Existence and compactness of solution of semilinear integro-differential equations with finite delay
UDC 517.9 We present some existence and uniqueness results for a class of functional integro-differential evolution equations  generated by the resolvent operator for which the semigroup is not necessarily compact. It  is proved that the set of solutions...
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Institute of Mathematics, NAS of Ukraine
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| author | Sahraoui, F. Ouahab, А. Sahraoui, F. Ouahab, А. |
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UDC 517.9
We present some existence and uniqueness results for a class of functional integro-differential evolution equations  generated by the resolvent operator for which the semigroup is not necessarily compact. It  is proved that the set of solutions is compact.  Our approach is based on fixed point theory.  Finally, some examples are given to illustrate the results. |
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DOI: 10.37863/umzh.v74i9.7106
UDC 517.9
F. Sahraoui1, A. Ouahab (Laboratory Math., Sidi-Bel-Abbès Univ., Algeria)
EXISTENCE AND COMPACTNESS OF SOLUTION
OF SEMILINEAR INTEGRO-DIFFERENTIAL EQUATIONS
WITH FINITE DELAY
IСНУВАННЯ ТА КОМПАКТНIСТЬ РОЗВ’ЯЗКУ
НАПIВЛIНIЙНИХ IНТЕГРО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ЗI СКIНЧЕННИМ ЗАПIЗНЕННЯМ
We present some existence and uniqueness results for a class of functional integro-differential evolution equations generated
by the resolvent operator for which the semigroup is not necessarily compact. It is proved that the set of solutions is compact.
Our approach is based on fixed point theory. Finally, some examples are given to illustrate the results.
Наведено деякi результати щодо iснування та єдиностi розв’язкiв деякого класу функцiональних iнтегро-диференцi-
альних еволюцiйних рiвнянь, породжених резольвентним оператором, де напiвгрупа необов’язково компактна. До-
ведено компактнiсть множини розв’язкiв. Наш пiдхiд ґрунтується на теорiї нерухомих точок. Крiм того, наведено
кiлька прикладiв, що iлюструють отриманi результати.
1. Introduction. Nonlinear evolution equations appeared in many fields of applied mathematics,
and also in other branches of science as material science, biological sciences, physics and mechanics.
For example, the nonlinear reaction-diffusion equations from heat transfers, Cahn – Hilliard equations
from material science, the nonlinear Klein – Gordon equations and nonlinear Schrödinger equations
from quantum mechanics and Navier – Stokes and Euler equations from fluid mechanics. See the
books [1 – 3, 10, 11].
The study of the existence of mild solutions for integro-differential equations and inclusions in
abstract spaces has been done by several authors, see the works [4 – 9, 13 – 15].
In this paper, we consider the following semilinear integro-differential problem:
u\prime (t) = Au(t) +
t\int
0
U(t - s)u(s)ds+ F
\left( t, ut, t\int
0
\rho (t, s, us)ds
\right) a.e. t \in \BbbR +,
u(t) = \phi (t), t \in J0,
(1.1)
where J0 = [ - r, 0], the operator A is the infinitesimal generator of a C0-semigroup \{ T (t)\} t\geq 0 on a
real Banach space (E, \| \cdot \| ) with domain D(A), F : \BbbR + \times \scrC (J0, E)\times E \rightarrow E, is a given function,
\rho : \Delta \times \scrC (J0, E) \rightarrow E is a continuous function, with \Delta = \{ (t, s) \in \BbbR + \times \BbbR +; s \geq t\} , and \phi \in
\in \scrC (J0, E). For any t \in \BbbR +, U(t) is a closed linear operator on E, with domain D(A) \subset D(U(t)),
which is independent of t.
For any function u defined on J = J0 \cup \BbbR + and any t \in \BbbR +, we denote by ut the element of
\scrC (J0, E) defined by
ut(\theta ) = u(t+ \theta ), \theta \in J0,
where ut(\cdot ) represents the history of the state from time t - r, up to the present time t.
1 Corresponding author, e-mail: douhy_fati@yahoo.fr.
c\bigcirc F. SAHRAOUI, A. OUAHAB, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9 1231
1232 F. SAHRAOUI, A. OUAHAB
Here we are interested to study the existence of the mild solutions for the above partial functional
integro-differential evolution equations where the semigroup is not necessarily compact and we prove
the compactness of the set of the solutions.
Using methods and theorems of functional analysis, we propose a set of sufficient conditions to
ensure the existence of solutions. Specifically, to establish our main results we have used the theory
of the resolvent operator in the sense of Grimmer and Kuratowski’s measure of noncompactness.
2. Preliminaries. In this section, we introduce some notations, definitions, and preliminary facts
which will be used throughout,
BC(J,E) =
\biggl\{
u \in \scrC (J,E)
\bigg/
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\| u(t)\| <\infty
\biggr\}
is a Banach space with the following norm:
\| u\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\| u(t)\| \forall u \in BC(J,E).
\scrB represents the Banach space D(A) equipped with the graph norm
\| u\| \scrB = \| Au\| \infty + \| u\| \infty \forall u \in \scrB .
The notation \scrC 1(\BbbR +, E) stands for the Banach space of all functions mapping \BbbR + into E which are
continuously differentiable, and the notation \scrC (\BbbR +,\scrB ) stands for the space of all functions from \BbbR +
into \scrB which are continuous.
We recall some knowledge on integro-differential equations and the related resolvent operators.
Definition 2.1 [16]. A resolvent operator for the problem (1.1) is a bounded linear operator
valued function, R(t) \in \scrL (E) for t \geq 0, satisfying the following properties:
(a) R(0) = I (the identity map of E ) and \| R(t)\| \leq Me\beta t for some constants M > 0 and
\beta \in \BbbR .
(b) For each u \in E, R(t)u is strongly continuous.
(c) For any u \in E, R(\cdot )u \in \scrC 1(\BbbR +, E) \cap \scrC (\BbbR +,\scrB ) and
R\prime (t)u = AR(t)u+
t\int
0
U(t - s)R(s)u ds = R(t)Au+
t\int
0
R(t - s)U(s)u ds.
Theorem 2.1 [16]. Suppose that \phi (0) \in D(A). Then (1.1) has a resolvent operator. Moreover,
if u is a solution of (1.1), then
u(t) =
\left\{ R(t)\phi (0) +
\int t
0
R(t - s)F
\biggl(
s, us,
\int s
0
\rho (s, r, ur)dr
\biggr)
ds, t \in \BbbR +,
\phi (t), t \in J0.
In order to prove our main results, we need to recall the important properties of Kuratowski
measure of noncompactness.
Definition 2.2 [12]. Let (E, d) be a metric space and \frakB (E) be the set of all bounded subsets
of E. The Kuratowski measure of noncompactness \alpha is a function defined on \frakB (E) by
\alpha (D) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \epsilon > 0/D has a finite cover by sets of diameter less or equal to \epsilon \} \forall D \in \frakB (E).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1233
Using the above definition, we can prove the following lemma.
Lemma 2.1 [12]. Let (E, d) be a complete metric space and D1, D2 be a bounded subsets of
E. Then:
(1) \alpha (D1) = 0 if and only if D1 is compact;
(2) \alpha (D1) = \alpha (D1);
(3) for any \lambda \in \BbbR , \alpha (\lambda D1) = | \lambda | \alpha (D1);
(4) for any u \in E, \alpha (\{ u\} \cup D1) = \alpha (D1);
(5) \alpha (\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(D1)) = \alpha (D1), where \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(D1) is the convex hull of D1;
(6) if D1 \subseteq D2, then \alpha (D1) \leq \alpha (D2);
(7) \alpha (D1 \cup D2) = \mathrm{m}\mathrm{a}\mathrm{x}\{ \alpha (D1), \alpha (D2)\} ;
(8) \alpha (D1 +D2) = \alpha (D1) + \alpha (D2), where D1 +D2 = \{ u+ v \in E/u \in D1 and v \in D2\} .
Theorem 2.2 [17]. Let E be a Banach space, V \subset E be a bounded open neighborhood of 0
and N : V \rightarrow E be a continuous operator satisfying:
(1) the Mönch condition: if C is a countable subset of V and C \subset \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(\{ 0\} \cup N(C)), then C
is relatively compact,
(2) the Leary – Schauder boundary condition: x \not = \gamma N(x) for all x \in \partial V and 0 < \gamma < 1.
Then \mathrm{F}\mathrm{i}\mathrm{x}(N) = \{ x \in E : x = N(x)\} is nonempty.
