Impulsive differential inclusions involving evolution operators in separable Banach spaces
We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\...
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| author | Benchohra, M. Nieto, J. J. Ouahab, A. Беньчохра, М. Ньєто, Дж. Дж. Оахаб, А. |
| author_facet | Benchohra, M. Nieto, J. J. Ouahab, A. Беньчохра, М. Ньєто, Дж. Дж. Оахаб, А. |
| author_sort | Benchohra, M. |
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| datestamp_date | 2020-03-18T19:31:34Z |
| description | We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for
the following first-order impulsive semilinear differential inclusions with initial and boundary conditions:
$$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$
$$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$
$$y(0) = a$$
and
$$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$
$$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$
$$Ly = a,$$
where $J = IR_+,\; 0 = t_0 < t_1 |
| first_indexed | 2026-03-24T02:27:06Z |
| format | Article |
| fulltext |
UDC 517.9
M. Benchohra (Univ. Sidi Bel-Abbès, Algérie),
J. J. Nieto (Univ. Santiago de Compostela, Spain),
A. Ouahab (Univ. Sidi Bel-Abbès, Algérie)
IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION
OPERATORS IN SEPARABLE BANACH SPACES
IМПУЛЬСНI ДИФЕРЕНЦIАЛЬНI ВКЛЮЧЕННЯ, ЩО МIСТЯТЬ
ОПЕРАТОРИ В СЕПАРАБЕЛЬНИХ БАНАХОВИХ ПРОСТОРАХ
We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for
the following first-order impulsive semilinear differential inclusions with initial and boundary conditions:
y′(t)−A(t)y(t) ∈ F (t, y(t)), for a.e. t ∈ J\{t1, . . . , tm, . . .},
y(t+k )− y(t−k ) = Ik(y(t
−
k )), k = 1, . . . ,
y(0) = a
and
y′(t)−A(t)y(t) ∈ F (t, y(t)), for a.e. t ∈ J\{t1, . . . , tm, . . .},
y(t+k )− y(t−k ) = Ik(y(t
−
k )), k = 1, . . . ,
Ly = a,
where J = IR+, 0 = t0 < t1 < . . . < tm < . . . ; (m ∈ IN), limk→∞ tk = ∞, A(t) is the infinitesimal generator of
a family of evolution operator U(t, s) on a separable Banach space E, and F is a set-valued mapping. The functions Ik
characterize the jump of solutions at the impulse points tk, k = 1, . . . . The mapping L : PCb → E is a bounded linear
operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions,
and the absolute retractness.
Наведено деякi результати про iснування м’яких розв’язкiв та вивчено топологiчну будову множин розв’язкiв для
наступних iмпульсних напiвлiнiйних диференцiальних включень першого порядку з початковими та граничними
умовами:
y′(t)−A(t)y(t) ∈ F (t, y(t)) для майже кожного t ∈ J\{t1, . . . , tm, . . .},
y(t+k )− y(t−k ) = Ik(y(t
−
k )), k = 1, . . . ,
y(0) = a
та
y′(t)−A(t)y(t) ∈ F (t, y(t)), для майже кожного t ∈ J\{t1, . . . , tm, . . .},
y(t+k )− y(t−k ) = Ik(y(t
−
k )), k = 1, . . . ,
Ly = a,
де J = IR+, 0 = t0 < t1 < . . . < tm < . . . ; (m ∈ IN), limk→∞ tk = ∞, A(t) — iнфiнiтезимальний генератор сiм’ї
операторiв еволюцiї U(t, s) на сепарабельному банаховому просторi E та F — багатозначне вiдображення. Функцiї
Ik характеризують стрибки розв’язкiв в точках iмпульсної дiї tk, k = 1, . . . . Вiдображення L : PCb → E є обме-
женим лiнiйним оператором. Також дослiджено компактнiсть множини розв’язкiв, деякi властивостi регулярностi
операторних розв’язкiв та абсолютну ретрактнiсть.
c© M. BENCHOHRA, J. J. NIETO, A. OUAHAB, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7 867
868 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
1. Introduction. Differential equations with impulses were considered for the first time by Milman
and Myshkis [47] and then followed by a period of active research which culminated with the
monograph by Halanay and Wexler [34]. Many phenomena and evolution processes in the field of
physics, chemical technology, population dynamics, and natural sciences may change state abruptly
or be subject to short-term perturbations. (See for instance [2, 42, 43] and the references therein.)
These perturbations may be seen as impulses. Impulsive problems arise also in various applications
in communications, chemical technology, mechanics (jump discontinuities in velocity), electrical
engineering, medicine, and biology. These perturbations may be seen as impulses. For instance,
in the periodic treatment of some diseases, impulses correspond to the administration of a drug
treatment. In environmental sciences, impulses correspond to seasonal changes of the water level of
artificial reservoirs. Their models are described by impulsive differential equations and inclusions.
Various mathematical results (existence, asymptotic behavior, . . . ) have been obtained so far (see
[4, 10, 12, 44, 51, 54, 55] and the references therein).
Given a real separable Banach space E with norm ‖ · ‖, consider the following problem:
y′(t)−A(t)y(t) ∈ F (t, y(t)), for a.e. t ∈ J\{t1, . . . , tm, . . .},
∆yt=tk = Ik(y(t−k )), k = 1, . . . ,
y(0) = a ∈ E,
(1)
where J = IR+ 0 = t0 < t1 < . . . < tm < tm+1 . . . (m ∈ IN), limk→∞ tk =∞. F : J×E → P(E)
is a multivalued map, A(t) is the infinitesimal generator of a family of evolution {U(t, s)}. We
always assume that the operator A(t) is closed and densely defined in its domain D(A(t)), which is
independent of t, Ik ∈ C(E,E), k = 1, . . . ,m, ∆y|t=tk = y(t+k )−y(t−k ), y(t+k ) = limh→0+ y(tk+h)
and y(t−k ) = limh→0+ y(tk − h) stand for the right and the left limits of y(t) at t = tk, respectively.
Later, we study the following impulsive boundary-value problems:
y′(t)−A(t)y(t) ∈ F (t, y(t)), for a.e. t ∈ J\{t1, . . . , tm, . . .},
y(t+k )− y(t−k ) = Ik(y(t−k )), k = 1, . . . ,
Ly = a,
(2)
where L : PCb → E is a bounded linear operator.
Many properties of solutions for differential equations and inclusions, such as stability or os-
cillation, require global properties of solutions. This is the main motivation to look for sufficient
conditions that ensure global existence of solutions for impulsive differential equations and inclu-
sions. In this direction, some questions have been discussed by Baghli and Benchohra [7 – 9], Graef
and Ouahab [26, 28], Guo [30, 31], Guo and Liu [32], Henderson and Ouahab [35 – 37], Marino et
al. [46], Ouahab [49, 50], Stamov and Stamova [56], Weng [60], and Yan [61, 62].
In case the space E is finite-dimensional and J is compact interval, some existence results of
solutions for problems (1) and (2) in the particular case Ay = λy, λ ∈ IR, have been obtained
in [11, 13, 27]. Very recently some existence results and solutions sets on unbounded interval was
studied by Djebali et al. [1, 21, 22]. For infinite dimensional space and A is infinitesimal generator
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 869
of a C0-semigroup and Ly = a − y(0) + y(b). Problems of existence and solutions sets of above
problems on bounded interval was solved by Djebali et al. [19, 20, 23].
The goal in this work is to complement and extend some recent results to the case of infinite-
dimensional spaces; moreover the right-hand side nonlinearity may be either convex or nonconvex.
Our approach here is based on a nonlinear alternative for compact u.s.c. maps. Then, we present
some existence results and investigate the compactness of solution set, some regularity of operator
solutions and absolute retract (in short AR) of solution is also proved.
