[y(t)− g(t, yt)] = f(yt), a.e. t ∈ (tm, T ),
y(t) = ym(t), t ∈ [tm−1 − r, tm−1], y(t+m) = Im(ym(t−m)).
The solution y of the problem (1.3) is then defined by
y(t) =
y1(t) if t ∈ [−r, t1],
y2(t) if t ∈ (t1, t2],
. . . . . . . . . . . . . . . . . .
ym+1(t) if t ∈ (tm, T ].
6. Second order impulsive neutral FDIs. In this section we study the initial value problem
(1.4). Its solution is defined in a similar maner. Let us introduce the following hypotheses:
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462 A. BELARBI, M. BENCHOHRA, A. OUAHAB
(A6) for all (t, s̄, x) ∈ [0, T ]× [0, T ]× IR and for all yt ∈ D we have
τ ′k(x)
Ik(y(s̄))− g(s̄, ys) + g(t, yt) +
t∫
s̄
v(s)ds
6= 1 for k = 1, . . . ,m, for all v ∈ F(y);
(A7) there exists a continuous nondecreasing function ψ : [0,∞) −→ (0,∞) and p ∈
∈ L1([0, T ], IR+) such that
|f(t, u)| ≤ p(t)ψ(‖u‖D) for a.e. t ∈ [0, T ] and each u ∈ D
with
∞∫
1
dγ
γ + ψ(γ)
= ∞.
Theorem 6.1. Assume that hypotheses (H1) – (H4), (H7) – (H8), (A1), and (A6) – (A7) are
satisfied. Then the IVP (1.4) has at least one solution on [−r, T ].
Proof. The details of the proof are left to the reader.
Acknowledgements. This work was done while M. Benchohra and A. Ouahab were visiting
the Abdus salam international centre for theoretical physics in Trieste (Italy). It is a pleasure
for them to acknowledge its financial support and the warm hospitality.
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Received 10.04.06,
after revision — 29.10.06
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