Lemma 2.2. Let E be a Banach space, V \subset E be a bounded open neighborhood of 0 and N :
V \rightarrow E be a continuous operator satisfying the Mönch condition. Then \mathrm{F}\mathrm{i}\mathrm{x}(N) is compact.
Proof. If \mathrm{F}\mathrm{i}\mathrm{x}(N) = \varnothing , it is clair that \mathrm{F}\mathrm{i}\mathrm{x}(N) is compact. Now, if \mathrm{F}\mathrm{i}\mathrm{x}(N) \not = \varnothing , let (xn)n\in \BbbN \subset
\subset \mathrm{F}\mathrm{i}\mathrm{x}(N). Then
(xn)n\in \BbbN \subset \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(\{ 0\} \cup N(V )).
By the Mönch condition, we get \{ xn : n \in \BbbN \} is relatively compact. Thus, there exists a subsequence
of (xn)n\in \BbbN that converges to some x \in V . By the continuity of N, we conclude that x = N(x) \in
\in \mathrm{F}\mathrm{i}\mathrm{x}(N).
Lemma 2.3 [22]. Let I = [0, a] be a compact interval in \BbbR , E be a real Banach space and
B = \{ u \in E; \| u - u(0)\| < b\} with b \in \BbbR +. Assume that f be a function from I\times B into a Banach
space F which satisfies the Carathéodory conditions and the condition: for any subset X of B,
\alpha (f(T \times X)) \leq \mathrm{s}\mathrm{u}\mathrm{p}
t\in T
h(t, \alpha (X))
for each closed subset T of I, where h : I \times \BbbR + \rightarrow \BbbR + is a Kamke function. Let K be a bounded
strongly measurable function from I \times I into the space of bounded linear mappings from F to E. If
V is an equicontinuous set of functions I \rightarrow B, then
\alpha
\left( \left\{
\int
T
K(t, s)f(s, u(s)ds; u \in V )
\right\}
\right) \leq
\int
T
\| K(s, t)\| h(s, \alpha (V (s)))ds.
Lemma 2.4 [18]. Let f, g \in L1(\BbbR +,\BbbR ), B, u \in C(\BbbR +,\BbbR +) such that
u(t) \leq B(t) +
t\int
0
f(s)u(s)ds+
t\int
0
f(s)
\left( s\int
0
g(\tau )d\tau
\right) ds \forall t \in \BbbR +.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1234 F. SAHRAOUI, A. OUAHAB
If B is nondecreasing function, then
u(t) \leq B(t)
\left( 1 +
t\int
0
f(s) \mathrm{e}\mathrm{x}\mathrm{p}
\left( s\int
0
(f(\tau ) + g(\tau )d\tau
\right) ds
\right) \forall t \in \BbbR +.
3. Main results.
Definition 3.1. A function u \in \scrC (J,E) is said to be a mild solution of the problem (1.1) if
u(t) =
\left\{ R(t)\phi (0) +
\int t
0
R(t - s)F
\biggl(
s, us,
\int s
0
\rho (s, r, ur)dr
\biggr)
ds, t \in \BbbR +,
\phi (t), t \in J0.
(3.1)
In order to give the first result of existence and uniqueness, we shall need the following hypothe-
ses:
(\scrH 1) The function F : J \times \scrC (J0, E) \times E \rightarrow E is Carathéodory and satisfies the following
conditions:
(i) There exists f1 \in L1(\BbbR +) such that
\| F (t, \phi , x) - F (t, \psi , y)\| \leq f1(t)(\| \phi - \psi \| \infty + \| x - y\| ) \forall \phi , \psi \in \scrC (J0, E) and x, y \in E.
(ii) The function g1 : t \in \BbbR + \rightarrow g1(t) = \| F (t, 0, 0)\| \in \BbbR + belongs to L1(\BbbR +).
(\scrH 2) The function \rho : \Delta \times \scrC (J0, E) \rightarrow E satisfies the following conditions:
(i) There exists f2 \in L1(\BbbR +) such that
\| \rho (t, s, \phi ) - \rho (t, s, \psi )\| \leq f2(s)\| \phi - \psi \| \infty \forall \phi , \psi \in \scrC (J0, E) and (t, s) \in \Delta .
(ii) There exists a constant N \geq 0 such that
s\int
0
\| \rho (s, r, 0)\| dr \leq N \forall s \in \BbbR +.
Theorem 3.1. Assume that the conditions (\scrH 1) and (\scrH 2) hold. Then the problem (1.1) has a
unique mild solution.
Proof. We put g = \mathrm{m}\mathrm{a}\mathrm{x}(f1, f2). Assume that \beta \geq 0. For some constant \lambda > 0, we introduce
the following real vectorial space:
BC\ast (J,E) =
\biggl\{
u \in \scrC (J,E)
\bigg/
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e - \beta te - \lambda
\int t
0 g(\xi )d\xi \| u(t)\|
\Bigr)
<\infty
\biggr\}
.
It is clair that BC(J,E) \subset BC\ast (J,E). This space, endowed with the norm
\| u\| \ast = \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e - \beta te - \lambda
\int t
0 g(\xi )d\xi \| u(t)\|
\Bigr)
for all u \in BC\ast (J,E), is a Banach space.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1235
We consider now the operator N : BC\ast (J,E) \rightarrow BC\ast (J,E) defined by
(Nu)(t) =
\left\{
R(t)\phi (0) +
\int t
0
R(t - s)F
\biggl(
s, us,
\int s
0
\rho (s, r, ur)dr
\biggr)
ds, t \in \BbbR +,
\phi (t), t \in J0.
It is clair that the fixed points of that operator are solutions of the problem (1.1). Using the Banach
fixed point theorem, we prove that N has a unique fixed point. Then we obtain to show that the
operator N is well defined and contractive.
Step 1: N is well defined.
In fact, let u \in BC\ast (J,E), then, for any t \in \BbbR +, we get
\| (Nu)(t)\| \leq Me\beta t\| \phi (0)\| +
+Me\beta t
t\int
0
e - \beta s
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) - F (s, 0, 0) + F (s, 0, 0)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf1(s)
\left( \| us\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
\rho (s, r, ur)dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\right) ds+
+Me\beta t
t\int
0
e - \beta s\| F (s, 0, 0)\| ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf1(s)\| us\| \infty ds+Me\beta t
t\int
0
e - \beta s\| F (s, 0, 0)\| ds+
+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
\| \rho (s, r, ur) - \rho (s, r, 0) + \rho (s, r, 0)\| dr
\right) ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf1(s)\| us\| \infty ds+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
f2(r)\| ur\| \infty dr
\right) ds+
+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
\| \rho (s, r, 0)\| dr
\right) ds+Me\beta t
t\int
0
e - \beta s\| F (s, 0, 0)\| ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sg(s)\| us\| \infty ds+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
g(r)\| ur\| \infty dr
\right) ds+
+MNe\beta t
t\int
0
e - \beta sf1(s)ds+Me\beta t
t\int
0
e - \beta s\| F (s, 0, 0)\| ds \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1236 F. SAHRAOUI, A. OUAHAB
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
\Bigl(
g(s)e\lambda
\int s
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int s
0 g(\xi )d\xi e - \beta s\| us\| \infty
\Bigr)
ds+
+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
e\beta r
\Bigl(
g(r)e\lambda
\int r
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int r
0 g(\xi )d\xi e - \beta r\| ur\| \infty
\Bigr)
dr
\right) ds+
+MNe\beta t
t\int
0
f1(s)ds+Me\beta t
t\int
0
\| f(s, 0, 0)\| ds \leq
\leq Me\beta t\| \phi (0)\| + M
\lambda
e\beta t\| u\| \ast
\Bigl(
e\lambda
\int t
0 g(\xi )d\xi - 1
\Bigr)
+
+
M
\lambda
e\beta t
t\int
0
e - \beta sf1(s)
\Bigl(
e\beta s\| u\| \ast
\Bigl(
e\lambda
\int s
0 g(\xi )d\xi - 1
\Bigr) \Bigr)
ds+MNe\beta t\| f1\| L1 +Me\beta t\| g1\| L1 \leq
\leq Me\beta t(\| \phi (0)\| +N\| f1\| L1 + \| g1\| L1) +
M
\lambda
\| u\| \ast e\beta te\lambda
\int t
0 g(\xi )d\xi +
M
\lambda
e\beta te\lambda
\int t
0 g(\xi )d\xi \| f1\| L1 .
Then
\| (Nu)(t)\| \leq Me\beta t(\| \phi (0)\| +N\| f1\| L1 + \| g1\| L1) +
M
\lambda
e\beta te\lambda
\int t
0 g(\xi )d\xi (\| u\| \ast + \| f1\| L1).
