Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces
In this paper, a recent nonlinear alternative for contraction maps in Frechet spaces, due to Frigon and ´ Granas, is combined with semigroups theory and used to investigate the controllability of some classes of semilinear functional and neutral functional differential equations in Banach spaces o...
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| Cite this: | Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces / A. Arara, M. Benchohra, A. Ouahab // Нелінійні коливання. — 2003. — Т. 6, № 3. — С. 291-308. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1769402025-02-09T11:38:26Z Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces Деякі результати про єдиність керованості для функціональних напівлінійних диференціальних рівнянь у просторах Фреше Некоторые результаты о единственности управляемости для функциональных полулинейных дифференциальных уравнений в пространствах Фреше Arara, A. Benchohra, M. Ouahab, A. In this paper, a recent nonlinear alternative for contraction maps in Frechet spaces, due to Frigon and ´ Granas, is combined with semigroups theory and used to investigate the controllability of some classes of semilinear functional and neutral functional differential equations in Banach spaces on the semiinfinite interval. Дослiджується керованiсть для деяких класiв напiвлiнiйних функцiональних i нейтральних функцiонально-диференцiальних рiвнянь у банахових просторах на напiвобмеженому iнтервалi за допомогою теорiї напiвгруп, а також нещодавно отриманої Фрiгоном i Гранасом нелiнiйної альтернативи для стискаючих вiдображень у просторах Фреше. 2003 Article Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces / A. Arara, M. Benchohra, A. Ouahab // Нелінійні коливання. — 2003. — Т. 6, № 3. — С. 291-308. — Бібліогр.: 10 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/176940 517.9 en Нелінійні коливання application/pdf Інститут математики НАН України |
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In this paper, a recent nonlinear alternative for contraction maps in Frechet spaces, due to Frigon and ´
Granas, is combined with semigroups theory and used to investigate the controllability of some classes of
semilinear functional and neutral functional differential equations in Banach spaces on the semiinfinite
interval. |
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Article |
| author |
Arara, A. Benchohra, M. Ouahab, A. |
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Arara, A. Benchohra, M. Ouahab, A. Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces Нелінійні коливання |
| author_facet |
Arara, A. Benchohra, M. Ouahab, A. |
| author_sort |
Arara, A. |
| title |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces |
| title_short |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces |
| title_full |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces |
| title_fullStr |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces |
| title_full_unstemmed |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces |
| title_sort |
some uniqueness result on controllability for functional semilinear differential equations in fréchet spaces |
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Інститут математики НАН України |
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2003 |
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https://nasplib.isofts.kiev.ua/handle/123456789/176940 |
| citation_txt |
Some uniqueness result on controllability for functional semilinear differential equations in Fréchet spaces / A. Arara, M. Benchohra, A. Ouahab // Нелінійні коливання. — 2003. — Т. 6, № 3. — С. 291-308. — Бібліогр.: 10 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
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2025-11-25T22:14:57Z |
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2025-11-25T22:14:57Z |
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| fulltext |
UDC 517.9
SOME UNIQUENESS RESULTS ON CONTROLLABILITY
FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL EQUATIONS
IN FRÉCHET SPACES
ДЕЯКI РЕЗУЛЬТАТИ ПРО ЄДИНIСТЬ КЕРОВАНОСТI
ДЛЯ ФУНКЦIОНАЛЬНИХ НАПIВЛIНIЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ У ПРОСТОРАХ ФРЕШЕ
A. Arara, M. Benchohra, and A. Ouahab
Laboratoire de Mathématiques, Université de Sidi Bel Abbès
BP 89, 22000 Sidi Bel Abbès, Algérie
e-mail: benchohra@univ-sba.dz
In this paper, a recent nonlinear alternative for contraction maps in Fréchet spaces, due to Frigon and
Granas, is combined with semigroups theory and used to investigate the controllability of some classes of
semilinear functional and neutral functional differential equations in Banach spaces on the semiinfinite
interval.
