Global existence results for neutral functional differential inclusions with state-dependent delay
We consider the existence of global solutions for a class of neutral functional differential inclusions with state-dependent delay. The proof of the main result is based on the semigroup theory and the Bohnenblust – Karlin fixed point theorem.
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| author | Alaidarous, E. Benchohra, M. Medjadj, I. Алаідарус, Е. Беньчохра, М. Медядй, І. |
| author_facet | Alaidarous, E. Benchohra, M. Medjadj, I. Алаідарус, Е. Беньчохра, М. Медядй, І. |
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| description | We consider the existence of global solutions for a class of neutral functional differential inclusions with state-dependent
delay. The proof of the main result is based on the semigroup theory and the Bohnenblust – Karlin fixed point theorem. |
| first_indexed | 2026-03-24T02:09:52Z |
| format | Article |
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UDC 517.9
E. Alaidarous (King Abdulaziz Univ., Saudi Arabia),
M. Benchohra, I. Medjadj (Djillali Liabes Univ. Sidi Bel-Abbès, Algeria)
GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL
DIFFERENTIAL INCLUSIONS WITH STATE-DEPENDENT DELAY
РЕЗУЛЬТАТИ ПРО ГЛОБАЛЬНЕ IСНУВАННЯ РОЗВ’ЯЗКIВ НЕЙТРАЛЬНИХ
ФУНКЦIОНАЛЬНИХ ДИФЕРЕНЦIАЛЬНИХ ВКЛЮЧЕНЬ IЗ ЗАТРИМКОЮ,
ЩО ЗАЛЕЖИТЬ ВIД СТАНУ
We consider the existence of global solutions for a class of neutral functional differential inclusions with state-dependent
delay. The proof of the main result is based on the semigroup theory and the Bohnenblust – Karlin fixed point theorem.
Розглянуто питання про iснування глобальних розв’язкiв одного класу нейтральних функцiональних диференцiаль-
них включень iз затримкою, що залежить вiд стану. Доведення основного результату базується на теорiї напiвгруп
та теоремi про нерухому точку Боненблюста та Карлiна.
1. Introduction. Neutral functional differential equations arise in many areas of applied mathematics
and for this reason these equations have received much attention in the last decades. The literature
relative to ordinary neutral functional differential equations is very extensive and refer to [8, 9,
11, 22, 32, 33]. Partial neutral differential equation with finite delay arise, for instance, from the
transmission line theory [38]. Wu and Xia have shown in [39] that a ring array of identical resistibly
coupled lossless transmission lines leads to a system of neutral functional differential equations with
discrete diffusive coupling which exhibits various types of discrete waves. For more results on partial
neutral functional differential equations and related issues we refer to Adimy and Ezzinbi [2], Hale
[20], Wu and Xia [38, 39] for finite delay equations, and Hern’andez and Henriquez [24, 25] for
unbounded delays. Functional differential equations with state-dependent delay appear frequently
in applications as model of equations and for this reason the study of this type of equations has
received a significant amount of attention in the last years, see for instance [1 – 3, 6, 10, 15, 35] and
the references therein. We also cite [4, 5, 16, 19, 23, 30, 31, 40] for the case neutral differential
equations with state-dependent delay. In [12, 13] Benchohra et al. considered the global existence of
mild solutions for some classes of functional evolutions equations on unbounded intervals.
In this work we prove the existence of solutions of a neutral functional differential inclusion. Our
investigations will be situated in the Banach space of real functions which are defined, continuous
and bounded on the real axis \BbbR . We will use Bohnenblust – Karlin fixed point theorem, combined
with the Corduneanu’s compactness criteria. More precisely we will consider the following problem:
d
dt
\bigl[
y(t) - g(t, y\rho (t,yt))
\bigr]
- A
\bigl[
y(t) - g(t, y\rho (t,yt))
\bigr]
\in F (t, y\rho (t,yt)) a.e. t \in J := [0,+\infty ), (1)
y(t) = \phi (t), t \in ( - \infty , 0], (2)
where F : J\times \scrB \rightarrow \scrP (E) is a multivalued map with nonempty compact values, \scrP (E) is the family of
all nonempty subsets of E, g : J\times \scrB \rightarrow E is given function, A : D(A) \subset E \rightarrow E is the infinitesimal
c\bigcirc E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11 1443
1444 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
generator of a strongly continuous semigroup \{ T (t)\} t\geq 0, \scrB is the phase space to be specified later,
\phi \in \scrB , \rho : J \times \scrB \rightarrow ( - \infty ,+\infty ) and (E, | \cdot | ) is a real separable Banach space. For any function y
defined on ( - \infty ,+\infty ) and any t \in J we denote by yt the element of \scrB defined by yt(\theta ) = y(t +
+ \theta ), \theta \in ( - \infty , 0]. Here yt(\cdot ) represents the history of the state from time - \infty , up to the present
time t. We assume that the histories yt belongs to some abstract phases \scrB , to be specified later.