We have proved that, for any t \in \BbbR +,
e - \beta te - \lambda
\int t
0 g(\xi )d\xi \| (Nu)(t)\| \leq
\leq Me - \lambda
\int t
0 g(\xi )d\xi
\bigl(
\| \phi (0)\| +N\| f1\| L1 + \| g1\| L1
\bigr)
+
M
\lambda
(\| u\| \ast + \| f1\| L1) \leq
\leq M
\bigl(
\| \phi (0)\| +N\| f1\| L1 + \| g1\| L1
\bigr)
+
M
\lambda
(\| u\| \ast + \| f1\| L1) < +\infty .
On the other hand, since \phi \in \scrC (J0, E), then \| Nu(t)\| <\infty for any t \in J0.
Hence,
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e - \beta te - \lambda
\int t
0 g(\xi )d\xi \| (Nu)(t)\|
\Bigr)
< +\infty ,
this means that the function Nu \in BC\ast (J,E).
Step 2: N is a contractive mapping.
In fact, let u, v \in BC\ast (J,E), then, for any t \in \BbbR +, we get
\| (Nu)(t) - (Nv)(t)\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
R(t - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) ds -
t\int
0
R(t - s)F
\left( s, vs, s\int
0
\rho (s, r, vr)dr
\right) ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1237
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
R(t - s)
\left[ F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) - F
\left( s, vs, s\int
0
\rho (s, r, vr)dr
\right) \right] ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq
t\int
0
\| R(t - s)\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) - F
\left( s, vs, s\int
0
\rho (s, r, vr)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds \leq
\leq Me\beta t
t\int
0
e - \beta sf1(s)
\left( \| us - vs\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
(\rho (s, r, ur) - \rho (s, r, vr))dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\right) ds \leq
\leq Me\beta t
t\int
0
e - \beta sf1(s)\| us - vs\| \infty ds+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
f2(r)\| ur - vr\| \infty dr
\right) ds \leq
\leq Me\beta t
t\int
0
e - \beta sg(s)\| us - vs\| \infty ds+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
g(r)\| ur - vr\| \infty dr
\right) ds \leq
\leq Me\beta t
t\int
0
e - \beta sg(s)\| us - vs\| \infty ds+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
g(r)\| ur - vr\| \infty dr
\right) ds \leq
\leq Me\beta t
t\int
0
\Bigl(
g(s)e\lambda
\int s
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int s
0 g(\xi )d\xi e - \beta s\| us - vs\| \infty
\Bigr)
ds+
+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
e\beta r
\Bigl(
g(r)e\lambda
\int r
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int r
0 g(\xi )d\xi e - \beta r\| ur - vr\| \infty dr
\Bigr) \right) ds \leq
\leq Me\beta t
t\int
0
\Bigl(
e\lambda
\int s
0 g(\xi )d\xi
\Bigr) \prime
ds\| u - v\| \ast +Me\beta t
t\int
0
f1(s)
\left( s\int
0
\Bigl(
e\lambda
\int r
0 g(\xi )d\xi
\Bigr) \prime
dr
\right) ds\| u - v\| \ast \leq
\leq M
\lambda
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \beta t+ t\int
0
g(s)ds
\right) \| u - v\| \ast +
M\| f1\| L1
\lambda
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \beta t+ t\int
0
g(s)ds
\right) \| u - v\| \ast .
Then
\mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t -
t\int
0
g(s)ds
\right) \| (Nu)(t) - (Nv)(t)\| \leq
\biggl(
M
\lambda
+
M\| f1\| L1
\lambda
\biggr)
\| u - v\| \ast , for all t \in J.
Therefore,
\| N(u) - N(v)\| \ast \leq
M(1 + \| f1\| L1)
\lambda
\| u - v\| \ast for all u, v \in BC\ast (J,E).
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1238 F. SAHRAOUI, A. OUAHAB
Then, for \lambda > M(1 + \| f1\| L1), N is a contraction. By Banach fixed point theorem, the unique fixed
point of N is the unique mild solution in BC\ast (J,E) of the problem (1.1).
Theorem 3.1 is proved.
For the next result, we present an application of the Mönch fixed point theorem type to prob-
lem (1.1).
Theorem 3.2. Let E be a separable Banach space and F : \BbbR + \times \scrC (J0, E) \times E \rightarrow E be a
Carathéodory function, such that (\scrH 2) is fulfilled, and the following condition holds:
(\scrH 3) \{ T (t)\} t\geq 0 is operator-norm continuous for t > 0.
(\scrH 4) There exist f \in L1(\BbbR +) such that f3 = e - \beta \cdot f \in L1(\BbbR +) with \beta \leq 0, and
\| F (t, \phi , x)\| \leq f(t)(\| \phi \| \infty + \| x\| + 1) for all \phi \in \scrC (J0, E), x \in E, a.e. t \in J.
(\scrH 5) There exists g2 \in L1(\BbbR +) such that, for all bounded D1 \in \scrC (J0, E), D2 \subset E, we have
\alpha (F (t,D1, D2)) \leq g2(t)
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
\alpha (D1(\theta )) + \alpha (D2)
\biggr)
for a.e. t \in J,
where
D1(\theta ) = \{ \phi (\theta ) : \phi \in D1\} , \theta \in J0.
(\scrH 6) There exists g\ast \in L1(\BbbR +) such that, for all bounded D \in \scrC (J0, E), we have
\alpha (\rho (t, s,D)) \leq g\ast (t) \mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
\alpha (D(\theta )) for a.e. (t, s) \in \Delta .
Then problem (1.1) has, at least, one mild solution and the solution set is compact.
Proof. We put g = \mathrm{m}\mathrm{a}\mathrm{x}(f, f2), For some constant \lambda > 0, we introduce the following real
vectorial space:
BC\ast (J,E) =
\biggl\{
u \in \scrC (J,E)
\bigg/
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e\beta te - \lambda
\int t
0 g(\xi )d\xi \| u(t)\|
\Bigr)
<\infty
\biggr\}
.
It is clair that BC(J,E) \subset BC\ast (J,E). This space, endowed with the norm
\| u\| \ast = \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e\beta te - \lambda
\int t
0 g(\xi )d\xi \| u(t)\|
\Bigr)
for all u \in BC\ast (J,E), is a Banach space.
We consider now the same operator N : BC\ast (J,E) \rightarrow BC\ast (J,E) defined in the proof of The-
orem 3.1. It is clair that the fixed points of that operator are solutions of the problem (1.1). Using
the Mönch fixed point theorem, we prove that N has, at least, one fixed point. Let us prove that the
conditions of the Mönch fixed point Theorem 2.2 are satisfied following several steps.
Step 1: N is well defined.
We follow the same manner as in step 1 of the proof of Theorem 3.1.
Step 2: N is continuous.
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1239
In fact, let (u(n))n\in \BbbN be a sequence in BC\ast (I,W ) such that u(n) \rightarrow u in BC\ast (I,W ). Then
\| (Nu(n))(t) - (Nu)(t)\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
R(t - s)F
\left( s, u(n)s ,
s\int
0
\rho (s, r, u(n)r )dr
\right) ds -
-
t\int
0
R(t - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
R(t - s)
\left[ F
\left( s, u(n)s ,
s\int
0
\rho (s, r, u(n)r )dr
\right) - F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \right] ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq
t\int
0
\| R(t - s)\|
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\left( s, u(n)s ,
s\int
0
\rho (s, r, u(n)r )dr
\right) - F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds. (3.2)
The sequence (Fn)n\in \BbbN , defined by Fn : t \in \BbbR + \rightarrow Fn(t) = F
\biggl(
t, u
(n)
t ,
\int t
0
\rho (t, r, u(n)r )
\biggr)
, satisfies
the conditions of the Lebesgue dominated convergence theorem. In fact, since F is a Carathéodory
function and \scrC (J0, E) \times E is separable, then F is measurable. For any n \in \BbbN , the function hn :
t \in \BbbR + \rightarrow hn(t) =
\biggl(
t, u
(n)
t ,
\int t
0
\rho (t, r, u(n)r )
\biggr)
is also measurable because u(n) and \rho are continuous.
As Fn is the composite function of two measurable functions hn and F, it follows that Fn is
measurable too.
Since the sequence (u(n))n\in \BbbN converges to u in BC\ast (J,E) and F is a Carathéodory function,
then, for any t \in \BbbR +,
Fn(t) = F
\left( t, u(n)t ,
t\int
0
\rho (t, r, u(n)r )
\right) - \rightarrow F
\left( t, ut, t\int
0
\rho (t, r, ur)dr
\right) .