2. Preliminaries. In this section, we recall from the literature some notations, definitions, and
auxiliary results which will be used throughout this paper. Let (E, ‖ · ‖) be a separable Banach space,
J = [0,∞) an interval in IR and Cb(J,E) the Banach space of all continuous and bounded functions
from J into E with the norm
‖y‖∞ = sup{‖y(t)‖ : t ∈ J}.
B(E) refers to the Banach space of linear bounded operators from E into E with the norm
‖N‖B(E) = sup{‖N(y)‖ : ‖y‖ = 1}.
A function y : J → E is called measurable provided for every open subset U ⊂ E, the set y−1(U) =
= {t ∈ J : y(t) ∈ U} is Lebesgue measurable. A measurable function y : J → E is Bochner
integrable if ‖y‖ is Lebesgue integrable. For properties of the Bochner integral, see, e.g., Yosida
[58]. In what follows, L1(J,E) denotes the Banach space of functions y : J −→ E, which are
Bochner integrable with norm
‖y‖1 =
∞∫
0
‖y(t)‖dt.
Denote by P(E) = {Y ⊂ E : Y 6= ∅}, Pcl(E) = {Y ∈ P(E) : Y closed}, Pb(E) = {Y ∈
∈ P(E) : Y bounded}, Pcv(E) = {Y ∈ P(E) : Y convex}, Pcp(E) = {Y ∈ P(E) : Y compact}.
2.1. Multivalued analysis. Let (X, d) and (Y, ρ) be two metric spaces and G : X → Pcl(Y )
be a multivalued map. A single-valued map g : X → Y is said to be a selection of G and we write
g ⊂ G whenever g(x) ∈ G(x) for every x ∈ X. G is called upper semicontinuous (u.s.c. for short)
on X if for each x0 ∈ X the set G(x0) is a nonempty, closed subset of X, and if for each open set
N of Y containing G(x0), there exists an open neighborhood M of x0 such that G(M) ⊆ Y. That
is, if the set G−1(V ) = {x ∈ X, G(x) ∩ V 6= ∅} is closed for any closed set V in Y. Equivalently,
G is u.s.c. if the set G+1(V ) = {x ∈ X, G(x) ⊂ V } is open for any open set V in Y.
G is said to be completely continuous if it is u.s.c. and, for every bounded subset A ⊆ X,
G(A) is relatively compact, i.e., there exists a relatively compact set K = K(A) ⊂ X such that
G(A) =
⋃
{G(x), x ∈ A} ⊂ K. G is compact if G(X) is relatively compact. It is called locally
compact if, for each x ∈ X, there exists U ∈ V(x) such that G(U) is relatively compact. G is
quasicompact if, for each subset A ⊂ X, compact, G(A) is relatively compact.
The following two results are easily deduced from the limit properties.
Lemma 2.1 (see, e.g., [6], Theorem 1.4.13). If G : X −→ Pcp(X) is u.s.c., then for any x0∈X,
lim sup
x→x0
G(x) = G(x0).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
870 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
Lemma 2.2 (see, e.g., [6], Lemma 1.1.9). Let (Kn)n∈IN ⊂ K ⊂ X be a sequence of subsets
where K is compact in the separable Banach space X. Then
co (lim sup
n→∞
Kn) =
⋂
N>0
co (
⋃
n≥N
Kn),
where coC refers to the closure of the convex hull of C.
Definition 2.1. A multivalued map F : [0,∞)→ Pcl(Y ) is said measurable provided for every
open U ⊂ Y, the set F+1(U) is Lebesgue measurable.
We have the following lemma.
Lemma 2.3 [16, 25]. The mapping F is measurable if and only if for each x ∈ Y, the function
ζ : J → [0,+∞) defined by
ζ(t) = dist(x, F (t)) = inf{‖x− y‖ : y ∈ F (t)}, t ∈ J,
is Lebesgue measurable.
The following two lemmas are needed in this paper. The first one is the celebrated Kuratowski –
Ryll – Nardzewski selection theorem.
Lemma 2.4 ([25], Theorem 19.7). Let Y be a separable metric space and F : J → P(Y ) a
measurable multivalued map with nonempty closed values. Then F has a measurable selection.
Lemma 2.5 [20, 63]. Let F : [0, b]→ Pcp(Y ) be a measurable multivalued map and u : [0, b]→
→ E a measurable function. Then there exists a measurable selection f of F such that for a.e.
t ∈ [0, b],
|u(t)− f(t)| ≤ d(u(t), F (t)).
We denote the graph of G to be the set Gr(G) = {(x, y) ∈ X × Y, y ∈ G(x)}.
Definition 2.2. G is closed if Gr(G) is a closed subset of X × Y, i.e., for every sequences
(xn)n∈IN ⊂ X and (yn)n∈IN ⊂ Y, if xn → x∗, yn → y∗ as n → ∞ with yn ∈ F (xn), then
y∗ ∈ G(x∗).
We recall the following two results; the first one is classical.
Lemma 2.6 ([18], Proposition 1.2). If G : X → Pcl(Y ) is u.s.c., then Gr(G) is a closed subset
of X×Y. Conversely, if G is locally compact and has nonempty compact values and a closed graph,
then it is u.s.c.
Lemma 2.7 [20, 40]. If G : X → Pcp(Y ) is quasicompact and has a closed graph, then G is
u.s.c.
Given a separable Banach space (E, ‖ · ‖), for a multivalued map F : J × E → P(E), denote
‖F (t, x)‖P := sup{‖v‖ : v ∈ F (t, x)}.
Definition 2.3. A multivalued map F is called a Carathéodory function if
(a) the function t 7→ F (t, x) is measurable for each x ∈ E;
(b) for a.e. t ∈ J, the map x 7→ F (t, x) is upper semicontinuous. Furthermore, F is L1 —
Carathéodory if it is locally integrably bounded, i.e., for each positive r, there exists hr ∈ L1(J, IR+)
such that
‖F (t, x)‖P ≤ hr(t), for a.e. t ∈ J and all ‖x‖ ≤ r.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 871
For each x ∈ C(J,E), the set
SF,x =
{
f ∈ L1(J,E) : f(t) ∈ F (t, x(t)) for a.e. t ∈ J
}
is known as the set of selection functions.
Remark 2.1. (a) For each x ∈ C(J,E), the set SF,x is closed whenever F has closed values. It
is convex if and only if F (t, x(t)) is convex for a.e. t ∈ J.
(b) From [38] (see also [45] when E is finite-dimensional), we know that SF,x is nonempty if
and only if the mapping t 7→ inf{‖v‖ : v ∈ F (t, x(t))} belongs to L1(J). It is bounded if and only
if the mapping t 7→ ‖F (t, x(t))‖P belongs to L1(J); this particularly holds true when F is L1 —
Carathéodory. For the sake of completeness, we refer also to Theorem 1.3.5 in [40] which states that
SF,x contains a measurable selection whenever x is measurable and F is a Carathéodory function.
Consider the Hausdorff pseudometric Hd : P(E)× P(E) −→ IR+ ∪ {∞} defined by
Hd(A,B) = max
{
sup
a∈A
d(a,B), sup
b∈B
d(A, b)
}
,
where d(A, b) = infa∈A d(a, b), d(a,B) = infb∈B d(a, b). (Pcl(X), Hd) is a generalized metric
space and (Pb,cl(E), Hd) is a metric space space (see [41]). Also, notice that if x0 ∈ E, then
d(x0, A) = inf
x∈A
d(x0, x) while Hd({x0}, A) = sup
x∈A
d(x0, x).