To our knowledge the literature on the global existence of neutral evolution inclusions is very
limited. Some of the exiting ones are obtained in the Fréchet space setting. The present results are
given in the Banach space setting, and hence are considered as a contribution of this class of problems.
2. Preliminaries. In this section we present briefly some notations and definition, and theorem
which are used throughout this work.
In this paper, we will employ an axiomatic definition of the phase space \scrB introduced by Hale
and Kato in [21] and follow the terminology used in [27]. Thus, (\scrB , \| \cdot \| \scrB ) will be a seminormed
linear space of functions mapping ( - \infty , 0] into E, and satisfying the following axioms:
(A1) If y : ( - \infty , b) \rightarrow E, b > 0, is continuous on J and y0 \in \scrB , then for every t \in J the
following conditions hold:
(i) yt \in \scrB ;
(ii) there exists a positive constant H such that | y(t)| \leq H\| yt\| \scrB ;
(iii) there exist two functions L(\cdot ),M(\cdot ) : \BbbR + \rightarrow \BbbR + independent of y with L continuous and
bounded, and M locally bounded such that
\| yt\| \scrB \leq L(t) \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| y(s)| : 0 \leq s \leq t
\bigr\}
+M(t)\| y0\| \scrB .
(A2) For the function y in (A1), yt is a \scrB -valued continuous function on J.
(A3) The space \scrB is complete.
Assume that
l = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
L(t) : t \in J
\bigr\}
, m = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
M(t) : t \in J
\bigr\}
.
Remark 2.1. 1. Condition (ii) is equivalent to | \phi (0)| \leq H\| \phi \| \scrB for every \phi \in \scrB .
2. Since \| \cdot \| \scrB is a seminorm, two elements \phi , \psi \in \scrB can verify \| \phi - \psi \| \scrB = 0 without
necessarily \phi (\theta ) = \psi (\theta ) for all \theta \leq 0.
3. From the equivalence of in the first remark, we can see that for all \phi , \psi \in \scrB such that
\| \phi - \psi \| \scrB = 0: we necessarily have that \phi (0) = \psi (0).
By BUC we denote the space of bounded uniformly continuous functions defined from ( - \infty , 0]
to E.
Let BC := BC([0,+\infty )) be the Banach space of all bounded and continuous functions from
[0,+\infty ) into E equipped with the standard norm
\| y\| BC = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,+\infty )
| y(t)| .
Let (E, d) be a metric space. We use the following notations:
\scrP cl(E) =
\bigl\{
Y \in \scrP (E) : Y closed
\bigr\}
, \scrP cv(E) =
\bigl\{
Y \in \scrP (E) : Y convex
\bigr\}
,
\scrP b(E) =
\bigl\{
Y \in \scrP (E) : Y bounded
\bigr\}
.
Consider Hd : \scrP (E)\times \scrP (E) - \rightarrow \BbbR + \cup \{ \infty \} , given by
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1445
Hd(\scrA ,\scrB ) = \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
a\in \scrA
d(a,\scrB ), \mathrm{s}\mathrm{u}\mathrm{p}
b\in \scrB
d(\scrA , b)
\biggr\}
,
where d(\scrA , b) = \mathrm{i}\mathrm{n}\mathrm{f}a\in \scrA d(a, b), d(a,\scrB ) = \mathrm{i}\mathrm{n}\mathrm{f}b\in \scrB d(a, b).
Definition 2.1. Let X, Y be Hausdorff topological spaces and F : X \rightarrow \scrP (Y ) is called upper
semicontinuous (u.s.c.) on X if for each x0 \in X, the set F (x0) is a nonempty closed subset of Y
and if for each open set N of Y containing F (x0), there exists an open neighborhood N0 of x0
such that F (N0) \subseteq N.
Let (E, \| \cdot \| ) be a Banach space. A multivalued map A : E \rightarrow \scrP (E) has convex (closed) values
if A(x) is convex (closed) for all x \in E. We say that A is bounded on bounded sets if A(B) is
bounded in E for each bounded set B of E, i.e.,
\mathrm{s}\mathrm{u}\mathrm{p}
x\in B
\bigl\{
\mathrm{s}\mathrm{u}\mathrm{p}\{ \| y\| : y \in A(x)\}
\bigr\}
<\infty .
F is said to be completely continuous if F (B) is relatively compact for every B \in \scrP b(E). If the
multivalued map F is completely continuous with non empty values, then F is u.s.c. if an only if F
has a closed graph (i.e., xn \rightarrow x\ast , yn \rightarrow y\ast , yn \in F (xn) implies y\ast \in F (x\ast )).
Definition 2.2. A function F : J \times \scrB - \rightarrow \scrP (E) is said to be an L1-Carathéodory multivalued
map if it satisfies:
(i) y \mapsto \rightarrow F (t, y) is upper semicontinuous for almost all t \in J ;
(ii) t \mapsto \rightarrow F (t, y) is measurable for each y \in \scrB ;
(iii) for every positive constant l there exists hl \in L1(J,\BbbR +)\bigm\| \bigm\| F (t, y)\bigm\| \bigm\| = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| v| : v \in F (t, y)
\bigr\}
\leq hl
for all | y| \leq l for almost all t \in J.