Since the sequence (u(n))n\in \BbbN is convergent, then (u(n))n\in \BbbN is bounded by some positive constant
M1. Let n \in \BbbN . By the Definition 2.1 and using the hypothesis (\scrH 4) and (\scrH 2), we get, for any
t \in \BbbR +,
\| Fn(t)\| \leq f(t)
\left( \| u(n)t \| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
\rho (t, r, u(n)r )dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| + 1
\right) \leq
\leq f(t)
\left( \| u(n)t \| \infty +
t\int
0
\bigm\| \bigm\| \bigm\| \rho (t, r, u(n)r ) - \rho (t, r, 0) + \rho (t, r, 0)
\bigm\| \bigm\| \bigm\| dr + 1
\right) \leq
\leq f(t)
\left( \| u(n)t \| \infty +
t\int
0
f2(r)\| u(n)r \| \infty dr +
t\int
0
\| \rho (t, r, 0)\| dr + 1
\right) \leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1240 F. SAHRAOUI, A. OUAHAB
\leq f(t)
\Bigl(
\| u(n)\| \infty + \| f2\| L1\| u(n)\| \infty dr +N + 1
\Bigr)
\leq
\leq f(t)(M1 + \| f2\| L1M1 +N + 1) = Kf(t),
with K = M1 + \| f2\| L1M1 + N + 1 is a positive constant. Since f is measurable, then Kf is
measurable. By the Lebesgue dominated convergence theorem, we obtain that the right-hand side of
the inequality (3.2) tends to 0 as n approaches to +\infty , this implies that
\bigm\| \bigm\| (Nu(n))(t) - (Nu)(t)
\bigm\| \bigm\| \rightarrow
\rightarrow 0 as n\rightarrow +\infty . Thus, N is continuous.
Step 3: N maps bounded sets of BC\ast (J,E) into bounded sets of BC\ast (J,E).
In fact, let d > 0 and Bd = \{ u \in BC\ast (J,E)/\| u\| \ast \leq d\} , we show that N(Bd) is bounded. By
the Definition 2.1 and using the hypothesis (\scrH 4) and (\scrH 2), we get, for any t \in \BbbR +,
\| (Nu)(t)\| \leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta s
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)
\left( \| us\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
\rho (s, r, ur)dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| + 1
\right) ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)\| us\| \infty ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, ur) - \rho (s, r, 0) + \rho (s, r, 0)\| dr
\right) ds+Me\beta t
t\int
0
e - \beta sf(s)ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)\| us\| \infty ds+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
f2(r)\| ur\| \infty dr
\right) ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, 0)\| dr
\right) ds+Me - \beta t\| f\| L1 \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sg(s)\| us\| \infty ds+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
g(r)\| ur\| \infty dr
\right) ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, 0)\| dr
\right) ds+Me - \beta t\| f\| L1 \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - 2\beta s
\Bigl(
g(s)e\lambda
\int s
0 g(\xi )d\xi
\Bigr)
e\beta se - \lambda
\int s
0 g(\xi )d\xi \| us\| \infty ds+
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1241
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
e - \beta r
\Bigl(
g(r)e\lambda
\int r
0 g(\xi )d\xi
\Bigr) \Bigl(
e\beta re - \lambda
\int r
0 g(\xi )d\xi \| ur\| \infty
\Bigr)
dr
\right) ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, 0)\| dr
\right) ds+Me - \beta t\| f\| L1 \leq
\leq Me\beta t\| \phi (0)\| + M
\lambda
e - \beta t\| u\| \ast
\Bigl(
e\lambda
\int t
0 g(\xi )d\xi - 1
\Bigr)
+
+
M
\lambda
e\beta t
t\int
0
e - \beta sf(s)
\Bigl(
e - \beta s\| u\| \ast
\Bigl(
e\lambda
\int s
0 g(\xi )d\xi - 1
\Bigr) \Bigr)
ds+MNe - \beta t\| f\| L1 +Me - \beta t\| f\| L1 \leq
\leq M\| \phi (0)\| +Me - \beta t(N + 1)\| f\| L1 +
M
\lambda
e - \beta te\lambda
\int t
0 g(\xi )d\xi \| u\| \ast (1 + \| f\| L1),
that is,
e\beta te - \lambda
\int t
0 g(\xi )d\xi \| (Nu)(t)\| \leq M(\| \phi (0)\| + (N + 1)\| f\| L1) +
M
\lambda
d(1 + \| f\| L1) = l1.
On the other hand, since \phi \in \scrC (J0, E), we have \| Nu(t)\| \leq \mathrm{s}\mathrm{u}\mathrm{p}t\in J0 \| \phi (t)\| = l2 for any t \in J0,
with l1 and l2 \in \BbbR +. Hence,
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\Bigl(
e\beta te - \lambda
\int t
0 g(\xi )d\xi \| (Nu)(t)\|
\Bigr)
\leq l1 + l2 = l.
This means that \| Nu\| \ast \leq l, which proves that N(Bd) \subset Bl. Thus N(Bd) is bounded.
Step 4: N maps bounded sets of BC\ast (J,E) into equicontinuous sets of BC\ast (J,E).
In fact, let d > 0 and Bd = \{ u \in BC\ast (J,E)/\| u\| \ast \leq d\} . We show that N(Bd) is equicontin-
uous. Let t1, t2 \in [a, b] with t1 < t2 and [a, b] is a compact interval in \BbbR +. By Definition 2.1 and
using the hypothesis (\scrH 4), (\scrH 3) and (\scrH 2), we get, for any u \in Bd,
\| (Nu)(t1) - (Nu)(t2)\| \leq
\leq \| (R(t1) - R(t2))\phi (0)\| +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t1\int
0
R(t1 - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) ds -
-
t2\int
0
R(t2 - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq \| (R(t1) - R(t2))\phi (0)\| +
t1\int
0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| (R(t1 - s) - R(t2 - s))F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds+
+
t2\int
t1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| R(t2 - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds \leq
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1242 F. SAHRAOUI, A. OUAHAB
\leq \| R(t1) - R(t2)\| \scrL (E)\| \phi (0)\| +
+
t1\int
0
\| R(t1 - s) - R(t2 - s)\| \scrL (E)f(s)
\left( \| us\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
\rho (s, r, ur)dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| + 1
\right) ds+
+
t2\int
t1
Me\beta (t2 - s)f(s)
\left( \| us\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
\rho (s, r, ur)dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| + 1
\right) ds \leq
\leq \| R(t1) - R(t2)\| \scrL (E)\| \phi (0)\| +
t1\int
0
\| R(t1 - s) - R(t2 - s)\| \scrL (E)f(s)\times
\times
\left( d+ s\int
0
\| \rho (s, r, ur) - \rho (s, r, 0) + \rho (s, r, 0)\| dr + 1
\right) ds+
+
t2\int
t1
Me\beta (t2 - s)f(s)
\left( d+ s\int
0
\| \rho (s, r, ur) - \rho (s, r, 0) + \rho (s, r, 0)\| dr + 1
\right) ds \leq
\leq \| R(t1) - R(t2)\| \scrL (E)\| \phi (0)\| +
+
t1\int
0
\| R(t1 - s) - R(t2 - s)\| \scrL (E)f(s)
\left( d+ s\int
0
f2(r)\| ur\| \infty dr +N + 1
\right) ds+
+
t2\int
t1
Me\beta (t2 - s)f(s)
\left( d+ s\int
0
f2(r)\| ur\| \infty dr +N + 1
\right) ds.
Then
\| (Nu)(t1) - (Nu)(t2)\| \leq \| R(t1) - R(t2)\| \scrL (E)\| \phi (0)\| +
+
t1\int
0
\| R(t1 - s) - R(t2 - s)\| \scrL (E)f(s)(d+ d\| f2\| L1 +N + 1)ds+
+
t2\int
t1
Me\beta (t2 - s)f(s)(d+ d\| f2\| L1 +N + 1)ds.
By hypothesis (\scrH 2), we have \| R(t1) - R(t2)\| \scrL (E) tends to 0 as t1 \rightarrow t2. This leads to the
right-hand side of the above inequality tends to 0 as t1 \rightarrow t2 independently of u. Thus, \| (Nu)(t1) -
- (Nu)(t2)\| \rightarrow 0 as t1 \rightarrow t2.
We denote by \omega T (u, \epsilon ) the modulus of continuity of u \in E on the interval [0, T ], i.e.,
\omega T (u, \epsilon ) = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\| u(t) - u(s)\| ; t, s \in [0, T ] and | t - s| \leq \epsilon
\bigr\}
.