Definition 2.4. A multivalued operator N : E → Pcl(E) is called
(a) γ-Lipschitz if there exists γ > 0 such that
Hd(N(x), N(y)) ≤ γd(x, y), for each x, y ∈ E,
(b) a contraction if it is γ-Lipschitz with γ < 1.
Notice that if N is γ−Lipschitz, then
∀x, y ∈ E : Hd(F (x), F (y)) ≤ γd(x, y).
For further readings and details on multivalued analysis, we refer to the books by Andres and
Górniewicz [3], Aubin and Cellina [5], Aubin and Frankowska [6], Deimling [18], Górniewicz [25],
Hu and Papageorgiou [38, 39], Kamenskii et al. [40], and Tolstonogov [57].
3. Evolution family.
Definition 3.1. A family of operators {U(t, s)}t≥s ⊂ B(E), with t, s ∈ IR or t, s ∈ IR+, is
called an evolution family if satisfying the conditions:
(i) U(t, s) = U(t, s) ◦ U(s, τ), for t ≥ s ≥ τ,
(ii) U(t, t) = I; here I denotes the identity operator in E,
(iii) for each x ∈ E, the function (t, s)→ U(t, s)x is continuous for t ≥ s.
In what follows, for the family {A(t), t ∈ J} of closed densely defined linear unbounded opera-
tors on the Banach space E we assume that it satisfies the following assumptions:
(i) the domain D(A(t)) is independent of t and is dense in E,
(ii) for t ≥ 0, the resolvent R(λ,A(t)) = (λI−A(t))−1 exists for all λ with Reλ ≤ 0, and there
is a constant M independent λ and t such that
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
872 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
‖R(λ,A(t))‖B(E) ≤M(1 + |λ|)−1, for Reλ ≤ 0,
(iii) there exist constants L > 0 and 0 < α ≤ 1 such that
‖(A(t)−A(s))A−1(θ)‖B(E) ≤ L|t− θ|α, for t, s, θ ∈ J,
(iv) the mapping (s, b] 3 t→ U(t, s) ∈ B(E) is continuous with respect to the uniform operator
topology of B(E). Moreover, this continuity is uniform with respect to s lying in sets bounded away
from t, i.e., as long as t− s ≥ β for any fixed β > 0.
Definition 3.2. The solution operator U(t, s) is called exponentially bounded if there are con-
stants L(U) > 0 and ω ≥ 0 such that
‖U(t, s)‖B(E) ≤ L(U)e−ω(t−s), t, s ≥ 0.
More details on evolution families can be fond in (see Engel and Nagel [24]) and Pazy [52].
4. Existence results. Consider the Banach space PCb = {y ∈ PC(J,E) : y is bounded}, where
PC(J,E) = {y : J → E, yk ∈ C((tk, tk+1], E), k = 0, . . . ,
y(t−k ) and y(t+k ) exist and satisfy y(tk) = y(t−k ) for k = 1, . . .}
and yk := y|(tk,tk+1]. Endowed with the norm
‖y‖b = sup{‖y(t)‖ : t ∈ J},
PCb is a Banach space. Next we define what we mean by a solution to problem (1).
Definition 4.1. A function y ∈ PC is said to be a mild solution of problem (1) if there exists
v ∈ L1(J,E) such that v(t) ∈ F (t, y(t)) a.e. on J, and
y(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, s)Ik(y(t−k )), for a.e. t ∈ J.
In what follows assume that the evolution family is exponentially bounded.
5. Existence and compactness of solutions set. In this section, we present a global existence
result and prove the compactness of solution set for the problem (1) by using a nonlinear alternative
for multivalued maps combined with a compactness argument. The nonlinearity is u.s.c. with respect
to the second variable and satisfies a Nagumo growth condition in all this part assume that E is a
reflexive Banach space. Hereafter we assume that the Carathéodory multivalued map F : J × E →
→ P(E) has compact and convex values.
Theorem 5.1. The impulsive functions Ik ∈ C(E,E) satisfy
(H1) there exist ck, dk > 0 such that
‖Ik(x)‖ ≤ ck‖x‖+ dk, for every x ∈ E, k = 1, 2, . . . ,
with
L(U)
∞∑
k=1
ck < 1 and
∞∑
k=1
dk <∞;
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 873
(H2) there exist a continuous nondecreasing function ψ : [0,∞) −→ (0,∞) and p ∈ L1(J, IR+)
such that
‖F (t, x)‖P ≤ p(t)ψ(‖x‖), for a.e. t ∈ J and each x ∈ IRn,
with
∞∫
0
m(s)ds <
∞∫
c
du
ψ(u)
,
where
m(s) =
L(U)p(s)
1− L(U)
∑∞
k=1
ck
and c =
L(U)
(
‖a‖+
∑∞
k=1
dk
)
1− L(U)
∑∞
k=1
ck
;
(H3) for every t− s > 0, U(t, s) is compact;
(H4) for every M > 0 and ε > 0 there exist an L1(J,E)−function b = b(M, ε) such that
F (t, y)− b ⊂ εB(0,M), for a.a. t ≥ T and all y ∈ B(0,M),
where
B(0,M) = {y ∈ E : ‖y‖ ≤M}.
Then problem (1) has at least one solution. Moreover, the solution set S(a) is compact and the
multivalued map S : a→ S(a) is u.s.c.
First, recall the well-known nonlinear alternative of Leray – Schauder for multivalued maps (see,
e.g., [29, 25]).
Lemma 5.1. Let X be a Banach space with C ⊂ X a convex. Assume U is a relatively open
subset of C with 0 ∈ U and G : U → Pcp,c(X) be an upper semicontinuous and compact map. Then
either,
(a) there is a point u ∈ ∂U and λ ∈ (0, 1) with u ∈ λG(u), or
(b) G has a fixed point in U.
Definition 5.1. Let E be a Banach space. A sequence (vn)n∈IN ⊂ L1([a, b], E) is said to be
semicompact if
(a) it is integrably bounded, i.e., there exists q ∈ L1([a, b], IR+) such that
‖vn(t)‖ ≤ q(t), for a.e. t ∈ [a, b] and every n ∈ IN,
(b) the image sequence (vn(t))n∈IN is relatively compact in E for a.e. t ∈ [a, b].
The following important result follows from Dunford – Pettis theorem (see [40], Proposition
4.2.1).
Lemma 5.2. Every semicompact sequence L1([a, b], E) is weakly compact in L1([a, b], E).
When the nonlinearity takes convex values, Mazur’s lemma, may be useful:
Lemma 5.3 [58]. Let E be a normed space and (xk)k∈IN ⊂ E a sequence weakly converging
to a limit x ∈ E. Then there exists a sequence of convex combinations ym =
∑m
k=1
αmkxk with
αmk > 0 for k = 1, 2, . . . ,m and
∑m
k=1
αmk = 1 which converges strongly to x.
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874 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
The following compactness criterion on unbounded domains is a simple extension of a compact-
ness criterion in Cb(J,E) (see [17, p. 62; 53]).
Lemma 5.4. A setM⊂ PCb is relatively compact if it satisfies the following conditions:
(a)M is uniformly bounded in PCb(J,E),
(b) the functions belonging to M are almost equicontinuous on J, i.e., equicontinuous on every
compact interval of J,
(c) the functions from M are equiconvergent, that is, given ε > 0, there exist T (ε) > 0 and
δ(ε) > 0 such that if x, y ∈M with ‖x(T )− y(T )‖ ≤ δ(ε), then |x(t)− y(t)| < ε for any t ≥ T (ε)
and x ∈M,
(d) for every t ∈ J, the set {x(t) : x ∈M} is relatively compact.
Proof [Proof of Theorem 5.1].