Definition 2.3. A function F : J \times \scrB - \rightarrow \scrP (E) is said to be an Carathéodory multivalued map
if it satisfies (i) and (ii).
The following two results are easily deduced from the limit properties.
Lemma 2.1 (see, e.g., [7], Theorem 1.4.13). If G : X \rightarrow \scrP (X) is u.s.c., then, for any x0 \in X,
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow x0
\mathrm{s}\mathrm{u}\mathrm{p}G(x) = G(x0).
Lemma 2.2 (see, e.g., [7], Lemma 1.1.9). Let (Kn)n\in \BbbN \subset K \subset X be a sequence of subsets
where K is compact in the separable Banach space X. Then
\mathrm{c}\mathrm{o}
\Bigl(
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}Kn
\Bigr)
=
\bigcap
N>0
\mathrm{c}\mathrm{o}
\Biggl( \bigcup
n\geq N
Kn
\Biggr)
,
where \mathrm{c}\mathrm{o}A refers to the closure of the convex hull of A.
The second one is due to Mazur (1933).
Lemma 2.3 (Mazur’s lemma [41]). Let E be a normed space and \{ xk\} k\in \BbbN \subset E be a sequence
weakly converging to a limit x \in E. Then there exists a sequence of convex combinations ym =
=
\sum m
k=1
\alpha mkxk with \alpha mk > 0 for k = 1, 2, . . . ,m and
\sum m
k=1
\alpha mk = 1, which converges strongly
to x.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1446 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
Lemma 2.4 [29]. Let E be a Banach space. Let F : J \times E \rightarrow \scrP cl,cv(E) be a L1-Carathéodory
multivalued map, and let \Gamma be a linear continuous from L1(J ;E) into C(J ;E), then the operator
\Gamma \circ SF : C(J,E) - \rightarrow \scrP cp,cv(C(J,X)), y \mapsto - \rightarrow (\Gamma \circ SF )(y) := \Gamma (SF,y)
is a closed graph operator in C(J ;X)\times C(J ;X).
Finally, we say that A has a fixed point if there exists x \in E such that x \in A(x).
For each y : ( - \infty ,+\infty ) \rightarrow E let the set SF,y known as the set of selectors from F defined by
SF,y =
\bigl\{
v \in L1(J ;E) : v(t) \in F (t, y\rho (t,yt)) a.e. t \in J
\bigr\}
.
For more details on multivalued maps we refer to the books of Deimling [18], Hu and Papageor-
giou [28], Górniewicz [34], and Perestyuk et al. [37].
Theorem 2.1 (Bohnenblust – Karlin fixed point [14]). Let B \in \scrP cl,cv(E), N : B \rightarrow \scrP cl,cv(B)
be a upper semicontinuous operator and N(B) is a relatively compact subset of E. Then N has at
least one fixed point in B.
Lemma 2.5 (Corduneanu [17]). Let D \subset BC([0,+\infty ), E). Then D is relatively compact if the
following conditions hold:
(a) D is bounded in BC.
(b) The function belonging to D is almost equicontinuous on [0,+\infty ), i.e., equicontinuous on
every compact of [0,+\infty ).
(c) The set D(t) := \{ y(t) : y \in D\} is relatively compact on every compact of [0,+\infty ).
(d) The function from D is equiconvergent, that is, given \epsilon > 0, responds T (\epsilon ) > 0 such that\bigm| \bigm| u(t) - \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow +\infty u(t)
\bigm| \bigm| < \epsilon , for any t \geq T (\epsilon ) and u \in D.
3. Existence of mild solutions. Now we give our main existence result for problem (1), (2).
Before starting and proving this result, we give the definition of the mild solution.
Definition 3.1. We say that a continuous function y : ( - \infty ,+\infty ) \rightarrow E is a mild solution of
problem (1), (2) if y(t) = \phi (t) for all t \in ( - \infty , 0], and the restriction of y(\cdot ) to the interval J is
continuous and there exists f(\cdot ) \in L1(J ;E) : f(t) \in F (t, y\rho (t,yt)) a.e. in J such that y satisfies the
integral equation
y(t) = T (t)
\bigl[
\phi (0) - g(0, \phi (0))
\bigr]
+ g(t, y\rho (t,yt)) +
t\int
0
T (t - s)f(s) ds, t \in J. (3)
Set
\scrR (\rho - ) =
\bigl\{
\rho (s, \phi ) : (s, \phi ) \in J \times \scrB , \rho (s, \phi ) \leq 0
\bigr\}
.
We always assume that \rho : J \times \scrB \rightarrow \BbbR is continuous. Additionally, we introduce following
hypothesis:
(H\phi ) The function t \rightarrow \phi t is continuous from \scrR (\rho - ) into \scrB and there exists a continuous and
bounded function \scrL \phi : \scrR (\rho - ) \rightarrow (0,\infty ) such that
\| \phi t\| \leq \scrL \phi (t)\| \phi \| for every t \in \scrR (\rho - ).