Дослiджується керованiсть для деяких класiв напiвлiнiйних функцiональних i нейтральних функ-
цiонально-диференцiальних рiвнянь у банахових просторах на напiвобмеженому iнтервалi за до-
помогою теорiї напiвгруп, а також нещодавно отриманої Фрiгоном i Гранасом нелiнiйної аль-
тернативи для стискаючих вiдображень у просторах Фреше.
1. Introduction. This paper is concerned with an application of a recent nonlinear alternative for
contraction maps in Fréchet spaces due to Frigon and Granas [1] to the controllability of some
classes of initial value problems for first and second order semilinear functional and neutral
functional differential equations in Fréchet spaces. In Section 3, we will consider the first order
semilinear functional differential equations
y′(t)−Ay(t) = f(t, yt) + (Bu)(t) a. e. t ∈ [0,∞), (1)
y(t) = φ(t), t ∈ [−r, 0], (2)
where r > 0, f : J × C([−r, 0], E) → E is a given function and φ ∈ C([−r, 0], E). Also
the control function u(·) ∈ L2(J, U), a Banach space of admissible control functions with U
as a Banach space, B is a bounded linear operator from U to E. Finally A is a densely defined
operator generating a semigroup {T (t)}, t ≥ 0, of bounded linear operators fromE intoE and
E is real Banach space with norm | · |. For any continuous function y defined on [−r,∞) and any
t ∈ [0,∞), we denote by yt the element ofC([−r, 0], E) defined by yt(θ) = y(t+θ), θ ∈ [−r, 0].
Here yt(·) represents the history of the state from time t−r, up to the present time t. In Section
4, we study the second order semilinear functional differential equations of the form
y′′(t)−Ay(t) = f(t, yt) + (Bu)(t) a. e. t ∈ [0,∞), (3)
c© A. Arara, M. Benchohra, and A. Ouahab, 2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 291
292 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η, (4)
where f and φ are as in problem (1), (2), η ∈ E, and A is a densely defined operator generating
a family of cosinus operators {C(t)}, t ≥ 0.
Sections 5 and 6 are concerned with the existence of solutions, of initial value problems
for first and second order semilinear neutral functional differential equations. In Section 5 we
consider the first order semilinear neutral functional differential equations of the form
d
dt
[y(t)− g(t, yt)] = Ay(t) + f(t, yt) + (Bu)(t) a. e. t ∈ [0,∞), (5)
y(t) = φ(t), t ∈ [−r, 0], (6)
where f, A, and φ are as in problem (1), (2) and g : J × C([−r, 0], E) → E. In Section 6 we
study the following second order problem
d
dt
[y′(t)− g(t, yt)] = Ay(t) + f(t, yt) + (Bu)(t) a. e. t ∈ [0,∞), (7)
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η, (8)
where f,A, η, and φ are as in problem (3), (4) and g is as in problem (5), (6). The last section
will be devoted to an example illustrating the abstract theory.
Recently the fixed point argument such as the Banach contraction principle and Schaefer’s
fixed point theorem were applied to the controllability, on compact intervals, of some classes
of semilinear differential, integrodifferential, and functional differential equations in Banach
spaces in the literature. We mention here the survey paper by Balachandran and Dauer [2] and
the references cited therein. In [3] the controllability of a class of first order evolution equations
on semiinfinite time horizon was studied by means of an application of Schauder – Tikhonov’s
fixed point theorem and the semigroup theory.
Our goal here is to give uniqueness results for the above problems. These results can be
considered as a contribution to the literature.
2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts
which are used throughout this paper.
C([−r, 0], E) is the Banach space of all continuous functions from [−r, 0] into E with the
norm
‖φ‖ := sup{|φ(θ)| : −r ≤ θ ≤ 0}.
B(E) is the Banach space of all linear bounded operator from E into E with norm
‖N‖B(E) := sup{|N(y)| : |y| = 1}.
A measurable function y : [0,∞) → E is Bochner integrable if and only if |y| is Lebesgue
integrable. (For properties of the Bochner integral, see for instance, Yosida [4].)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 293
L1([0,∞), E) denotes the Banach space of functions y : [0,∞) −→ E which are Bochner
integrable normed by
‖y‖L1 =
∞∫
0
|y(t)|dt.