Step 1. Existence of solutions. Consider the operator N : PCb → P(PCb) defined for y ∈ PCb
by
N(y) =
h ∈ PCb : h(t) =
U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
+
∑
0<tk<t
U(t, tk)Ik(y(tk)), for a.e. t ∈ J,
(3)
where v ∈ SF,y = {v ∈ L1(J,E) : v(t) ∈ F (t, y(t)), a.e. t ∈ J}. Note that, from [59], Theorem
5.10 or [45], the set SF,y is nonempty if and only if the mapping t 7→ inf{‖v‖ : v ∈ F (t, y(t))}
belongs to L1(J). It is further bounded if and only if the mapping t 7→ ‖F (t, y(t))‖P belongs to
L1(J); this particularly holds true when F satisfies (H2). Moreover, fixed points of the operator N
are mild solutions of problem (1). We shall show that N satisfies the assumptions of Lemma 5.1.
Finally notice that since SF,y is convex (because F has convex values), then N takes convex values.
Claim 1. N(PCb) ⊂ PCb. Indeed, if y ∈ PCb and h ∈ N(y) then there exists v ∈ SF,y such
that
h(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)), for a.e. t ∈ J. (4)
Since v ∈ L1(J), we have
‖h(t)‖ ≤ L(U)‖a‖+
t∫
0
‖F (s, y(s))‖Pds+ L(U)
∑
0<tk<t
‖Ik(y(tk))‖ ≤
≤ L(U)‖a‖+
t∫
0
p(s)ψ(‖y(s)‖)ds+ L(U)
∑
0<tk<t
(ck‖y(tk)‖+ dk).
Hence
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‖h‖PCb ≤ L(U)‖a‖+ L(U)ψ(‖y‖PCb)
∞∫
0
p(s)ds+ L(U)
∞∑
k=1
ck‖y‖PCb +
∞∑
k=1
dk.
This shows that N sends bounded sets into bounded sets in PCb.
Claim 2. N sends bounded sets in PCb into almost equicontinuous sets of PCb.
Let r > 0, Br := {y ∈ PCb : ‖y‖∞ ≤ r} be a bounded set in PCb, τ1, τ2 ∈ J, τ1 < τ2, and
y ∈ Br. For each h ∈ N(y), we have
‖h(τ2)− h(τ1)‖ ≤ ‖U(τ1, 0)− U(τ2, 0)‖B(E)‖a‖+
+L(U)
τ2∫
τ1
‖v(s)‖ds
τ1∫
0
‖U(τ1, s)− U(τ2, s)‖B(E)‖v(s)‖ds+
+L(U)
∑
τ1<tk<τ2
‖Ik(y(tk))‖ ≤
≤ L(U)ψ(r)
τ2∫
τ1
p(s)ds+ L(U)
∑
τ1<tk<τ2
(ckr + dk).
Since
∑∞
k=1
ck < ∞,
∑∞
k=1
dk < ∞ and p ∈ L1(J, IR+), the right-hand term tends to zero as
|τ1− τ2| → 0, proving equicontinuity for the case where t 6= t−i , i = 1, . . . . To prove equicontinuity
at t = ti for some i ∈ IN∗, fix ε0 > 0 such that {tj : j 6= i} ∩ [ti − ε0, ti + ε0] = ∅. Then for each
0 < ε < ε0, we have the estimates
‖h(ti)− h(ti − ε)‖ ≤ ‖U(ti − ε, 0)− U(ti, 0)‖B(E)‖a‖+
ti∫
ti−ε
‖v(s)‖ds ≤
≤ ‖U(ti − ε, 0)− U(ti, 0)‖B(E)‖a‖+ ψ(r)
ti∫
ti−ε
p(s)ds.
Since p ∈ L1(J, IR+), the right-hand term tends to 0 as ε→ 0. The equicontinuity at t+i , i = 1, . . . , is
proved in the same way. Now we to show that N maps Bq into a precompact set in E. Let 0 < t ≤ b
and let 0 < ε < t. For y ∈ Bq, define
hε(t) = U(t− ε, 0)a+ U(ε, 0)
t−ε∫
0
U(t− ε, 0)f(s)ds+
+U(ε, 0)
∑
0<tk<t−ε
U(t− ε, tk)Ik(y(tk)).
Then
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876 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
‖h(t)− hε(t)‖ ≤
t∫
t−ε
‖U(t, s)‖B(E)p(s)ds+
∑
t−ε<tk<t
‖U(t, tk)‖B(E)(ckr + dk),
which tends to 0 as ε → 0. Therefore, there are precompact sets arbitrarily close to the set H(t) =
= {h(t) : h ∈ N(y)}. This set is then precompact in E.
Claim 3. We now show the stability of the set N(B(0, r)), i.e., we show that for every ε >
> 0, there exist T (ε), δ(ε) > 0 such that if h ∈ N(x) and h∗ ∈ N(y) with x, y ∈ B(0, r) and
‖h(T (ε))−h∗(T (ε))‖ ≤ δ(ε), then ‖h(t)−h∗(t)‖ ≤ T (ε) for every t ≥ T and each h ∈ N(B(0, r)).
Fix ε > 0 and choose T1 = T1(ε) such that
F (t, y)− b(t) ⊂ εα
8L(U)
B(0, 1), for a.a. t ≥ T1 and all y ∈ B(0, 1).
Since
∑∞
k=1
ck <∞,
∑∞
k=1
dk <∞ there exist k0 such that
L(U)
∞∑
k=k0
(ckr + dk) ≤
ε
8
.
From (H2) we have
F (t, y) ⊂ p(t)ψ(r)B(0, 1), for a.a. t ∈ [0, T2], T2 = max(T1, k0).
We choose T = T (ε) ≥ T2 so large that for t ≥ T and s ≤ T2 we have
‖U(t, s)‖B(E) ≤ L(U)e−α(T−T2) <
ε
8(‖p‖L1 + 1)(ψ(r) + 1)
(
r
∑∞
k=1
ck +
∑∞
k=1
dk + 1
) .
Hence ∥∥∥∥∥∥
∑
0<tk<t
U(t, tk)[Ik(x(tk))− Ik(y(tk))]
∥∥∥∥∥∥ ≤ 2L(U)
∑
0<tk<k0
e−α(T−tk)(ckr + dk)+
+2L(U)
∞∑
k=k0
(ckr + dk).
Then ∥∥∥∥∥∥
∑
0<tk<t
U(t, tk)[Ik(x(tk))− Ik(y(tk))]
∥∥∥∥∥∥ ≤ ε
2
. (5)
Letting h ∈ N(x) and h∗ ∈ N(y) for some x, y ∈ B(0, r), there exist v1 ∈ SF,x and v2 ∈ SF,y such
that
h(t) = U(t, 0)a+
t∫
0
U(t, s)v1(s)ds+
∑
0<tk<t
U(t, tk)Ik(x(tk))
and
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h∗(t) = U(t, 0)a+
t∫
0
U(t, s)v2(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)).
Consequently, for t ≥ T,∥∥∥∥∥∥
t∫
0
U(t, s)[v1(s)− v2(s)]ds
∥∥∥∥∥∥ ≤
∥∥∥∥∥∥∥
∫
[0,t]∩{s≤T2}
U(t, s)[v1(s)− v2(s)]ds
∥∥∥∥∥∥∥+
+
∥∥∥∥∥∥∥
∫
[0,t]∩{s>T2}
U(t, s)[v1(s)− v2(s)]ds
∥∥∥∥∥∥∥ ≤
≤
∥∥∥∥∥∥∥
∫
[0,t]∩{s≤T2}
U(t, s)[v1(s)− b(s)]ds
∥∥∥∥∥∥∥+
+
∥∥∥∥∥∥∥
∫
[0,t]∩{s≤T2}
U(t, s)[v2(s)− b(s)]ds
∥∥∥∥∥∥∥+
+
∥∥∥∥∥∥∥
∫
[0,t]∩{s>T2}
U(t, s)[v1(s)− v2(s)]ds
∥∥∥∥∥∥∥ ≤
ε
4L(U)
t∫
0
L(U)e−α(t−s)ds+
+
∫
[0,t]∩{s>T2}
‖U(t, s)‖B(E)‖v1(s)‖ds+
∫
[0,t]∩{s>T2}
‖U(t, s)‖B(E)‖v2(s)‖ds ≤
≤ ε
4L(U)
t∫
0
L(U)e−α(t−s)ds+
ε‖p‖L1ψ(r)
4(‖p‖L1 + 1)(ψ(r) + 1)
(
r
∞∑
k=1
ck +
∞∑
k=1
dk + 1
) .