Remark 3.1. The condition (H\phi ), is frequently verified by functions continuous and bounded.
For more details, see, for instance, [27].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1447
Lemma 3.1 [26]. If y : ( - \infty ,+\infty ) \rightarrow E is a function such that y0 = \phi , then
\| ys\| \scrB \leq (M + \scrL \phi )\| \phi \| \scrB + l \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| y(\theta )| ; \theta \in [0,\mathrm{m}\mathrm{a}\mathrm{x}\{ 0, s\} ]
\bigr\}
, s \in \scrR (\rho - ) \cup J,
where \scrL \phi = \mathrm{s}\mathrm{u}\mathrm{p}t\in \scrR (\rho - ) \scrL \phi (t).
Let us introduce the following hypotheses:
(H1) The semigroup T (t) is compact for t > 0, and there is a positive constant M such that
\| T (t)\| B(E) \leq M.
(H2) The multifunction F : J \times \scrB - \rightarrow \scrP (E) is Carathéodory with compact, closed and convex
values.
(H3) There exists a continuous function k : J \rightarrow [0,+\infty ) such that
Hd(F (t, u), F (t, v)) \leq k(t)\| u - v\| \scrB
for each t \in J and for all u, v \in \scrB and
d(0, F (t, 0)) \leq k(t)
with
k\ast := \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
t\int
0
k(s) ds <\infty . (4)
(H4) The function g(t, \cdot ) is continuous on J and there exists a constant kg > 0 such that\bigm| \bigm| g(t, u) - g(t, v)
\bigm| \bigm| \leq kg\| u - v\| \scrB for each u, v \in \scrB
and
g\ast := \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
| g(t, 0)| <\infty .
(H5) For each t \in J and any bounded set B \subset \scrB , the set \{ g(t, u) : u \in B\} is relatively compact
in E.
(H6) For any bounded set B \subset \scrB , the function \{ t\rightarrow g(t, u) : u \in B\} is equicontinuous on each
compact interval of [0,+\infty ).
Set
\Omega =
\bigl\{
y : ( - \infty ,+\infty ) \rightarrow E : y| ( - \infty ,0] \in \scrB and y| [0,+\infty ) \in BC
\bigr\}
.
Remark 3.2. By the condition (H4) we deduce that
| g(t, u)| \leq kg\| u\| \scrB + g\ast , t \in J, u \in \scrB .
Theorem 3.1. Assume that (H1) - (H6) and (H\phi ) hold. If l(Mk\ast +kg) < 1, then the problem (1),
(2) has at least one mild solution.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1448 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
Proof. Transform the problem (1), (2) into a fixed point problem. Consider the operator
N : \Omega \rightarrow \scrP (\Omega ) defined by
N(y) :=
\left\{
h \in \Omega : h(t) =
\left\{
\phi (t), if t \in ( - \infty , 0],
T (t)
\bigl[
\phi (0) - g(0, \phi (0))
\bigr]
+
+g(t, y\rho (t,yt)) +
\int t
0
T (t - s)f(s) ds, if t \in J
\right\}
where f \in SF,y\rho (t,yt) .
Let x(\cdot ) : ( - \infty ,+\infty ) \rightarrow E be the function defined by
x(t) =
\left\{ \phi (t), if t \in ( - \infty , 0],
T (t)\phi (0), if t \in J.
Then x0 = \phi . For each z \in \Omega with z(0) = 0, we denote by z the function
z(t) =
\left\{ 0, if t \in ( - \infty , 0],
z(t), if t \in J,
if y(\cdot ) satisfies (3) we can decompose it as y(t) = z(t) + x(t), t \in J, which implies yt = zt + xt
for every t \in J and the function z(\cdot ) satisfies
z(t) = g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) - T (t)g(0, \phi (0)) +
t\int
0
T (t - s)f(s) ds, t \in J,
where f \in SF,z\rho (t,zt+xt)
+x\rho (t,zt+xt)
. Set
\Omega 0 = \{ z \in \Omega : z(0) = 0\}
and let
\| z\| \Omega 0 = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
| z(t)| : t \in J
\bigr\}
z \in \Omega 0.
\Omega 0 is a Banach space with the norm \| \cdot \| \Omega 0 .
We define the operator \scrA : \Omega 0 \rightarrow \scrP (\Omega 0) by
\scrA (z) :=
\left\{
h \in \Omega 0 : h(t) =
\left\{
0, if t \leq 0,
g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) -
- T (t)g(0, \phi (0)) +
\int t
0
T (t - s)f(s) ds, if t \in J
\right\}
where f \in SF,z\rho (t,zt+xt)
+x\rho (t,zt+xt)
.