We say that a family {C(t) : t ∈ R} of operators in B(E) is a strongly continuous cosine
family if:
1) C(0) = I (I is the identity operator in E);
2) C(t+ s) + C(t− s) = 2C(t)C(s) for all s, t ∈ R;
3) the map t 7−→ C(t)y is strongly continuous for each y ∈ E.
The strongly continuous sine family {S(t) : t ∈ R}, associated to the given strongly conti-
nuous cosine family {C(t) : t ∈ R}, is defined by
S(t)y =
t∫
0
C(s)yds, y ∈ E, t ∈ R.
The infinitesimal generator A : D(A) ⊆ E −→ E of a cosine family {C(t) : t ∈ R} is defined
by
Ay =
d2
dt2
C(t)y
∣∣∣
t=0
.
For more details on strongly continuous cosine and sine families, we refer the reader to the
books of Fattorini [5] and Goldstein [6] and to the papers of Travis and Webb [7, 8]. For properti-
es of semigroup theory, we refer the interested reader to the books of Engel and Nagel [9] and
Pazy [10].
For more details on the following notions we refer to [4]. Let X be a Fréchet space with a
family of seminorms {‖ · ‖n), n ∈ N}. Let Y ⊂ X, we say that Y is bounded if for every n ∈ N,
there exists Mn > 0 such that
‖y‖n ≤ Mn for all y ∈ Y.
To X , we associate a sequence of Banach spaces {(Xn, ‖ · ‖n)} as follows. For every n ∈ N,
we consider the equivalence relation ∼n defined by ∼n if and only if ‖x− y‖n = 0. We denote
Xn = (X/ ∼n, ‖ · ‖) the quotient space, the completion of Xn with respect to ‖ · ‖. To every
Y ⊂ X, we associate a sequence {Y n} of subsets Y n ⊂ Xn as follows. For every x ∈ X, we
denote [x]n the equivalence class of x of subset Xn and define Y n = {[x]n : x ∈ Y }. We
denote Y
n
, intn(Y n), and ∂nY n, respectively, the closure, the interior and the boundary of Y n
with respect to ‖ · ‖ in Xn. We assume that the family of seminorms {‖ · ‖n} verifies
‖x‖1 ≤ ‖x‖2 ≤ ‖x‖3 ≤ . . . for every x ∈ X.
Definition 2.1 A function f : X → X is said to be a contraction if for each n ∈ N there exists
kn ∈ (0, 1) such that
‖f(x)− f(y)‖n ≤ kn‖x− y‖n for all x, y ∈ X.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
294 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
Theorem 2.1 (Nonlinear Alternative, [1]). Let X be a Fréchet space and Y ⊂ X a closed
subset in Y let N : Y → X be a contraction such that N(Y ) is bounded. Then one of the
following statements holds:
C1) N has a unique fixed point;
C2) there exists λ ∈ [0, 1), n ∈ N, and x ∈ ∂nY n such that ‖x− λN(x)‖n = 0.
In what follows, we will assume that the function f : [0,∞) × C([−r, 0], E) → E is an L1-
Carathéodory function, i. e.,
i) t 7−→ f(t, u) is measurable for each u ∈ C([−r, 0], E);
ii) u 7−→ f(t, u) is continuous for almost all t ∈ [0,∞);
iii) for each q > 0, there exists hq ∈ L1
loc([0,∞),R+) such that
|f(t, u)| ≤ hq(t) for all ‖u‖ ≤ q and for almost all t ∈ [0,∞).
3. Controllability for first order semilinear FDEs. The main result of this section concerns
the IVP (1), (2). Before stating and proving it, we first give a definition of a mild solution of the
IVP.
Definition 3.1. A function y ∈ C([−r,∞), E) is said a mild solution of (1), (2) if y(t) =
= φ(t), t ∈ [−r, 0], and for each t ∈ [0,∞),
y(t) = T (t)φ(0) +
0∫
t
T (t− s)f(s, ys)ds+
0∫
t
T (t− s)(Bu)(s)ds.