Hence ∥∥∥∥∥∥
t∫
0
U(t, s)[v1(s)− v2(s)]ds
∥∥∥∥∥∥ ≤ ε
2
. (6)
From (5) and (6) we conclude that
‖h(t)− h∗(t)‖ ≤ ε, for each t ≥ T (ε).
Then N(B(0, r)) is equiconvergent. With Lemma 5.4 and Claims 1 – 3, we conclude that N is
completely continuous.
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878 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
Claim 4. N is u.s.c.
To this end, we show that N has a closed graph. Let hn ∈ N(yn) such that hn −→ h and
yn −→ y, as n → +∞. Then there exists M > 0 such that ‖yn‖ ≤ M. We shall prove that
h ∈ N(y). hn ∈ N(yn) means that there exists vn ∈ SF,yn such that for each t ∈ J
hn(t) = U(t, 0)a+
t∫
0
U(t, s)vn(s)ds+
∑
0<tk<t
U(t, tk)Ik(yn(tk)).
(H2) implies that vn(t) ∈ p(t)ψ(M)B(0, 1). Then (vn)n∈IN is integrably bounded in L1(J,E) by
using that E is reflexive space we conclude that {vn} is semicompact. By Lemma 5.2, there exists a
subsequence, still denoted (vn)n∈IN, which converges weakly to some limit v ∈ L1(J,E). Moreover,
the mapping Γ: L1(J,E)→ PCb(J,E) defined by
Γ(g)(t) =
t∫
0
U(t, s)g(s)ds
is a continuous linear operator. Then it remains continuous if these spaces are endowed with their
weak topologies [14, 15]. Therefore for a.e. t ∈ J, yn(t) converge to y(t) and by continuity of Ik,
we get
h(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)).
It remains to prove that v ∈ F (t, y(t)), for a.e. t ∈ J. Lemma 5.3 yields the existence of αni ≥
≥ 0, i = 1, . . . , k(n), such that
∑k(n)
i=1
αni = 1 and the sequence of convex combinations gn(·) =
=
∑k(n)
i=1
αni vi(·) converges strongly to v in L1. Using Lemma 2.2, we obtain that
v(t) ∈
⋂
n≥1
{gk(t) : k ≥ n}, for a.e. t ∈ J ⊂
⊂
⋂
n≥1
co{vk(t), k ≥ n} ⊂
⊂
⋂
n≥1
co{
⋃
k≥n
F (t, yk(t))} = co(lim sup
k→∞
F (t, yk(t))). (7)
However, the fact that the multivalued x → F (., x) is u.s.c. and has compact values together with
Lemma 2.1 imply that
lim sup
n→∞
F (t, yn(t)) = F (t, y(t)), for a.e. t ∈ J.
This with (7) yield that v(t) ∈ coF (t, y(t)). Finally F (., .) has closed, convex values, hence v(t) ∈
∈ F (t, y(t)), for a.e. t ∈ J. Thus h ∈ N(y), proving that N has a closed graph. Finally, with Lemma
2.7 and the compactness of N, we conclude that N is u.s.c.
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 879
Claim 5. A priori bounds on solutions.
Let y ∈ PCb be such that y ∈ λN(y) for some λ ∈ (0, 1). Then there exists v ∈ SF,y such that
y(t) = λU(t, 0)a+ λ
t∫
0
U(t, s)v(s)ds+ λ
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for a.e. t ∈ J.
Arguing as in Claim 1, we get
‖y(t)‖ ≤ L(U)‖a‖+ L(U)
t∫
0
p(s)ψ(‖y(s)‖)ds+ L(U)
∑
0<tk<t
(ck‖y(tk)‖+ dk), for a.e. t ∈ J.
Letting α(t) = sup{‖y(s)‖ : s ∈ [0, t]} and using the increasing character of ψ, we get
α(t) ≤ L(U)‖a‖+ L(U)
t∫
0
p(s)ψ(α(s))ds+ L(U)
∑
0<tk<t
(ckα(t) + dk).
Hence
α(t) ≤ L(U)
1− L(U)
∑∞
k=1
ck
‖a‖+
t∫
0
p(s)ψ(α(s))ds+
∞∑
k=1
dk
.
Denoting by β(t) the right-hand side of the above inequality, we have
‖y(t)‖ ≤ α(t) ≤ β(t), t ∈ J,
as well as
β(0) =
L(U)
(
‖a‖+
∑∞
k=1
dk
)
1− L(U)
∑∞
k=1
ck
,
and
β′(t) =
L(U)p(t)ψ(α(t))
1− L(U)
∑∞
k=1
ck
≤ L(U)p(t)ψ(β(t))
1− L(U)
∑∞
k=1
ck
·
From (H1), this implies that for t ∈ J
Γ(z(t)) =
β(t)∫
β(0)
ds
ψ(s)
≤ L(U)
1− L(U)
∑∞
k=1
ck
∞∫
0
p(s)ds <
∞∫
β(0)
ds
ψ(s)
= Γ(+∞).
Thus
β(t) ≤ Γ−1
L(U)‖p‖L1
1− L(U)
∑∞
k=1
ck
, for every t ∈ J,
where Γ(z) =
∫ z
β(0)
du
ψ(u)
. As a consequence
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880 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
‖y‖PCb ≤ Γ−1
L(U)‖p‖L1
1− L(U)
∑∞
k=1
ck
:= M̃.
Finally, let
U := {y ∈ PCb : ‖y‖PCb < M̃ + 1}
and consider the operator N : U → Pcv,cp(PCb). From the choice of U, there is no y ∈ ∂U such
that y ∈ λN(y) for some λ ∈ (0, 1). As a consequence of the multivalued version of the nonlinear
alternative of Leray – Schauder (Lemma 5.1), N has a fixed point y in U which is a solution of
problem (1).
Step 2. Compactness of the solution set.
For each a ∈ E, let
S(a) = {y ∈ PCb : y is a solution of problem (1)}.
From Step 1, there exists M̃ such that for every y ∈ S(a), ‖y‖PCb ≤ M̃. Since N is com-
pletely continuous, N(S(a)) is relatively compact in PCb. Let y ∈ S(a); then y ∈ N(y) hence
S(a) ⊂ N(S(a)). It remains to prove that S(a) is a closed subset in PCb. Let {yn : n ∈ IN} ⊂ S(a)
be such that (yn)n∈IN converges to y. For every n ∈ IN, there exists vn such that vn(t) ∈
∈ F (t, yn(t)), a.e. t ∈ J and
yn(t) = U(t, 0)a+
t∫
0
U(t, s)vn(s)ds+
∑
0<tk<t
U(t, tk)Ik(yn(tk)). (8)
Arguing as in Claim 4, we can prove that there exists v such that v(t) ∈ F (t, y(t)) and
y(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)), for a.e. t ∈ J.
Therefore y ∈ S(a) which yields that S(a) is closed, hence compact subset in PCb. Finally, we
prove that S(.) is u.s.c. by proving that the graph of S(.)