The operator A maps \Omega 0 into \Omega 0, indeed the map \scrA (z) is continuous on [0,+\infty ) for any
z \in \Omega 0, h \in \scrA (z) and for each t \in J we have
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1449
| h(t)| \leq
\bigm| \bigm| g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt))
\bigm| \bigm| +M
\bigm| \bigm| g(0, \phi (0))\bigm| \bigm| +M
t\int
0
| f(s)| ds \leq
\leq M(kg\| \phi \| \scrB + g\ast ) + kg\| z\rho (t,zt+xt) + x\rho (t,zt+xt)\| \scrB + g\ast +
+M
t\int
0
| F (s, 0)| ds+M
t\int
0
k(s)\| z\rho (s,zs+xs) + x\rho (s,zs+xs)\| \scrB ds \leq
\leq M
\bigl(
kg\| \phi \| \scrB + g\ast
\bigr)
+ kg
\bigl(
l| z(t)| + (m+ \scrL \phi + lMH)\| \phi \| \scrB
\bigr)
+ g\ast +
+Mk\ast +M
t\int
0
k(s)
\bigl(
l| z(s)| + (m+ \scrL \phi + lMH)\| \phi \| \scrB
\bigr)
ds.
Set
C1 :=
\bigl(
m+ \scrL \phi + lMH
\bigr)
\| \phi \| \scrB ,
C2 :=M
\bigl(
kg\| \phi \| \scrB + g\ast
\bigr)
+ kgC1 + g\ast +Mk\ast .
Then we have
| h(t)| \leq C2 + kgl| z(t)| +MC1
t\int
0
k(s) ds+M
t\int
0
l| z(s)| k(s) ds \leq
\leq C2 + kgl\| z\| \Omega 0 +MC1k
\ast +Ml\| z\| \Omega 0k
\ast .
Hence, \scrA (z) \in \Omega 0.
Moreover, let r > 0 be such that
r \geq C2 +MC1k
\ast
1 - l(Mk\ast + kg)
,
and Br be the closed ball in \Omega 0 centered at the origin and of radius r. Let z \in Br and t \in [0,+\infty ).
Then
| h(t)| \leq C2 + kglr +MC1k
\ast +Mk\ast lr.
Thus
\| h\| \Omega 0 \leq r,
which means that the operator \scrA transforms the ball Br into itself.
Now we prove that \scrA : Br \rightarrow \scrP (Br) satisfies the assumptions of Bohnenblust – Karlin fixed
point theorem. The proof will be given in several steps.
Step 1. We shall show that the operator \scrA is closed and convex valued. This will be given in
several claims.
Claim 1. \scrA (z) is closed for each z \in Br.
Let (hn)n\geq 0 \in \scrA (z) such that hn \rightarrow \~h in Br. Then for hn \in Br there exists fn \in
\in SF,z\rho (t,zt+xt)
+x\rho (t,zt+xt)
such that for each t \in J,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1450 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
hn(t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g(0, \phi (0)) +
t\int
0
T (t - s)fn(s) ds.
We shall use the fact that F has compact values and from hypotheses (H2), (H3), and Mazur’s
lemma we may pass a subsequence if necessary to get that fn converges to f \in L1(J,E) and
hence f \in SF,y. Indeed, Lemma 2.3 yields the existence of \alpha n
i \geq 0, i = n, . . . , k - n), such that\sum k(n)
i=1
\alpha n
i = 1 and the sequence of convex combinations gn(\cdot ) =
\sum k(n)
i=1
\alpha n
i fi(\cdot ) converges strongly
to f \in L1. Since F takes convex values, using Lemma 2.2, we obtain that for a.e. t \in J
f(t) \in
\bigcap
n\geq 1
\{ gn(t)\} \subset
\subset
\bigcap
n\geq 1
\mathrm{c}\mathrm{o}
\bigl\{
fk(t), k \geq n
\bigr\}
\subset
\bigcap
n\geq 1
\mathrm{c}\mathrm{o}
\left\{ \bigcup
k\geq n
F (t, zk
\rho (zkt +xt)
+ x\rho (t,zkt +xt)
)
\right\} =
= \mathrm{c}\mathrm{o}
\Bigl(
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}F
\bigl(
t, zk
\rho (zkt +xt)
+ x\rho (t,zkt +xt)
\bigr) \Bigr)
.
Since F is u.s.c. with compact values, then by Lemma 2.1, we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}F
\bigl(
t, zn\rho (znt +xt)
+ x\rho (t,znt +xt)
\bigr)
= F
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
for a.e. t \in J.
This implies that f(t) \in \mathrm{c}\mathrm{o}F (t, z\rho (t,zt+xt) + x\rho (t,zt+xt)). Since F (\cdot , \cdot ) has closed, convex values,
we deduce that f(t) \in F (t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) for a.e. t \in J.
Let f \in SF,z\rho (s,zs+xs)+x\rho (s,zs+xs)
. Then, for each t \in J,
hn(t) \rightarrow \~h(t) = g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) - T (t)g(0, \phi (0)) +
t\int
0
T (t - s)f(s) ds.
So, \~h \in \scrA (z).
Claim 2. \scrA (z) is convex for each z \in Br.
Let h1, h2 \in \scrA (z), the there exists f1, f2 \in SF,z\rho (t,zt+xt)
+x\rho (t,zt+xt)
such that, for each t \in J we
have
hi(t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g
\bigl(
0, \phi (0)
\bigr)
+
t\int
0
T (t - s)fi(s) ds, i = 1, 2.