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1243
For any K \subset E, we put
\omega T (K, \epsilon ) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \omega T (u, \epsilon ); u \in K\} and \omega T
0 (K) = \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0
\omega T (K, \epsilon ).
Let us consider the measure of noncompactness \mu defined on the family of bounded subsets of
BC\ast (J,E) by
\mu (K) = \omega T
0 (K) + \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) + \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
u\in K
\| u(t)\|
for any K bounded subset of BC\ast (J,E), where \lambda > 0 and G = \mathrm{m}\mathrm{a}\mathrm{x}\{ g2, g\ast \} .
Step 5: We prove that the Mönch condition is satisfied. Let K be a bounded countable subset of
BC\ast (J,E) such that K \subset \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(\{ 0\} \cup N(K)\} ). Suppose that K \subset Bd = \{ u \in BC\ast (J,E)/\| u\| \ast \leq
\leq d\} , where d > 0. We have to show that K is relatively compact. To do this, it suffices to prove
that \mu (K) = 0.
This will be given in several claims:
Claim 1: \omega T
0 (K) = 0.
In fact, using the properties of the function \omega T
0 (\cdot ) (see [21]), and the fact that N(Bd) is equicon-
tinuous, we get
\omega T
0 (K) \leq \omega T
0
\bigl(
\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(\{ 0\} \cup N(K))
\bigr)
= \omega T
0 (N(K)) = 0.
Hence, we infer that \omega T
0 (K) = 0.
Claim 2: \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\biggl(
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- \beta t - \lambda
\int t
0
G(\xi )d\xi
\biggr)
\alpha (K(t))
\biggr)
= 0.
In fact, let t \in \BbbR +. We put K(t) = \{ u(t);u \in K\} and Kt = \{ ut;u \in K\} . By hypothesis (\scrH 5),
(\scrH 6) and applying the Lemma 2.3, we get
\alpha (K(t)) \leq \alpha (N(K(t))) = \alpha
\left\{ R(t)\phi (0) +
t\int
0
R(t - s)F
\left( s,Ks,
s\int
0
\rho (s, r,Kr)dr
\right) ds
\right\} \leq
\leq \alpha
\left\{
t\int
0
R(t - s)F
\left( s,Ks,
s\int
0
\rho (s, r,Kr)dr
\right) ds
\right\} \leq
\leq
t\int
0
\| R(t - s)\| g2(s)
\left( \mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Ks(\theta ))) + \alpha
\left( s\int
0
\rho (s, r,Kr)dr
\right) \right) ds \leq
\leq
t\int
0
Me\beta (t - s)g2(s)
\left( \mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Ks(\theta ))) + \alpha
\left( s\int
0
\rho (s, r,Kr)dr
\right) \right) ds \leq
\leq Me\beta t
t\int
0
e - \beta sg2(s)
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Ks(\theta )))
\biggr)
ds+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1244 F. SAHRAOUI, A. OUAHAB
+Me\beta t
t\int
0
e - \beta sg2(s)
\left( s\int
0
g\ast (r)
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Kr(\theta )))dr
\biggr) \right) ds \leq
\leq Me\beta t
t\int
0
e - \beta sG(s)
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Ks(\theta )))
\biggr)
ds+
+Me\beta t
t\int
0
e - \beta sg2(s)
\left( s\int
0
G(r)
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Kr(\theta )))dr
\biggr) \right) ds \leq
\leq Me\beta t
t\int
0
G(s)e\lambda
\int t
0 G(\xi )d\xi e - \beta se - \lambda
\int s
0 G(\xi )d\xi
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Ks(\theta )))
\biggr)
ds+
+Me\beta t
t\int
0
e - \beta sg2(s)
\left( s\int
0
e\beta rG(r)e\lambda
\int t
0 G(\xi )d\xi e - \beta se - \lambda
\int r
0 G(\xi )d\xi
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
\theta \in J0
(\alpha (Kr(\theta )))dr
\biggr) \right) ds \leq
\leq Me\beta t \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) t\int
0
G(s)e\lambda
\int s
0 G(\xi )d\xi ds+
+Me\beta t \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) t\int
0
e - \beta sg2(s)
\left( s\int
0
e\beta rG(r)e\lambda
\int r
0 G(\xi )d\xi dr
\right) ds \leq
\leq M
\lambda
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \beta t+ \lambda
t\int
0
G(\xi )d\xi
\right) \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) +
+
M
\lambda
\mathrm{e}\mathrm{x}\mathrm{p}
\left( \beta t+ \lambda
t\int
0
G(\xi )d\xi
\right) \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) \| g2\| L1 .
The above inequality reduces to
\alpha (K(t)) \leq M
\lambda
(1 + \| g2\| L1) \mathrm{e}\mathrm{x}\mathrm{p}
\left( \beta t+ \lambda
t\int
0
G(\xi )d\xi
\right) \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) .
This proves that for any t \in \BbbR +, we have
\mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t)) \leq M
\lambda
(1 + \| g2\| L1) \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) .
That means
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1245
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) \leq
\leq M
\lambda
(1 + \| g2\| L1) \mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) .
Taking \lambda > M(1 + \| g2\| L1), we obtain 0 <
M
\lambda
(1 + \| g2\| L1) < 1, and therefore
\mathrm{s}\mathrm{u}\mathrm{p}
t\in J
\left( \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \beta t - \lambda
t\int
0
G(\xi )d\xi
\right) \alpha (K(t))
\right) = 0.
Claim 3: \mathrm{s}\mathrm{u}\mathrm{p}u\in K \| u(t)\| \rightarrow 0 as t\rightarrow +\infty .
In fact, let u \in K \subset Bd. According to the step 3, we get, for any t \in \BbbR +,
\| u(t)\| =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| R(t)\phi (0) +
t\int
0
R(t - s)F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta s
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\left( s, us, s\int
0
\rho (s, r, ur)dr
\right) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)
\left( \| us\| \infty +
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
s\int
0
\rho (s, r, ur)dr
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| + 1
\right) ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)\| us\| \infty ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, ur) - \rho (s, r, 0) + \rho (s, r, 0)\| dr
\right) ds+Me\beta t
t\int
0
e - \beta sf(s)ds \leq
\leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
e - \beta sf(s)ds+Me\beta t
t\int
0
e - \beta sf(s)
\left( s\int
0
\| \rho (s, r, 0)\| dr
\right) ds+
+Me\beta t
t\int
0
e - \beta sf(s)
\left( \| us\| \infty +
s\int
0
f2(r)\| ur\| \infty dr
\right) ds \leq
\leq Me\beta t\| \phi (0)\| +M(1 +N)e\beta t
t\int
0
e - \beta sf(s)ds+
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1246 F. SAHRAOUI, A. OUAHAB
+Me\beta t
t\int
0
e - \beta sf(s)
\left( \| us\| \infty +
s\int
0
f2(r)\| ur\| \infty dr
\right) ds.
Then
e - \beta t\| u(t)\| \leq M\| \phi (0)\| +M(1 +N)
t\int
0
e - \beta sf(s)ds+
+M
t\int
0
e - \beta sf(s)
\left( \| us\| \infty +
s\int
0
f2(r)\| ur\| \infty dr
\right) ds \leq
\leq M\| \phi (0)\| +M(1 +N)
t\int
0
e - \beta sf(s)ds+M
t\int
0
f(s)e - \beta s\| us\| \infty ds+
+Me - \beta t
t\int
0
f(s)
\left( s\int
0
f2(r)e
- \beta r\| ur\| \infty dr
\right) ds.
Putting
B(t) =M\| \phi (0)\| +M(1 +N)
t\int
0
e - \beta sf(s)ds,
we get
\| u(t)\| \leq B(t) +M
t\int
0
f(s)
\left( e - \beta s\| us\| \infty +
s\int
0
f2(r)e
- \beta r\| ur\| \infty dr
\right) ds. (3.3)
Set
V (t) = \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,t]
e\beta s\| u(s)\| .
Thus, the inequality (3.3) implies
V (t) \leq B(t) +M
t\int
0
f(s)
\left( V (s) +
s\int
0
f2(r)V (r)dr
\right) ds.
Applying the Lemma 2.4, we obtain
V (t) \leq B(t)
\left( 1 +
t\int
0
f(s) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
(f(\tau ) + f2(\tau ))d\tau
\right) ds
\right) .
Hence
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1247
e - \beta t\| u(t)\| \leq B(t)
\left( 1 +
t\int
0
f(s) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
(f(\tau ) + f2(\tau ))d\tau
\right) ds
\right) .