ΓS := {(a, y) : y ∈ S(a)}
is closed. Let (an, yn) ∈ ΓS be such that (an, yn) → (a, y) as n → ∞. Since yn ∈ S(an), there
exists vn ∈ L1(J,E) such that
yn(t) = U(t, 0)an +
t∫
0
U(t, s)vn(s)ds+
∑
0<tk<t
U(t, tk)Ik(yn(tk)), t ∈ J.
Arguing as in Claim 4, we can prove that there exists v ∈ SF,y such that
y(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)), for a.e. t ∈ J.
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 881
Thus, y ∈ S(a). Now, we show that S(.) maps bounded sets into relatively compact sets of PCb.
Let B be a compact set in E and let {yn} ⊂ S(B). Then there exists {an} ⊂ B such that
yn(t) = U(t, 0)an +
t∫
0
U(t, s)vn(s)ds+
∑
0<tk<t
U(t, tk)Ik(yn(tk)), t ∈ J,
where vn ∈ SF,yn , n ∈ IN. Since {an} ⊂ B is a bounded sequence and B is compact set, there
exists a subsequence of {an} converging to a. As in Claims 2, 3, we can show that {yn : n ∈ IN} is
equicontinuous on every compact of J and equiconvergent at ∞. As a consequence of Lemma 5.4,
we conclude that there exists a subsequence of {yn} converging to y in PC. By a similar argument
of Claim 4, we can prove that
y(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(tk)), for a.e. t ∈ J,
where v ∈ SF,y. Thus, y ∈ S(B). This implies that S(.) is u.s.c., ending the proof of Theorem 5.1.
Remark 5.1. (R1) We can replace the compactness of U(., .) by F (J × E) is compact.
(R2) We can replace the reflexivity of E by assuming that there exist PM : J → Pwkc(E) such
that
F (t, x) ⊂ PM (t), for a.a. t ∈ J.
6. Existence result in Fréchet space. In this section, we prove existence of solutions under
Hausdorff – Lipschitz conditions. Let E be a Fréchet space with the topology generated by a family
of seminorms ‖ · ‖n. First, we start with the following definition.
Definition 6.1. A multivalued map F : E → E is called an admissible contraction with con-
stant {kn}n∈IN if for each n ∈ IN, there exists kn ∈ (0, 1) such that
(a) Hdn(F (x), F (y)) ≤ kn|x− y|n for all x, y ∈ E, where Hd is the Hausdorff distance,
(b) for every x ∈ E and every ε > 0, there exists y ∈ F (x) such that
‖x− y‖n ≤ dn(x, F (x)) + ε, for every n ∈ IN.
A subset A ⊂ E is bounded if for every n ∈ IN, there exists Mn > 0 such that |x|n ≤ Mn, for
every x ∈ A. Our main tool will be the following nonlinear alternative of Frigon for multivalued
contractions [33]:
Lemma 6.1. Let E be a Fréchet space, U ⊂ E an open neighborhood of the origin, and
let N : U → E be a bounded admissible multivalued contraction. Then either one of the following
statements holds:
(a) N has a fixed point,
(b) there exists λ ∈ [0, 1) and x ∈ ∂U such that x ∈ λN(x).
We now present our second existence result for problem (1).
Theorem 6.1. Let (E, ‖ · ‖) be a Banach space. Suppose the multivalued map F : J × E →
→ Pcp(E) is such that t→ F (t, .) is a measurable and
(HF3) for each k = 1, 2, . . . , there exist lk ∈ L1(Jk, IR+) such that
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882 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
Hd(F (t, x), F (t, y)) ≤ lk(t)‖x− y‖, for x, y ∈ E and a.e. t ∈ Jk
and
F (t, 0) ⊂ lk(t)B(0, 1), for a.e. t ∈ Jk,
(HF4)
∑∞
k=1
‖Ik(0)‖ <∞ and there exist constants ck ≥ 0 such that L(U)
∑∞
k=1
ck < 1 and
‖Ik(x)− Ik(y)‖ ≤ ck‖x− y‖, for each x, y ∈ E.
Then problem (1) has at least one mild solution.
Here and hereafter Jk = [0, tk].
Remark 6.1. (a) Note that (HF4) implies (H1) with dk = ‖Ik(0)‖.
(b) (HF3) implies that the nonlinearity F has at most linear growth
‖F (t, x)‖P ≤ lk(t)(1 + ‖x‖), lk ∈ L1(Jk, IR
+), for a.e. t ∈ Jk, x ∈ E
and thus (H2) is satisfied. However, F is not Carathéodory and may take nonconvex values.
Proof. We begin by defining a family of seminorms on PC, thus rendering PC a Fréchet space.
Let τ be a sufficiently large real parameter, say
1
τ
+ L(U)
∞∑
k=1
ck < 1.
For each n ∈ IN, define in PC the seminorms
‖y‖n = sup{e−τLn(t)‖y(t)‖ : 0 ≤ t ≤ tn},
where
Ln(t) =
t∫
0
ln(s)ds.
Thus PC =
⋂
n≥1 PCn, where PCn = {y : Jn → E such that y is continuous everywhere except
for some tk at which y(t−k ) and y(t+k ) exist and y(t−k ) = y(tk), k = 1, 2, . . . , n− 1}. Then PC is a
Fréchet space with the family of seminorms {‖ · ‖n}. In order to transform problem (1) into a fixed
point problem, we define the operator N : PC → P(PC) by
N(y) =
h ∈ PC : h(t) = U(t, 0)a+
t∫
0
U(t, s)v(s)ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )), a.e. t ∈ J
,
where v ∈ SF,y = {v ∈ L1
loc(J,E) : v(t) ∈ F (t, y(t)), t ∈ J}. Clearly, the fixed points of the
operator N are solutions of problem (1). We use the Frigon nonlinear alternative to prove that N has
a fixed point.
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 883
Step 1. A priori estimates.
Given t ∈ Jn, let y ∈ λN(y) for some λ ∈ (0, 1]. Then there exists v ∈ SF,y such that
‖y(t)‖ ≤ L(U)‖a‖+ L(U)
t∫
0
‖v(s)‖ds+ L(U)
∑
0<tk<t
‖Ik(y(tk))‖ ≤
≤ L(U)‖a‖+ L(U)
t∫
0
ln(s)(1 + ‖y(s)‖)ds+
+L(U)
n∑
k=1
ck‖y(t−k )‖+ L(U)
n∑
k=1
‖Ik(0)‖.
Consider the function µ defined on Jn by
µ(t) = sup{‖y(s)‖ : 0 ≤ s ≤ t}.
By the previous inequality, we have for t ∈ Jn
µ(t) ≤ L(U)
1− L(U)
∑n
k=1
ck
‖a‖+
n∑
k=1
‖Ik(0)‖+
t∫
0
ln(s)(1 + µ(s))ds
.
Let us take the right-hand side of the above inequality as β(t). Then we have
β(0) =
L(U)(‖a‖+
∑∞
k=1
‖Ik(0)‖)
1− L(U)
∑n
k=1
ck
= c,
µ(t) ≤ β(t), t ∈ Jn,
and
β′(t) =
L(U)ln(t)(1 + µ(t))
1−
∑∞
k=1
ck
≤ L(U)ln(t)(1 + β(t))
1− L(U)
∑∞
k=1
ck
, t ∈ Jn.
Integrating we get
β(t)∫
β(0)
ds
1 + s
≤ L(U)
1− L(U)
∑∞
k=1
ck
tn∫
0
ln(s)ds =: Mn.
Hence β(t) ≤ Kn := (1 + β(0))eMn and as a consequence
‖y(t)‖ ≤ µ(t) ≤ β(t) ≤ Kn, t ∈ Jn.