Let 0 \leq \delta \leq 1. Then we have, for each t \in J,\bigl(
\delta h1 + (1 - \delta )h2
\bigr)
(t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g
\bigl(
0, \phi (0)
\bigr)
+
+
t\int
0
T (t - s)
\bigl[
\delta f1(s) + (1 - \delta )f2(s)
\bigr]
ds.
Since F has convex values, one has
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GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1451
\delta h1 + (1 - \delta )h2 \in \scrA (z).
Step 2. \scrA (Br) \subset Br this is clear.
Step 3. \scrA (Br) is equicontinuous on every compact interval [0, b] of [0,+\infty ) for b > 0.
Let \tau 1, \tau 2 \in [0, b], h \in \scrA (z) with \tau 2 > \tau 1, we have\bigm| \bigm| h(\tau 2) - h(\tau 1)
\bigm| \bigm| \leq
\leq
\bigm| \bigm| g\bigl( \tau 2, z\rho (\tau 2,z\tau 2+x\tau 2 )
+ x\rho (\tau 2,z\tau 2+x\tau 2 )
\bigr)
- g
\bigl(
\tau 1, z\rho (\tau 1,z\tau 1+x\tau 1 )
+ x\rho (\tau 1,z\tau 1+x\tau 1 )
\bigr) \bigm| \bigm| +
+
\bigm\| \bigm\| T (\tau 2) - T (\tau 1)
\bigm\| \bigm\|
B(E)
| g(0, \phi (0))| +
+
\tau 1\int
0
\bigm\| \bigm\| T (\tau 2 - s) - T (\tau 1 - s)
\bigm\| \bigm\|
B(E)
| f(s)| ds+
+
\tau 2\int
\tau 1
\| T (\tau 2 - s)\| B(E)| f(s)| ds \leq
\leq
\bigm| \bigm| g\bigl( \tau 2, z\rho (\tau 2,z\tau 2+x\tau 2 )
+ x\rho (\tau 2,z\tau 2+x\tau 2 )
\bigr)
- g
\bigl(
\tau 1, z\rho (\tau 1,z\tau 1+x\tau 1 )
+ x\rho (\tau 1,z\tau 1+x\tau 1 )
\bigr) \bigm| \bigm| +
+
\bigm\| \bigm\| T (\tau 2) - T (\tau 1)
\bigm\| \bigm\|
B(E)
(kg\| \phi \| \scrB + g\ast )+
+
\tau 1\int
0
\| T (\tau 2 - s) - T (\tau 1 - s)\| B(E)(k(s)\| z\rho (s,zs+xs) + x\rho (s,zs+xs)\| \scrB + | F (s, 0)| ) ds+
+
\tau 2\int
\tau 1
\| T (\tau 2 - s)\| B(E)(k(s)\| z\rho (s,zs+xs) + x\rho (s,zs+xs)\| \scrB + | F (s, 0)| ) ds \leq
\leq
\bigm| \bigm| g\bigl( \tau 2, z\rho (\tau 2,z\tau 2+x\tau 2 )
+ x\rho (\tau 2,z\tau 2+x\tau 2 )
\bigr)
- g
\bigl(
\tau 1, z\rho (\tau 1,z\tau 1+x\tau 1 )
+ x\rho (\tau 1,z\tau 1+x\tau 1 )
\bigr) \bigm| \bigm| +
+C1
\tau 1\int
0
\bigm\| \bigm\| T (\tau 2 - s) - T (\tau 1 - s)
\bigm\| \bigm\|
B(E)
k(s) ds+
+rl
\tau 1\int
0
\bigm\| \bigm\| T (\tau 2 - s) - T (\tau 1 - s)
\bigm\| \bigm\|
B(E)
k(s) ds+
+
\tau 1\int
0
\bigm\| \bigm\| T (\tau 2 - s) - T (\tau 1 - s)
\bigm\| \bigm\|
B(E)
k(s) ds+ C1
\tau 2\int
\tau 1
\| T (\tau 2 - s)\| B(E)k(s) ds+
+ rl
\tau 2\int
\tau 1
\| T (\tau 2 - s)\| B(E)k(s) ds+
\tau 2\int
\tau 1
\| T (\tau 2 - s)\| B(E)k(s) ds.
When \tau 2 \rightarrow \tau 1, the right-hand side of the above inequality tends to zero, since (H6) and T (t) is a
strongly continuous operator and the compactness of T (t) for t > 0, implies the continuity in the
uniform operator topology (see [36]), this proves the equicontinuity.
Step 4. \scrA (Br) is relatively compact on every compact interval of [0,\infty ).
Let t \in [0, b] for b > 0 and let \varepsilon be a real number satisfying 0 < \varepsilon < t. For z \in Br we define
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1452 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
h\varepsilon (t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (\varepsilon )
\bigl(
T (t - \varepsilon )g(0, \phi (0))
\bigr)
+
+T (\varepsilon )
t - \varepsilon \int
0
T (t - s - \varepsilon )f(s) ds.