Definition 3.2. The system (1), (2) is said to be infinite controllable if for any continuous
function φ on [−r, 0] and any x1 ∈ E and for each n ∈ N there exists a control u ∈ L2([0, n], U)
such that the mild solution y(·) of (1) satisfies y(n) = x1.
Let us introduce the following hypotheses which are assumed hereafter:
H1) there exists a continuous nondecreasing function ψ : [0,∞) −→ (0,∞) and p ∈
∈ L1
loc([0,∞),R+) such that
|f(t, u)| ≤ p(t)ψ(‖u‖) for a. e. t ∈ [0,∞) and each u ∈ C([−r, 0], E)
with
∞∫
1
ds
ψ(s)
= ∞;
H2) for all R > 0 there exists lR ∈ L1
loc([−r,∞),R+) such that
|f(t, u)− f(t, u)| ≤ lR(t)‖u− u‖ for all u, u ∈ C([−r, 0], E) with ‖u‖, ‖u‖ ≤ R;
H3) there exists M ≥ 1 such that
|T (t)‖B(E) ≤ M for each t ≥ 0;
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 295
H4) for every n > 0 the linear operator W : L2(Jn, U) → E (Jn = [0, n]), defined by
Wu =
n∫
0
T (n− s)Bu(s)ds,
has an inverse operator W−1 which takes values in L2(Jn, U)\KerW and there exist positive
constants M, M1 such that ‖B‖ ≤ M and ‖W−1‖ ≤ M1.
For each n ∈ N we define in C([−r,∞), E) the seminorms by
‖y‖n = sup{e−τL∗n(t)|y(t)| : t ≤ n},
where L∗n(t) =
t∫
0
l̄n(s)ds, l̄n(t) = max(Mln(t), nMM2M1‖ln‖L1([0,n])), and ln is the function
from H2). Then C([−r,∞), E) is a Fréchet space with the family of seminorms {‖ · ‖n}. In what
follows we will choose τ sufficiently large.
Theorem 3.1. Suppose that hypotheses H1) – H4) are satisfied. Then problem (1), (2) has a
unique solution.
Proof. Using hypothesis H4) for each y(·) and each n ∈ N define the control
uny (t) = W−1
x1 − T (n)φ(0)−
n∫
0
T (n− s)f(s, ys)ds
(t).
Transform the problem (1), (2) into a fixed point problem. Consider the operatorN : C([−r,∞),
E) → C([−r,∞), E) defined by
N(y)(t) =
φ(t), if t ∈ [−r, 0];
T (t)φ(0) +
t∫
0
T (t− s)(Buny )(s)ds+
t∫
0
T (t− s)f(s, ys)ds, if t ∈ [0,∞).
Clearly, the fixed points of the operator N are solutions of the problem (1), (2). Let y be a
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
296 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
possible solution of the problem (1), (2). Given n ∈ N and t ≤ n, then
|y(t)| ≤ |T (t)||φ(0)|+
t∫
0
|T (t− s)||(Buny )(s)|ds+
t∫
0
|T (t− s)||f(s, ys)|ds ≤
≤ M |φ(0)|+M
t∫
0
‖B‖|uny (s)|ds+M
t∫
0
p(s)ψ(‖ys‖)ds ≤
≤ M‖φ‖+ nMMM1
‖x1‖+M‖φ‖+ nM
t∫
0
p(s)ψ(‖ys‖)ds
+
+ M
t∫
0
p(s)ψ(‖ys‖)ds ≤
≤ M‖φ‖+ nMMM1[‖y1‖+M‖φ‖] +
+ max{n2MM2M1,M}
t∫
0
p(s)ψ(‖ys‖)ds.
We consider the function µ defined by
µ(t) = sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ n.
Let t∗ ∈ [−r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ [0, n], by the previous inequality we have,
for t ∈ [0, n],
µ(t) ≤ M‖φ‖+ nMMM1[‖x1‖+M‖φ‖] + max{n2MM2M1,M}
t∫
0
p(s)ψ(µ(s))ds.