Therefore,
\| u(t)\| \leq e\beta tB(t)
\left( 1 +
t\int
0
f(s) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
(f(\tau ) + f2(\tau ))d\tau
\right) ds
\right) .
Since \beta < 0 and by the condition e - \beta \cdot f \in L1(\BbbR +) cited in (\scrH 4), we conclude that the right-hand
side of the above inequality tends to 0 as t tends to +\infty , and, therefore, \mathrm{s}\mathrm{u}\mathrm{p}u\in K \| u(t)\| \rightarrow 0 as
t\rightarrow +\infty .
By Claims 1, 2, and 3, we obtain \mu (K) = 0. Thus, we find that K is relatively compact.
Step 6: A priori bounds. We prove now the existence of V a bounded open subset of BC\ast (J,E)
containing 0 and satisfying the Leary – Schauder boundary condition: u \not = \gamma N(u) for all u \in \partial V
and 0 < \gamma < 1.
Let u \in BC\ast (J,E) and u = \gamma Nu for some \gamma \in ]0, 1[. Then we get
\| u(t)\| \leq Me\beta t\| \phi (0)\| +Me\beta t
t\int
0
\Bigl(
g(s)e\lambda
\int s
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int s
0 g(\xi )d\xi e - \beta s\| us\| \infty
\Bigr)
ds+
+Me\beta t
t\int
0
e - \beta sf1(s)
\left( s\int
0
e\beta r
\Bigl(
g(r)e\lambda
\int r
0 g(\xi )d\xi
\Bigr) \Bigl(
e - \lambda
\int r
0 g(\xi )d\xi e - \beta r\| ur\| \infty
\Bigr)
dr
\right) ds+
+MNe\beta t
t\int
0
f1(s)ds+Me\beta t
t\int
0
\| f(s, 0, 0)\| ds.
Then
e
\beta t - \lambda
t\int
0
g(s)ds
\| u(t)\| \leq M\| \phi (0)\| +M
t\int
0
g(s)
\Bigl(
e - \lambda
\int s
0 g(\xi )d\xi e\beta s\| us\| \infty
\Bigr)
ds+
+M
t\int
0
e - \beta sf1(s)
\left( s\int
0
e - \beta rg(r)
\Bigl(
e - \lambda
\int r
0 g(\xi )d\xi e\beta r\| ur\| \infty
\Bigr)
dr
\right) ds+
+MN
t\int
0
f1(s)ds+M
t\int
0
\| f(s, 0, 0)\| ds.
Since we have
us(\theta ) =
\left\{ \phi (s+ \theta ), if s+ \theta \in [ - r, 0],
u(s+ \theta ), if s+ \theta \geq 0,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1248 F. SAHRAOUI, A. OUAHAB
and, hence,
e\beta t - \lambda
\int t
0 g(s)ds\| u(t)\| \leq M\| \phi (0)\| +M
t\int
0
g(s)\| \phi \| \infty ds+
+
t\int
0
g(s)
\Biggl(
e - \lambda
\int s
0 g(\xi )d\xi e\beta s \mathrm{s}\mathrm{u}\mathrm{p}
r\in [0,s]
\| u(r)\|
\Biggr)
ds+
+M\| \phi \| \infty
t\int
0
e - \beta sf1(s)
s\int
0
g(r)drds+
+M
t\int
0
e - \beta sf1(s)
\left( s\int
0
g(r)
\Biggl(
e - \lambda
\int r
0 g(\xi )d\xi e\beta r \mathrm{s}\mathrm{u}\mathrm{p}
\xi \in [0,r]
\| u(\xi )\|
\Biggr)
dr
\right) ds+
+MN
t\int
0
f1(s)ds+M
t\int
0
\| f(s, 0, 0)\| ds.
Set
V\ast (t) = \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,t]
e
\beta s - \lambda
s\int
0
g(\tau )d\tau
\| u(t)\| .
Then
V\ast (t) \leq M\| \phi (0)\| +M
t\int
0
g(s)\| \phi \| \infty ds+
+M\| \phi \| \infty
t\int
0
e - \beta sf1(s)
s\int
0
g(r)drds+MN
t\int
0
f1(s)ds+
+M
t\int
0
\| f(s, 0, 0)\| ds+
t\int
0
g(s)V\ast (s)ds+M
t\int
0
e - \beta sf1(s)
\left( s\int
0
g(r)V\ast (r)dr
\right) ds.
Applying the Lemma 2.4, we obtain
V\ast (t) \leq B\ast (t)
\left( 1 +
t\int
0
f(s) \mathrm{e}\mathrm{x}\mathrm{p}
\left( t\int
0
(g(\tau ) + f3(\tau ))d\tau
\right) ds
\right) ,
where
B\ast (t) =M\| \phi (0)\| +M
t\int
0
g(s)\| \phi \| \infty ds+M\| \phi \| \infty
t\int
0
e - \beta sf1(s)
s\int
0
g(r)drds+MN
t\int
0
f1(s)ds.
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1249
Consequently, we have
\| u\| \ast \leq B\infty (1 + \| f\| \infty \mathrm{e}\mathrm{x}\mathrm{p}(\| g\| \infty + \| f2\| \infty )) :=M\ast .
Set V = \{ u \in BC\ast (J,E) : \| u\| \ast < M\ast + 1\} . So, V is a bounded open neighborhood of 0 and
the operator N : V \rightarrow BC(J,E) satisfies the conditions of Mönch fixed point Theorem 2.2. Hence
N has at least a fixed point u \in V which is a solution to problem (1.1). It is clear that \mathrm{F}\mathrm{i}\mathrm{x}(N) \subset V.
By Lemma 2.2, \mathrm{F}\mathrm{i}\mathrm{x}(N) is also compact.
Theorem 3.2 is proved.
4. Applications. In this part, we give some applications on our results of this paper, for
this we assume that we have a bounded domain G of \BbbR d with a smooth boundary \Gamma = \partial G =
= \Gamma 0 \cup \Gamma 1, \Gamma 0 \cap \Gamma 1 = \varnothing . Let us consider the following internally damped wave equation:
\partial ttu - \Delta u+ a(x)\partial tu(x, t) = 0, (x, t) \in G\times (0,\infty ),
u(x, t) = 0, (x, t) \in \Gamma 0 \times (0,\infty ),
\partial u
\partial \nu
+ u = 0, (x, t) \in \Gamma 1 \times (0,\infty ),
u(x, t) = u0(x, t), ut(x, t) = u1(x, t), (x, t) \in G\times [ - r, 0],
(4.1)
where a : G \rightarrow \BbbR + be a positive continuous function and u0 \in \scrC
\bigl(
[ - r, 0], L2(G)
\bigr)
, u1 \in L2(G). By
putting v = \partial tu, we write the system (4.1) into the following problem:
\partial tu - v = 0, x \in G, t > 0,
\partial tv - \Delta u+ a(x)v = 0, x \in G, t > 0,
u(x, t) = 0, x \in \Gamma 0, t > 0,
\partial u
\partial \nu
+ u = 0, x \in \Gamma 1, t > 0,
u(x, t) = u0(x, t), v(x, t) = u1(x, t), x \in G, \in [ - r, 0].
(4.2)
The operator Ad = - \Delta is strict positive and auto-adjoint in H = L2(G), D(Ad) = H1
0 (G). We
shall use the semigroup method to demonstrate the global existence and uniqueness of solution, for
this purpose we rewrite the system (4.2) as an evolution equation for
\scrU \prime (t) = \scrA \scrU (t), t > 0,
\scrU (t) = \scrU 0(t), t \in [ - r, 0],
where \scrU (\cdot ) =
\biggl(
u(\cdot , \cdot )
v(\cdot , \cdot )
\biggr)
, \scrU 0(t) =
\biggl(
u0(\cdot , t)
u1(\cdot , t)
\biggr)
and \scrA : D(\scrA ) \subset \scrH \rightarrow \scrH is defined by
\scrA
\Biggl(
u(\cdot , t)
v(\cdot , t)
\Biggr)
=
\Biggl(
v
- Adu(\cdot , t) - a(\cdot )v
\Biggr)
with domain
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1250 F. SAHRAOUI, A. OUAHAB
D(\scrA ) =
\Biggl\{ \Biggl(
u
v
\Biggr)
\in \scrH : u \in H2(G) \cap H1
\Gamma 0
(G), and
\partial u
\partial \nu
= 0 on \Gamma 1
\Biggr\}
in the Hilbert space \scrH = H1
\Gamma 0
(G) \times L2(G), where H1
\Gamma 0
(G) := \{ u \in H1(G) : u = 0 on \Gamma 0\} . We
equipped \scrH with the scalar product\Biggl\langle \Biggl(
u1
v1
\Biggr)
,
\Biggl(
u2
v2
\Biggr) \Biggr\rangle
\scrH
=
\int
G
(\nabla u1\nabla u2 + v1v2)dx+
\int
\Gamma 1
u1u2d\Gamma ,
and the norm defined by
\| (u, v)\| 2\scrH =
\int
G
(| \nabla u| 2 + v2)dx+
\int
\Gamma 1
u2d\Gamma .