Note that the set \Biggl\{
g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t - \varepsilon )g(0, \phi (0))+
+
t - \varepsilon \int
0
T (t - s - \varepsilon )f(s) ds : z \in Br
\Biggr\}
is bounded, \bigm| \bigm| \bigm| \bigm| \bigm| g\bigl( t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t - \varepsilon )g(0, \phi (0))+
+
t - \varepsilon \int
0
T (t - s - \varepsilon )f(s) ds
\bigm| \bigm| \bigm| \bigm| \bigm| \leq r.
Since T (t) is a compact operator for t > 0, and (H5) we have that the set\bigl\{
h\varepsilon (t) : z \in Br
\bigr\}
is precompact in E for every \varepsilon , 0 < \varepsilon < t. Moreover, for every z \in Br we have
\bigm| \bigm| h(t) - h\varepsilon (t)
\bigm| \bigm| \leq M
t\int
t - \varepsilon
| f(s)| ds \leq
\leq M
t\int
t - \varepsilon
k(s) ds+MC1
t\int
t - \varepsilon
k(s) ds+ rM
t\int
t - \varepsilon
lk(s) ds\rightarrow 0 as \varepsilon \rightarrow 0.
Therefore, the set \{ h(t) : z \in Br\} is precompact, i.e., relatively compact.
Step 5. \scrA has closed graph.
Let \{ zn\} be a sequence such that zn \rightarrow z\ast , hn \in \scrA (zn) and hn \rightarrow h\ast . We shall show that
h\ast \in \scrA (z\ast ).
hn \in \scrA (zn) means that there exists fn \in SF,zn
\rho (t,znt +xt)
+x\rho (t,znt +xt)
such that
hn(t) = g
\bigl(
t, zn\rho (t,znt +xt)
+ x\rho (t,znt +xt)
\bigr)
- T (t)g(0, \phi (0)) +
t\int
0
T (t - s)fn(s) ds,
we must prove that there exists f\ast
h\ast (t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g(0, \phi (0)) +
t\int
0
T (t - s)f\ast (s) ds.
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GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1453
Consider the linear and continuous operator K : L1(J,E) \rightarrow Br defined by
K(v)(t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g(0, \phi (0)) +
t\int
0
T (t - s)v(s) ds.
We have\bigm| \bigm| K(fn)(t) - K(f\ast )(t)
\bigm| \bigm| = \bigm| \bigm| \bigl( hn(t) - g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) + T (t)g(0, \phi (0))
\bigr)
-
-
\bigl(
h\ast (t) - g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt)) + T (t)g(0, \phi (0))
\bigr) \bigm| \bigm| \leq
\leq \| hn - h\ast \| \infty \rightarrow 0 as n\rightarrow \infty .
From Lemma 2.2 it follows that K \circ SF is a closed graph operator and from the definition of K has
hn(t) \in K \circ SF,zn
\rho (t,znt +xt)
+x\rho (t,znt +xt)
.
As zn \rightarrow z\ast and hn \rightarrow h\ast , there exist f\ast \in SF,z\ast
\rho (t,z\ast t +xt)
+x\rho (t,z\ast +xt)
such that
h\ast (t) = g
\bigl(
t, z\rho (t,zt+xt) + x\rho (t,zt+xt)
\bigr)
- T (t)g(0, \phi (0)) +
t\int
0
T (t - s)f\ast (s) ds.
Hence the mutivalued operator \scrA is upper semicontinuous.
Step 6. \scrA (Br) is equiconvergent.
Let z \in Br, we have, for h \in \scrA (z),
| h(t)| \leq
\bigm| \bigm| g(t, z\rho (t,zt+xt) + x\rho (t,zt+xt))
\bigm| \bigm| +M | g(0, \phi (0))| +M
t\int
0
| f(s)| ds \leq
\leq M
\bigl(
kg\| \phi \| \scrB + g\ast
\bigr)
+ kg
\bigl(
l| z(t)| + (m+ \scrL \phi + lMH)\| \phi \| \scrB
\bigr)
+ g\ast +
+Mk\ast +M
t\int
0
k(s)
\bigl(
l| z(s)| + (m+ \scrL \phi + lMH)\| \phi \| \scrB
\bigr)
ds \leq
\leq C2 + kgl\| z\| BC\prime
0
+MC1k
\ast +Ml\| z\| BC\prime
0
k\ast .
Then we obtain
| h(t)| \rightarrow l \leq C2 + kglr +MC1k
\ast +Mlrk\ast as t\rightarrow +\infty .
Hence, \bigm| \bigm| h(t) - h(+\infty )
\bigm| \bigm| \rightarrow 0 as t\rightarrow +\infty .
As a consequence of Steps 1 – 4, with Lemma 2.5, we can conclude that \scrA : Br \rightarrow \scrP (Br) is
continuous and compact. From Schauder’s theorem, we deduce that \scrA has a fixed point z\ast . Then
y\ast = z\ast + x is a fixed point of the operators N, which is a mild solution of the problem (1), (2).