If t∗ ∈ [−r, 0], then µ(t) = ‖φ‖ and the previous inequality holds. Let us take the right-hand
side of the above inequality as v(t). Then we have
c = v(0) = M‖φ‖+ nMMM1[‖x1‖+M‖φ‖], µ(t) ≤ v(t), t ∈ [0, n],
and
v′(t) = max{n2MM2M1,M}p(t)ψ(µ(t)) a. e. t ∈ [0, n].
Using the nondecreasing character of ψ we get
v′(t) ≤ max{n2MM2M1,M}p(t)ψ(v(t)) a. e. t ∈ [0, n].
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 297
This implies that for each t ∈ [0, n]
v(t)∫
v(0)
ds
ψ(s)
≤ max{n2MM2M1,M}
t∫
0
p(s)ds < ∞.
Thus from H1) there exists a constant Kn such that v(t) ≤ Kn, t ∈ [0, n], and hence µ(t) ≤
≤ Kn, t ∈ [0, n]. Since for every t ∈ [0, n], ‖yt‖ ≤ µ(t), we have
‖y‖n ≤ max{‖φ‖,Kn} := Mn.
Set
Y = {y ∈ C([−r,∞), E) : sup{|y(t)| : t ≤ n} ≤ Mn + 1 for all n ∈ N}.
Clearly, Y is a closed subset of C([−r,∞), E). We shall show that N : Y → C([−r,∞), E)
is a contraction operator. Indeed, consider y, y ∈ C([−r,∞), E), thus for each t ∈ [0, n] and
n ∈ N,
|N(y)(t)−N(y)(t)| =
∣∣∣∣∣
t∫
0
T (t− s)[(Buny )(s)− (Buny )(s)]ds+
+
t∫
0
T (t− s)[f(s, ys)− f(s, ys)]ds
∣∣∣∣∣ ≤
≤M
t∫
0
‖B‖|(uny )(s)− (uny )(s)|ds+
+
t∫
0
ln(s)eτL(s)e−τL(s)M‖yt − yt‖dt ≤
≤M
t∫
0
l̄n(s)eτL
∗
n(s)dt‖y − y‖n +
+ MM
t∫
0
|W−1
[
y1 − T (n)φ(0)−
n∫
0
T (n− s)f(ω, y(ω))dω
]
ds−
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3
298 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
− W−1
[
y1 − T (n)φ(0)−
n∫
0
T (n− s)f(ω, y(ω))dω
]
ds ≤
≤MMM1
t∫
0
M
n∫
0
|f(ω, y(ω))− f(ω, y(ω))|dωds+
+
M
τ
t∫
0
(eτL
∗
n(s))′dt‖y − y‖n ≤
≤ 1
τ
eτL
∗
n(t)‖y − y‖n +
1
τ
eτL
∗(t)‖y − y‖n.
Therefore,
‖N(y)−N(y)‖n ≤
2
τ
‖y − y‖n,
showing that for τ sufficiently large, the operator N is a contraction for all n ∈ N. From the
choice of Y there is no y ∈ ∂Y n such that y = λN(y) for some λ ∈ (0, 1). As a consequence of
the nonlinear alternative type [1], we deduce that N1 has a unique fixed point which is a mild
solution to (1), (2).
4. Controllability for second order semilinear FDEs. In this section we give a uniqueness
result for the IVP (3), (4).
Definition 4.1. A function y ∈ C([−r,∞), E) is said to be a mild solution of (3), (4) if
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η, and
y(t) = C(t)φ(0) + S(t)η +
0∫
t
S(t− s)(Bu)(s)ds+
0∫
t
S(t− s)f(s, ys)ds.
Definition 4.2. The system (3), (4) is said to be infinite controllable if for any continuous
function φ on [−r, 0] and any x1 ∈ E and for each n ∈ N there exists a control u ∈ L2([0, n], U)
such that the mild solution y(·) of (3) satisfies y(n) = x1.