4.1. Well-posedness.
Proposition 4.1. \scrA is m-dissipative in the Hilbert space \scrH .
Proof. Let \scrU \in D(\scrA ), then
\langle \scrA \scrU ,\scrU \rangle \scrH =
\int
G
(\nabla u\nabla v + (\Delta u - a(x)v)v dx+
\int
\Gamma 1
uvd\Gamma =
=
\int
G
\nabla u\nabla v dx+
\int
G
v\Delta udx -
\int
G
a(x)v2dx+
\int
\Gamma 1
uvd\Gamma .
By Green formula, we obtain
\langle \scrA \scrU ,\scrU \rangle \scrH = -
\int
G
a(x)v2dx \leq 0.
Hence \scrA is dissipative.
Now, we show that R(I - \scrA ) = \scrH . For any f = (f1, f2) \in H1
\Gamma 0
(G)\times L2(G), we consider the
following problem:
u - v = f1, x \in G, t > 0,
v - \Delta u+ a(x)v = f2, x \in G, t > 0,
u(x, t) = 0, x \in \Gamma 0, t > 0,
\partial u
\partial \nu
+ u = 0, x \in \Gamma 1, t > 0.
From the first and the second equation of the above system, we get - \Delta u+(1+a)u = f2+(1+a)f1.
We associate this problem with the following bilinear form on H1
\Gamma 0
(G)\times H1
\Gamma 0
(G) :
B(u, v) =
\int
G
\nabla u\nabla v dx+
\int
G
(1 + a)uv dx+
\int
\Gamma 1
uv dx.
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EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1251
By the Hölder inequality, there exists C > 0 such that
| B(u, v)| \leq C\| u\| H1
\Gamma 0
(G)\| v\| H1
\Gamma 0
(G) and | B(u, u)| \geq \| u\| 2H1
\Gamma 0
(G).
Then, by the Lax – Milgram theorem, there is a unique solution u \in H1
\Gamma 0
(G) such that\int
G
\nabla u\nabla v dx+
\int
G
(1 + a)uv dx+
\int
\Gamma 1
uv dx =
\int
G
(f2 + (1 + a)f1)v dx \forall v \in H1
\Gamma 0
(G).
Now, the \phi \in C\infty
0 (G), then
\int
\Gamma 1
u\phi dx = 0.
Therefore,\int
G
\nabla u\nabla \phi dx+
\int
\Omega
(1 + a)u\phi dx =
\int
G
(f2 + (1 + a)f1)\phi dx \forall \phi \in C\infty
0 (G).
Then
-
\int
G
\phi \Delta udx+
\int
G
(1 + a)u\phi dx =
\int
G
(f2 + (1 + a)f1)\phi dx \forall \phi \in C\infty
0 (G).
Hence - \Delta u+ (1 + a)u = f2 + (1 + a)f1. Since f2 + (1 + a)f1 \in L2(G), then
- \Delta u+ (1 + a)u \in L2(G) =\Rightarrow \Delta u = (1 + a)u - f2 - (1 + a)f1 \in L2(G).
Hence u \in H1(G) and \Delta u \in L2(G). Using the fact that G is smooth, thus by regularity theorem,
u \in H2(G). So, u \in H2(G) \cap H1
\Gamma 0
(G).
Now, we want to show that
\partial u
\partial \nu
+ u = 0 on \Gamma 1. (4.3)
Indeed, for every v \in H1
\Gamma 0
(G), we have\int
G
\nabla u\nabla v dx+
\int
G
(1 + a)uv dx+
\int
\Gamma 1
uv dx =
\int
G
(f2 + (1 + a)f1)v dx.
Since u \in H2(G), we can apply Green’s formula and we get
-
\int
G
\Delta uv dx+
\int
\Gamma
\partial u
\partial \nu
v dx+
\int
G
(1 + a)uv dx+
\int
\Gamma 1
uv dx =
=
\int
G
(f2 + (1 + a)f1)v dx -
\int
G
(\Delta uv + (1 + a)uv) dx+
\int
\Gamma 1
\biggl(
\partial u
\partial \nu
+ u
\biggr)
v dx =
=
\int
G
(f2 + (1 + a)f1)v dx.
Therefore,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1252 F. SAHRAOUI, A. OUAHAB\int
\Gamma 1
\biggl(
\partial u
\partial \nu
+ u
\biggr)
v dx = 0 for all v \in H1
\Gamma 0
(G).
Thus, the condition (4.3) holds. Taking v = u - f1, we find that \scrU = (u, v) \in D(\scrA ) is the solution
of the equation (I +\scrA )\scrU = f. Consequently, \scrA is m-dissipative.
Proposition 4.1 is proved.
As an application of Theorem 3.1, we consider the following nonlinear wave equation with finite
delay:
\partial ttu - \Delta u+ a(x)\partial tu(x, t) =
t\int
0
g(t - s)
\bigl[
\Delta u(\cdot , s) + a(x)\partial su(\cdot , s)
\bigr]
ds+
+ F
\left( t, ut, t\int
0
\rho (t, s, us)ds
\right) , (x, t) \in G\times (0,\infty ),
(4.4)
u(x, t) = 0, (x, t) \in \Gamma 0 \times (0,\infty ),
\partial u
\partial \nu
+ u = 0, (x, t) \in \Gamma 1 \times (0,\infty ),
u(x, t) = u0(x, t), ut(x, t) = u1(x, t), (x, t) \in G\times [ - r, 0],
where F : \BbbR + \times \scrC (J0, L2(G)) \times L2(G) is a Carathéodory function and g \in C1
b (\BbbR +,\BbbR ) = \{ f \in
\in \scrC (\BbbR +,\BbbR ) : \| f \prime \| \infty <\infty \} . We introduce the following hypotheses:
(\scrH 1) There exists f1 \in L2(\BbbR +) such that
\| F (t, \phi , u) - F (t, \psi , v)\| L2(G) \leq f1(t)(\| \phi - \psi \| \infty + \| u - v\| L2(G))
\forall \phi , \psi \in \scrC (J0, L2(G)), u, v \in L2(G).
(\scrH 2) The function g1 : t \in \BbbR + \rightarrow g1(t) = \| F (t, 0, 0)\| \in \BbbR + belongs to L2(\BbbR +).
(\scrH 3) The function \rho : \Delta \times \scrC (J0, L2(G)) \rightarrow L2(G) satisfies the following conditions:
(i) There exists f2 \in L2(\BbbR +) such that
\| \rho (t, s, \phi ) - \rho (t, s, \psi )\| \leq f2(s)\| \phi - \psi \| \infty \forall \phi , \psi \in \scrC (J0, L2(G)) and (t, s) \in \Delta .
(ii) There exists a constant N \geq 0, such that
s\int
0
\| \rho (s, r, 0)\| dr \leq N \forall s \in \BbbR +.
We transform the system (4.4) into the following form:
\scrU \prime (t) -
t\int
0
B(t - s)\scrA \scrU (s)ds = \scrA \scrU (t) + F\ast (t,\scrU t), t > 0,
(4.5)
\scrU (t) = \scrU 0(t), t \in [ - r, 0],
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
EXISTENCE AND COMPACTNESS OF SOLUTION OF SEMILINEAR INTEGRO-DIFFERENTIAL . . . 1253
where F\ast : \BbbR + \times \scrC (J0,\scrH ) \rightarrow \scrH be a function given by
F\ast (t,\scrU t) =
\left( 0
F
\biggl(
t, ut,
\int t
0
\rho (t, s, us)ds
\biggr) \right) .
Let B(t) : D(\scrA ) \subset \scrH \rightarrow \scrH be the operator defined by
B(t)\scrU = g(t)\scrA \scrU , \scrU \in D(\scrA ).
Theorem 4.1. Assume that the conditions (\scrH 1) – (\scrH 3) are satisfied. Then the problem (4.5) has
a unique solution.