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1454 E. ALAIDAROUS, M. BENCHOHRA, I. MEDJADJ
4. An example. Consider the following neutral functional partial differential inclusion
\partial
\partial t
\bigl[
z(t, x) - g(t, z(t - \sigma (t, z(t, 0)), x))
\bigr]
- \partial 2
\partial x2
\bigl[
z(t, x) - g(t, z(t - \sigma (t, z(t, 0)), x))
\bigr]
\in
\in
t\int
- \infty
f(s, z(t - \sigma (s, z(s, 0)), x)) ds, x \in [0, \pi ], t \in J := [0,+\infty ), (5)
z(t, 0) = z(t, \pi ) = 0, t \in J, (6)
z(\theta , x) = \~z(\theta , x), t \in ( - \infty , 0] x \in [0, \pi ], (7)
where f is a given multivalued map, g a given function, and \sigma : \BbbR \rightarrow \BbbR +. Take E = L2[0, \pi ] and
define A : E \rightarrow E by A\omega = \omega \prime \prime with domain
D(A) =
\bigl\{
\omega \in E, \omega , \omega \prime are absolutely continuous \omega \prime \prime \in E, \omega (0) = \omega (\pi ) = 0
\bigr\}
.
Then
A\omega =
\infty \sum
n=1
n2(\omega , \omega n)\omega n, \omega \in D(A),
where \omega n(s) =
\sqrt{}
2
\pi
\mathrm{s}\mathrm{i}\mathrm{n}ns, n = 1, 2, . . . , is the orthogonal set of eigenvectors in A. It is well know
(see [36]) that A is the infinitesimal generator of an analytic semigroup T (t), t \geq 0, in E and is
given by
T (t)\omega =
\infty \sum
n=1
\mathrm{e}\mathrm{x}\mathrm{p}( - n2t)(\omega , \omega n)\omega n, \omega \in E.
Since the analytic semigroup T (t) is compact for t > 0, there exists a positive constant M such that
\| T (t)\| B(E) \leq M.
Let \scrB = BCU(\BbbR - ;E) and \phi \in \scrB , then (H\phi ) is satisfied with
\rho (t, \varphi ) = t - \sigma (\varphi ), t \in J.
Set
y(t)(x) = z(t, x), (t, x) \in J \times [0, \pi ],
F (t, \varphi )(x) =
t\int
- \infty
f(s, \varphi ) ds, (t, x) \in J \times [0, \pi ],
\phi (t)(x) = \~z(t, x), (t, x) \in ( - \infty , 0]\times [0, \pi ].
Hence, the problem (1), (2) in an abstract formulation of the problem (5) – (7), and if the condi-
tions (H1) – (H6) are satisfied, Theorem 3.1 implies that the problem (5) – (7) has at least one mild
solutions.
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GLOBAL EXISTENCE RESULTS FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS . . . 1455
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Received 20.08.14,
after revision — 11.07.18
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|
| id | umjimathkievua-article-1648 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:52Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4b/f29508fd4f55b6f92c34854c3c4b6f4b.pdf |
| spelling | umjimathkievua-article-16482019-12-05T09:22:19Z Global existence results for neutral functional differential inclusions with state-dependent delay Результати про глобальне iснування розв’язкiв нейтральних функцiональних диференцiальних включень iз затримкою, що залежить вiд стану Alaidarous, E. Benchohra, M. Medjadj, I. Алаідарус, Е. Беньчохра, М. Медядй, І. We consider the existence of global solutions for a class of neutral functional differential inclusions with state-dependent delay. The proof of the main result is based on the semigroup theory and the Bohnenblust – Karlin fixed point theorem. Розглянуто питання про iснування глобальних розв’язкiв одного класу нейтральних функцiональних диференцiальних включень iз затримкою, що залежить вiд стану. Доведення основного результату базується на теорiї напiвгруп та теоремi про нерухому точку Боненблюста та Карлiна. Institute of Mathematics, NAS of Ukraine 2018-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1648 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 11 (2018); 1443-1456 Український математичний журнал; Том 70 № 11 (2018); 1443-1456 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1648/630 Copyright (c) 2018 Alaidarous E.; Benchohra M.; Medjadj I. |
| spellingShingle | Alaidarous, E. Benchohra, M. Medjadj, I. Алаідарус, Е. Беньчохра, М. Медядй, І. Global existence results for neutral functional differential inclusions with state-dependent delay |
| title | Global existence results for neutral functional differential
inclusions with state-dependent delay |
| title_alt | Результати про глобальне iснування розв’язкiв нейтральних функцiональних диференцiальних включень iз затримкою, що залежить вiд стану |
| title_full | Global existence results for neutral functional differential
inclusions with state-dependent delay |
| title_fullStr | Global existence results for neutral functional differential
inclusions with state-dependent delay |
| title_full_unstemmed | Global existence results for neutral functional differential
inclusions with state-dependent delay |
| title_short | Global existence results for neutral functional differential
inclusions with state-dependent delay |
| title_sort | global existence results for neutral functional differential
inclusions with state-dependent delay |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1648 |
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