Theorem 4.1. Assume H1), H2) and the condition:
H5) there exists a constant M1 ≥ 1 such that
‖C(t)‖B(E) ≤ M1 for all t ∈ R;
H6) for every n > 0 the linear operator W : L2(Jn, U) → E, defined by
Wu =
n∫
0
S(n− s)Bu(s)ds,
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SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 299
has an inverse W−1 which takes values in L2(Jn, U)\KerW and there exist positive constants
M
∗
, M
∗
1 such that ‖B‖ ≤ M
∗
and ‖W−1‖ ≤ M
∗
1 are satisfied. Then the IVP (3), (4) has a
unique mild solution.
Proof. Using hypothesis H6) for each y(·) and each n ∈ N define the control
uny (t) = W−1
x1 − C(n)φ(0)− S(n)η −
n∫
0
S(n− s)f(s, ys)ds
(t).
Consider the operator N1 : C([−r,∞), E) → C([−r,∞), E) defined by
N(y)(t) =
φ(t), if t ∈ [−r, 0];
C(t)φ(0) + S(t)η +
t∫
0
S(t− s)(Buny )ds+
+
t∫
0
S(t− s)f(s, ys)ds, if t ∈ [0,∞).
Clearly, the fixed points of the operator N2 are mild solutions of the problem (3), (4).
Let y be a possible solution of (3), (4). Thus for all t ≤ n, n ∈ N, we have
y(t) = C(t)φ(0) + S(t)η +
t∫
0
S(t− s)(Buny )ds+
t∫
0
S(t− s)f(s, ys)ds.
This implies, by H1) and H4, that for each t ∈ [0, n] we have
|y(t)| ≤M1‖φ‖+ nM1|η|+ nM1
t∫
0
‖B‖|uny (s)|ds+ nM1
t∫
0
p(s)ψ(‖ys‖)ds ≤
≤M1‖φ‖+ nM1|η|+ nM∗M1M∗1
[
‖x1‖+M1|φ(0)|+ nM1η +
+ nM1
t∫
0
p(s)ψ(‖ys‖)ds
]
+ nM1
t∫
0
p(s)ψ(‖ys‖)ds ≤
≤M1‖φ‖+ nM1|η|+ nM∗M1M∗1 [‖x1‖+M1|φ(0)|+ nM1η] +
+ max{n2M2
1M
∗
M
∗
1, nM1}
t∫
0
p(s)ψ(‖ys‖)ds.
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300 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
We consider the function µ defined by
µ(t) = sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ n.
Let t∗ ∈ [−r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ [0, n], by the previous inequality we have,
for t ∈ [0, n],
µ(t) ≤M1‖φ‖+ nM1|η|+ nM∗M1M∗1 [‖x1‖+M1|φ(0)|+ nM1η] +
+ max{n2M2
1M
∗
M
∗
1, nM1}
t∫
0
p(s)ψ(µ(s))ds.
If t∗ ∈ [−r, 0], then µ(t) = ‖φ‖ and the previous inequality holds. Let us take the right-hand
side of the above inequality as v(t). Then we have
c = v(0) = M1‖φ‖+nM1|η|+nM∗M1M∗1 [‖x1‖+M1|φ(0)|+nM1η], µ(t) ≤ v(t), t ∈ [0, n],
and
v′(t) = max{n2M2
1M
∗
M
∗
1, nM1}p(t)ψ(µ(t)) a. e. t ∈ [0, n].
Using the nondecreasing character of ψ we get
v′(t) ≤ max{n2M2
1M
∗
M
∗
1, nM1}p(t)ψ(v(t)) a. e. t ∈ [0, n].
This implies that for each t ∈ [0, n]
v(t)∫
v(0)
ds
ψ(s)
≤ max{n2M2
1M
∗
M
∗
1, nM1}
t∫
0
p(s)ds < ∞.
Thus from H1) there exists a constant Kn such that v(t) ≤ Kn, t ∈ [0, n], and hence µ(t) ≤
≤ Kn, t ∈ [0, n]. Since for every t ∈ [0, n], ‖yt‖ ≤ µ(t), we have
‖y‖n ≤ max{‖φ‖,Kn} := Mn.