Proof. From Proposition 4.1, the operator \scrA is m-dissipative, then by the Lumer – Phillips
theorem [19, 20], \scrA generates a strongly continuous semigroup of contractions on \scrH and closed.
This implies that B(t) is a closed operator. For every \Phi = (\phi , \psi ), \Phi = (\phi , \psi ) \in \scrC (J0,\scrH ), we get
\bigm\| \bigm\| F\ast (t,\Phi ) - F\ast
\bigl(
t,\Phi
\bigr) \bigm\| \bigm\| 2
\scrH =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\Bigl(
t, \phi ,
t\int
0
\rho (t, s, \phi )ds
\Bigr)
- F
\Bigl(
t, \phi ,
t\int
0
\rho
\bigl(
t, s, \phi
\bigr)
ds
\Bigr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
L2(G)
\leq
\leq f21 (t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
t\int
0
\rho (t, s, \phi )ds -
t\int
0
\rho
\bigl(
t, s, \phi
\bigr)
ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
2
L2(G)
+ f21 (t)
\bigm\| \bigm\| \phi - \phi
\bigm\| \bigm\| 2
\infty \leq
\leq f21 (t)\| g\| 2L2
\bigm\| \bigm\| \phi - \phi
\bigm\| \bigm\| 2
\infty + f21 (t)
\bigm\| \bigm\| \phi - \phi
\bigm\| \bigm\| 2
\infty .
Then \bigm\| \bigm\| F\ast (t,\Phi ) - F\ast
\bigl(
t,\Phi
\bigr) \bigm\| \bigm\|
\scrH \leq f1(t)(\| g\| 2L2 + 1)
1
2
\bigm\| \bigm\| \phi - \phi
\bigm\| \bigm\|
\infty .
Hence \bigm\| \bigm\| F\ast (t,\Phi ) - F\ast
\bigl(
t,\Phi
\bigr) \bigm\| \bigm\| 2
\scrH \leq f1(t)(\| g\| L2 + 1)
1
2
\bigm\| \bigm\| \Phi - \Phi
\bigm\| \bigm\|
\infty for all \Phi ,\Phi \in \scrC (J0,\scrH ),
where \bigm\| \bigm\| \Phi - \Phi
\bigm\| \bigm\|
\infty = \mathrm{s}\mathrm{u}\mathrm{p}
t\in [ - r,0]
\bigm\| \bigm\| \Phi (\cdot , t) - \Phi (\cdot , t)
\bigm\| \bigm\|
\scrH .
Applying Theorem 3.1, we conclude that the problem (4.5) has a unique mild solution.
Theorem 4.1 is proved.
In the end of this example, we give some functions satisfying the conditions of Theorem 4.1:
Let h \in Cb(\BbbR +,\BbbR ), we define F1, F2 : \BbbR + \times \scrC (\BbbR +, L
2(G)) \rightarrow L2(G)H by
F1(t, \phi ) = h(t) \mathrm{s}\mathrm{i}\mathrm{n}(\phi ( - r, x)) \forall x \in G
and
F2(t, \phi ) = h(t) \mathrm{s}\mathrm{i}\mathrm{n}(\phi ( - r, x)) +
t\int
0
h(t - s) \mathrm{c}\mathrm{o}\mathrm{s}(s+ \phi ( - r, x))ds \forall x \in G.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1254 F. SAHRAOUI, A. OUAHAB
4.2. Semilinear parabolic problem. As an application of the Theorem 3.2, we consider the
following problem:
\partial tu(x, t) + \BbbA \ast (x,D)u(x, t) = -
t\int
0
g(t - s)\BbbA \ast (x,D)u(x, s)ds+
+ F
\left( t, ut, t\int
0
\rho (t, s, us)ds
\right) , (x, t) \in G\times (0,\infty ),
D\nu u(x, t) = 0, (x, t) \in \Gamma \times (0,\infty ), | \nu | \leq m, (4.6)
u(x, t) = u0(x, t), (x, t) \in G\times [ - r, 0],
where G \subset \BbbR d is a bounded domain with a smooth boundary \partial G = \Gamma , \BbbA \ast (x,D) =
=
\sum
| \nu | \leq 2m
a\nu (x)D
\nu u is a strong elliptic operator with coefficients a\nu \in C2m(G), F : \BbbR + \times
\times \scrC ([ - r, 0], L2(G))\times L2(G) \rightarrow L2(G), is a given function, \rho : \Delta \times \scrC ([ - r, 0], L2(G)) \rightarrow L2(G) is
a continuous function and g \in C1
b (\BbbR +, \BbbR ). We define the operator A : D(A) \subset L2(G) \rightarrow L2(G) by
Au = - \BbbA \ast (\cdot , D)u \forall u \in D(A) = H2m(G) \cap Hm
0 (G).
In [19] (Theorem 7.3.7) we have the following theorem.
Theorem 4.2. Under the assumption that \BbbA \ast is a strong elliptic operator with smooth coef-
ficients, then the operator A generates an analytic semigroup on L2. Moreover, the semigroups
(S(t))t\geq 0 associated to A is equicontinuous.
For every t \in \BbbR +, we define u(t) = u(\cdot , t). Hence the problem (4.6) can be rewritten as follows:
u\prime (t) - Au =
t\int
0
g(t - s)Au(s)ds+ F
\left( t, ut, t\int
0
\rho (t, s, us)ds
\right) , t \in \BbbR +,
u(t) = u0(t), t \in [ - r, 0].
(4.7)
If we assume that F and \rho satisfied the conditions (\scrH 4) – (\scrH 6) of the Theorem 3.2, the problem (4.7)
has at least one mild solution.
For examples if F is compact and \rho (t, s, \cdot ) is Lipschitz function, then (\scrH 4) – (\scrH 6) hold.
For examples if \rho is compact and F (t, \cdot , \cdot ) is Lipschitz function, then (\scrH 4) – (\scrH 6) hold.
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|
| id | umjimathkievua-article-7106 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:28Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f5/9ce29f2b2ccad1b229c4fb9ef5cab1f5.pdf |
| spelling | umjimathkievua-article-71062023-01-07T13:45:36Z Existence and compactness of solution of semilinear integro-differential equations with finite delay Existence and compactness of solution of semilinear integro-differential equations with finite delay Sahraoui, F. Ouahab, А. Sahraoui, F. Ouahab, А. Mild solutions, resolvent operator, Banach fixed point, measure of non-compactness, attractivity. UDC 517.9 We present some existence and uniqueness results for a class of functional integro-differential evolution equations&nbsp; generated by the resolvent operator for which the semigroup is not necessarily compact.&nbsp;It&nbsp; is proved that the set of solutions is compact.&nbsp;&nbsp;Our approach is based on fixed point theory.&nbsp;&nbsp;Finally, some examples are given to illustrate the results. УДК 517.9 Існування та компактність розв'язку напівлінійних інтегро-диференціальних рівнянь зі скінченним запізненням Наведено деякі результати&nbsp; щодо існування та єдиності розв'язків&nbsp; деякого класу функціональних інтегро-диференціальних еволюційних рівнянь, породжених резольвентним оператором, де напівгрупа необов'язково компактна.&nbsp;&nbsp;Доведено компактність множини розв'язків.&nbsp;&nbsp;Наш підхід ґрунтується на теорії нерухомих точок.&nbsp;&nbsp;Крім того, наведено кілька прикладів, що ілюструють отримані результати. Institute of Mathematics, NAS of Ukraine 2022-11-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7106 10.37863/umzh.v74i9.7106 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 9 (2022); 1231 - 1255 Український математичний журнал; Том 74 № 9 (2022); 1231 - 1255 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7106/9299 Copyright (c) 2022 Fatiha Sahraoui, Abdelghani OUAHAB |
| spellingShingle | Sahraoui, F. Ouahab, А. Sahraoui, F. Ouahab, А. Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_alt | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_full | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_fullStr | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_full_unstemmed | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_short | Existence and compactness of solution of semilinear integro-differential equations with finite delay |
| title_sort | existence and compactness of solution of semilinear integro-differential equations with finite delay |
| topic_facet | Mild solutions resolvent operator Banach fixed point measure of non-compactness attractivity. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7106 |
| work_keys_str_mv | AT sahraouif existenceandcompactnessofsolutionofsemilinearintegrodifferentialequationswithfinitedelay AT ouahaba existenceandcompactnessofsolutionofsemilinearintegrodifferentialequationswithfinitedelay AT sahraouif existenceandcompactnessofsolutionofsemilinearintegrodifferentialequationswithfinitedelay AT ouahaba existenceandcompactnessofsolutionofsemilinearintegrodifferentialequationswithfinitedelay |