Therefore
‖y‖n ≤ Kn ∀n ∈ IN∗.
Let
U = {y ∈ PC : ‖y‖n < Kn + 1, for all n ∈ IN}.
Clearly that U is open set in PC.
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884 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
Step 2. We show N : U → Pcl(PC) is an admissible multivalued contraction, where U ⊂ PC
is some open subset. Firstly we prove that there exists γ < 1 such that
Hd(N(y), N(y)) ≤ γ‖y − y‖n, for each y, y ∈ PCn.
Let y, y ∈ PCn and h ∈ N(y). Then there exists v ∈ SF,y such that
h(t) = U(t, 0)a+
t∫
0
U(t, s)v(s) ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for a.e. t ∈ Jn.
(HF3) implies that
Hd(F (t, y(t)), F (t, y(t)) ≤ ln(t)‖y(t)− y(t)‖, for a.e. t ∈ Jn.
Hence, there is some w ∈ F (t, y(t) such that
‖v(t)− w‖ ≤ ln(t)‖y(t)− y(t)‖, t ∈ Jn.
Consider the multivalued map Un : Jn → P(IRn) defined by
Un(t) = {w ∈ F (t, y(t)) : ‖v(t)− w‖ ≤ ln(t)‖y(t)− y(t)‖, for a.e. t ∈ Jn}.
Then Un(t) is a nonempty set and Theorem III.4.1 in [16] tells us that Un is measurable. Moreover,
the multivalued intersection operator Vn(.) = Un(.)∩F (., y(.)) is measurable. Therefore, by Lemma
2.5, there exists a function t 7→ vn(t), which is a measurable selection for Vn, that is vn(t) ∈
∈ F (t, y(t)) and
‖v(t)− vn(t)‖ ≤ ln(t)‖y(t)− y(t)‖, for a.e. t ∈ Jn.
Define h by
h(t) = U(t, 0)a+
t∫
0
U(t, s)vn(s) ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )).
Then we have, for t ∈ Jn,
‖h(t)− h(t)‖ ≤ L(U)
t∫
0
‖v(s)− vn(s)‖ ds+ L(U)
∑
0<tk<t
‖Ik(y(t−k ))− Ik(y(t−k ))‖ ≤
≤ L(U)
t∫
0
ln(s)‖y(s)− y(s)‖ds+ L(U)
∑
0<tk<t
ck‖y(tk)− y(tk)‖ ≤
≤
t∫
0
L(U)ln(s)eτLn(s)e−τLn(s)‖y(s)− y(s)‖ds+
+
∑
0<tk<t
cke
τLn(t)e−τLn(t)‖y(tk)− y(tk)‖ ≤
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 885
≤
t∫
0
ln(s)eτLn(s)ds‖y − y‖n + L(U)
∑
0<tk<t
cke
τL(t)‖y − y‖n ≤
≤
t∫
0
1
τ
(eτLn(s))′ds‖y − y‖n +
n∑
k=1
cke
τLn(t)‖y − y‖n ≤
≤ eτLn(t)
(
1
τ
+
n∑
k=1
ck
)
‖y − y‖n.
Thus
e−τLn(t)‖h(t)− h(t)‖ ≤
(
1
τ
+ L(U)
n∑
k=1
ck
)
‖y − y‖n.
By an analogous relation, obtained by interchanging the roles of y and y, we finally arrive at the
estimate
Hdn(N(y), N(y)) ≤
(
1
τ
+ L(U)
n∑
k=1
ck
)
‖y − y‖n.
In addition, since F is compact valued, we can prove that N has compact values too. Let x ∈ U and
ε > 0. If x 6∈ N(x), then dn(x,N(x)) 6= 0. Since N(x) is compact, there exists y ∈ N(x) such that
dn(x,N(x)) = ‖x− y‖n and we have
‖x− y‖n ≤ dn(x,N(x)) + ε.
If x ∈ N(x), then we may take y = x. Therefore N is an admissible operator contraction.
Clearly, U is a open subset of PC and there is no y ∈ ∂U such that y ∈ λN(y) and λ ∈ (0, 1).
By Lemma 6.1 and Steps 1, 2, N has at least one fixed point y solution to problem (1).
In this part we prove that if we work in Banach space and F is globally Lipschitz, the solution
set of the problem (2) is an absolute retract.
Definition 6.2. A space X is called an absolute retract (in short X ∈ AR) provided that for
every space Y, every closed subset B ⊆ Y and any continuous map f : B → X, there exists a
continuous extension f̃ : Y → X of f over Y, i.e., f̃(x) = f(x) for every x ∈ B. In other words,
for every space Y and for any embedding f : X −→ Y, the set f(X) is a retract of Y.
Proposition 6.1 [48]. Let C be a closed, convex subset of a Banach space E and let N : C →
→ Pcp,cv(C) be a contraction multivalued map. Then Fix (N) is a nonempty, compact AR-space.
Our contribution is the following theorem.
Theorem 6.2. Let F : J × E → Pcp,cv(E) be multivalued. Assume the following conditions:
(A1) the function F : J × E → Pcp(E) satisfies:
for fixed y, the multifunction t 7→ F (t, y) is measurable,
(A2) there exists p ∈ L1(IR+, (0,∞)) such that
Hd(F (t, z1), F (t, z2)) ≤ p(t)‖z1 − z2‖, for all z1, z2 ∈ E
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886 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
and
F (t, 0) ⊂ p(t)B(0, 1), for a.e. t ∈ J,
are satisfied.
Then the solution set S(a) ∈ AR.
Proof. It is clear that all solutions of problem (1) are fixed points of the multivalued opera-
tor N defined in Theorem 5.1. Using the fact that F (., .) has convex and compact values and by
(HF4),(A1) (A2), for every y ∈ Ω we have N(y) ∈ Pcv,cp(Ω). By some Bielecki-type norm on Ω
we can prove that N is a contraction. Hence, from Proposition 6.1, the solution set S(a) = Fix (N)
is a nonempty, compact AR-space.
7. Boundary-value problem on unbounded interval. The following conditions will be needed
in the sequel
(C1) The operator L∗, defined by
L∗x = L(U(., 0)x), x ∈ E,
has bounded inverse L−1∗ : E → E.
Lemma 7.1. Let f : J → E be a continuous function. If y ∈ PC is a solution of the problem
y′(t)−A(t)y(t) = f(t), for a.e. t ∈ J,
y(t−k )− y(tk) = Ik(y(tk)), t 6= tk, k = 1, . . . ,
Ly = a,
(9)
then it is given by
y(t) = U(t, 0)L−1∗
a− ∑
0<tk<t
U(t, tk)Ik(y(t−k ))−
t∫
0
U(t, s)f(s)ds
+
+
t∫
0
U(t, s)f(s))ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for t ∈ J. (10)
Proof. Let y be a solution of problem (9), then
y(t) = U(t, 0)y(0) +
∑
0<tk<t
U(t, tk)Ik(y(tk) +
t∫
0
U(t, s)f(s)ds.
Since L∗y = L(U(., 0)y) and Ly = a thus
L(U(t, 0)y(0) +
∑
0<tk<t
U(t, tk)Ik(y(tk) +
t∫
0
U(t, s)f(s)ds) = a.
Hence
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 887
L∗
y(0) +
∑
0<tk<t
U(t, tk)Ik(y(tk) +
t∫
0
U(t, s)f(s)ds
= a.
This implies that
y(0) = L−1∗
a− ∑
0<tk<t
U(t, tk)Ik(y(tk)−
t∫
0
U(t, s)f(s)ds
.