Set
Y = {y ∈ C([−r,∞), E) : sup{|y(t)| : t ≤ n} ≤ Mn + 1 for all n ∈ N}.
Thus
‖y‖n ≤ max{‖φ‖,K∗n} := M∗n.
Let
Y = {y ∈ C([−r,∞), E) : sup{|y(t)| : t ≤ n} ≤ M∗n + 1 for all n ∈ N}.
For each n ∈ N, we define in C([0,∞), E) the seminorms by
‖y‖n = sup{e−τL̃n(t)|y(t)| : t ≤ n},
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SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 301
where L̃n(t) =
∫ t
0
l̃n(s)ds and l̃n(t) = max(nM1ln(t), nM∗M2
1M
∗
1 ‖ln‖L1([0,n])). As in Theorem
3.1 we can show that N1 : Y → C([−r,∞), E) defined by
N1(y)(t) =
φ(t), if t ∈ [−r, 0];
C(t)φ(0) + S(t)η +
t∫
0
S(t− s)(Bu)(s)ds+
+
t∫
0
S(t− s)f(s, ys)ds, if t ∈ [0,∞),
is a contraction operator. From the choice of Y there is no y ∈ ∂Y such that y = λN1(y) for
some λ ∈ (0, 1). As a consequence of the nonlinear alternative [1] we deduce that N2 has a
unique fixed point which is a mild solution to (3), (4).
5. Controllability for first order semilinear neutral FDEs. Let us start by defining what we
mean by a solution of IVP (5), (6).
Definition 5.1. A function y ∈ C([−r,∞), E) is said to be a mild solution of (5), (6) if
y(t) = φ(t), t ∈ [−r, 0], the restriction of y(·) to the interval [0,∞) is continuous, and for each
t ∈ [0,∞) the function AT (t− s)g(s, ys), s ∈ [0, t), and
y(t) = T (t)[φ(0)− g(0, φ)] + g(t, yt) +
t∫
0
AT (t− s)g(s, ys)ds+
+
t∫
0
T (t− s)(Bu)(s)ds+
∫ t
0
T (t− s)f(s, ys)ds.
Let us introduce the following hypotheses which are assumed hereafter:
A1) for each R > 0, there exists a function l̃R ∈ L1
loc(J,E) ∩ C(J,E) such that |g(t, u) −
−g(t, u)| ≤ l̃R(t)‖u− u‖, t ∈ [0,∞), u, u ∈ C([−r, 0], E), with ‖u‖, ‖u‖ ≤ R;
A2) there exists a constant L > 0 such that
|g(t, u)| ≤ L for each t ∈ [0,∞), u ∈ C([−r, 0], E);
A3) A is the infinitesimal generator of the semigroup {T (t) : t ≥ 0} such that
‖AT (t)‖B(E) ≤ M2 for some M2 > 0;
A4) for every n > 0 the linear operator W : L2(Jn, U) → E, defined by
Wu =
n∫
0
T (n− s)Bu(s)ds,
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302 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
has an inverse W−1 which takes values in L2(Jn, U)\KerW and there exist positive constants
M, M1 such that ‖B‖ ≤ M and ‖W−1‖ ≤ M1.
Let Ln(t) =
∫ t
0
l̂n(s)ds, where
l̂n(t) = max(l̃n(t),M2 l̃n(t), nM1MMM2‖l̃n‖, nM1M
2MM2‖ln‖).
For each n ∈ N we define in C([−r,∞), E) the seminorms by
‖y‖n = sup{e−τLn(t)|y(t)| : t ≤ n}.
Then C([−r,∞), E) is a Fréchet space with a family of seminorms {‖ · ‖n}.
Theorem 5.1. Assume that hypotheses H1), H2) and A1) – A4) hold. If for each n ∈ N we
have (
4
τ
+ sup
t∈[0,n]
l̃n(t)
)
< 1,
then the problem (5), (6) has a unique mild solution.