Hence we obtain
y(t) = U(t, 0)L−1∗
a− ∑
0<tk<t
U(t, tk)Ik(y(tk))−
t∫
0
U(t, s)f(s)ds
+
+
∑
0<tk<t
U(t, tk)Ik(y(tk)) +
t∫
0
U(t, s)f(s)ds, for t ∈ J,
proving the lemma. This lemma leads us to the definition of a mild solution.
Definition 7.1. A function y ∈ PCb is said to be a mild solution of problem (2) if there exists
f ∈ L1(J,E) such that f(t) ∈ F (t, y(t)) a.e. on J, and
y(t) = U(t, 0)L−1∗
a− ∑
0<tk<t
U(t, tk)Ik(y(tk))−
t∫
0
U(t, s)f(s)ds
+
+
t∫
0
U(t, s)f(s))ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for t ∈ J.
Consider the following assumptions:
(B1) Let E be reflexive Banach space and F : J × E −→ Pcp,cv(E) is an integrably bounded
multivalued map, i.e., there exists p ∈ L1(J,E) such that
‖F (t, x)‖P ≤ p(t), for every x ∈ E and a.e. t ∈ J.
(B2) There exist constants ak, bk > 0 and α ∈ [0, 1) such that
‖Ik(x)‖ ≤ ak‖x‖α + bk for every x ∈ E, k = 1, . . . ,
with
∞∑
k=1
ak <∞,
∞∑
k=1
bk <∞.
One can relax assumption (B1) with the following sublinear growth condition:
(B′1) there exist p, q ∈ L1(J, IR+) and β ∈ [0, 1− α) such that
‖F (t, x)‖P ≤ q(t) + p(t)‖x‖β, for every x ∈ E and a.e. t ∈ J.
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888 M. BENCHOHRA, J. J. NIETO, A. OUAHAB
Theorem 7.1. Assume F : J × E → Pcp,cv(E) is a Carathéodory map satisfying (H3), (H4),
(C1) and (B1) – (B3). Then problem (2) has at least one solution. If further E is a reflexive space,
then the solution set is compact in PC.
Proof. It is clear that all solutions of problem (2) are fixed points of the multivalued operator
N1 : PCb → P(PCb) defined by
N1(y):=
h ∈ PCb : h(t)=
U(t, 0)L−1∗
a− ∑
0<tk<t
U(t, tk)I(y(t−k ))−
t∫
0
U(t, s)f(s)ds
+
+
t∫
0
U(t, s)f(s))ds+
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for t ∈ J
,
where f ∈ SF,y. Notice that the set SF,y is nonempty (see Remark 2.1 (b)). Since, for each y ∈ PCb,
the nonlinearity F takes convex values, the selection set SF,y is convex and therefore N1 has convex
values. As in Theorem 5.1 we can prove that the operator N1 is completely continuous and u.s.c.
Now we prove only the priori bounded of solution for the problem (2). Let y ∈ PCb such that
y ∈ N1(y). Then there exists f ∈ SF,y such that
y(t) = λU(t, 0)L−1∗
a− ∞∑
0<tk<t
U(t, tk)I(y(tk))−
t∫
0
U(t, s)f(s)ds
+
+λ
t∫
0
U(t, s)f(s))ds+ λ
∑
0<tk<t
U(t, tk)Ik(y(t−k )), for t ∈ J, λ ∈ (0, 1).
Then
‖y(t)‖ ≤ L2(U)‖L−1∗ ‖B(E)
∞∑
k=1
(ak‖y(tk)‖α + bk) +
b∫
0
‖f(s)‖ds
+
+L(U)
t∫
0
‖f(s)‖ds+ L(U)
∞∑
k=1
(ak‖y(t−k )‖α + bk),
and so
‖y‖PC ≤ L(U)‖L−1∗ ‖B(E)
( ∞∑
k=1
(ak‖y‖αPC + bk) + ‖p‖L1
)
+
+L(U)‖p‖L1 + L(U)
m∑
k=1
(ak‖y‖αPC + bk).
If ‖y‖PC > 1, then since 0 ≤ α < 1, we have
‖y‖1−αPC ≤ L(U)‖L−1∗ ‖B(E)
( ∞∑
k=1
(ak + bk) + ‖p‖L1
)
+
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IMPULSIVE DIFFERENTIAL INCLUSIONS INVOLVING EVOLUTION OPERATORS . . . 889
+L(U)‖p‖L1 + L(U
∞∑
k=1
(ak + bk).
Hence
‖y‖PC ≤
(
L2(U)‖L−1∗ ‖B(E)
(
m∑
k=1
(ak + bk) + ‖p‖L1
)
+
+ L(U)‖p‖L1 + L(U)
∞∑
k=1
(ak + bk)
) 1
1−α
:= M.
Therefore
‖y‖PC ≤ max(1,M) := M̃.
Let
U := {y ∈ PCb : ‖y‖PC < M̃ + 1},
and consider the operator N : U → Pcv,cp(PCb). From the choice of U, there is no y ∈ ∂U such
that y ∈ λN1(y) for some λ ∈ (0, 1). As a consequence of the Leray – Schauder nonlinear alternative
(Lemma 5.1), we deduce that N has a fixed point y in U which is a solution of problem (2). Also
by the sam method used in Theorem 5.1 we can prove that the solution set of the problem (2) is
compact. In this part we present existence result of problem (2) with right-hand side not necessarily
convex.
Theorem 7.2. Assume that the conditions (HF3), (HF4) are satisfied. If
L(U)(‖L−1∗ ‖B(E)L(U) + 1)
∞∑
k=1
ck < 1,
then the problem (2) has at least one solution.
Proof. By the same method as used in Theorem 6.1 we can prove that N1 has at least one fixed
point which is solution of the problem (2) in PCb.
Acknowledgement. This paper was completed while M. Benchohra and A. Ouahab visited the
department of mathemátical analysis of the university of Santiago de Compostela. They would like
to thank the department of its hospitality and support.
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| id | umjimathkievua-article-2625 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:06Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/49/3b5dd4c626628f852f219bc099dc9c49.pdf |
| spelling | umjimathkievua-article-26252020-03-18T19:31:34Z Impulsive differential inclusions involving evolution operators in separable Banach spaces Імпульснi диференцiальнi включення, що мiстять оператори в сепарабельних банахових просторах Benchohra, M. Nieto, J. J. Ouahab, A. Беньчохра, М. Ньєто, Дж. Дж. Оахаб, А. We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$y(0) = a$$ and $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$Ly = a,$$ where $J = IR_+,\; 0 = t_0 < t_1 Наведено деякi результати про iснування м’яких розв’язкiв та вивчено топологiчну будову множин розв’язкiв для наступних iмпульсних напiвлiнiйних диференцiальних включень першого порядку з початковими та граничними умовами: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{для майже кожного} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$y(0) = a$$ та $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{для майже кожного} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$Ly = a,$$ де $J = IR_+,\; 0 = t_0 < t_1 Institute of Mathematics, NAS of Ukraine 2012-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2625 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 7 (2012); 867-891 Український математичний журнал; Том 64 № 7 (2012); 867-891 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2625/2009 https://umj.imath.kiev.ua/index.php/umj/article/view/2625/2010 Copyright (c) 2012 Benchohra M.; Nieto J. J.; Ouahab A. |
| spellingShingle | Benchohra, M. Nieto, J. J. Ouahab, A. Беньчохра, М. Ньєто, Дж. Дж. Оахаб, А. Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title | Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title_alt | Імпульснi диференцiальнi включення, що мiстять оператори в сепарабельних банахових просторах |
| title_full | Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title_fullStr | Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title_full_unstemmed | Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title_short | Impulsive differential inclusions involving evolution operators in separable Banach spaces |
| title_sort | impulsive differential inclusions involving evolution operators in separable banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2625 |
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