Proof. Using hypothesis A4) for each y(·) and each n ∈ N define the control
uny (t) = W−1
[
x1 − T (n)[φ(0)− g(0, φ)] + g(n, yn) +
+
n∫
0
AT (n− s)g(s, ys)ds+
n∫
0
AT (n− s)f(s, ys)ds
]
(t).
Transform the problem (5), (6) into a fixed point problem. Consider the operator N2 :
C([−r,∞), E) → C([−r,∞), E) defined by
N3(y)(t) :=
φ(t), if t ∈ [−r, 0];
T (t)[φ− g(0, φ)] + g(t, yt) +
t∫
0
A(t− s)g(s, ys) ds+
+
t∫
0
T (t− s)(Buny )(s) ds+
+
t∫
0
T (t− s)f(s, ys) ds, if t ∈ [0,∞).
Remark 5.1. It is clear that the fixed points of N2 are mild solutions to (5), (6).
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SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 303
Let y be a possible solution of the problem (5), (6). Given n ∈ N and t ≤ n, we have
y(t) = T (t)[φ(0)− g(0, φ(0))] + g(t, yt) +
t∫
0
AT (t− s)g(s, ys)ds+
+
t∫
0
T (t− s)(Buny )(s) ds+
t∫
0
T (t− s)f(s, ys) ds.
This implies by H3), A2), A3), and A4) that for each t ∈ [0, n] we have
|y(t)| ≤M |φ(0)|+M |g(0, φ(0))|+ |g(t, yt)|+
+ M
t∫
0
‖B‖|uny (s)|ds+ nM2L+M
0∫
t
p(s)ψ(‖ys‖)ds ≤
≤M‖φ‖∞ + (M + 1)L+ nM2L+ nMMM1[‖x1‖+
+ M‖φ‖+ (M + 1)L+ nM2L] +
+ max{nMM2M1,M}
t∫
0
p(s)ψ(‖ys‖)ds.
Consider the function µ defined by
µ(t) = sup{|y(s)| : −r ≤ s ≤ t}, 0 ≤ t ≤ n.
Let t∗ ∈ [−r, t] be such that µ(t) = |y(t∗)|. If t∗ ∈ [0, n], by the previous inequality we have,
for t ∈ [0, n],
µ(t) ≤M‖φ‖∞ + (M + 1)L+ nM2L+ nMMM1
[
‖x1‖+M‖φ‖+ (M + 1)L+
+ nM2L
]
+ max{nMM2M1,M}
t∫
0
p(s)ψ(‖ys‖)ds.
If t∗ ∈ [−r, 0], then µ(t) = ‖φ‖ and the previous inequality holds. Let us take the right-hand
side of the above inequality as v(t). Then we have
c = v(0) = M‖φ‖∞ + (M + 1)L+ nM2L+ nMMM1[‖x1‖+
+ M‖φ‖+ (M + 1)L+ nM2L], µ(t) ≤ v(t), t ∈ [0, n],
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304 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
and
v′(t) = max{nMM2M1,M}p(t)ψ(µ(t)) a. e. t ∈ [0, n].
Using the nondecreasing character of ψ we get
v′(t) ≤ max{nMM2M1,M}p(t)ψ(v(t)) a. e. t ∈ [0, n].
This implies that for each t ∈ [0, n]
v(t)∫
v(0)
ds
ψ(s)
≤ max{nMM2M1,M}
t∫
0
p(s)ds < ∞.
Thus from H1) there exists a constant Kn such that v(t) ≤ Kn, t ∈ [0, n], and hence µ(t) ≤
≤ Kn, t ∈ [0, n]. Since for every t ∈ [0, n], ‖yt‖ ≤ µ(t), we have
‖y‖n ≤ max{‖φ‖,Kn} := M ′n.
Let
Y1 = {y ∈ C([−r,∞), E) : sup{|y(t)| : t ≤ n} ≤ M ′n + 1 for all n ∈ N}.
We shall show that N3 : Y → C([−r,∞), E) is a contraction operator. Indeed, consider y, y ∈
∈ C([−r,∞), E), thus for each t ∈ [0,∞) such that t ≤ n, n ∈ N,
|N3(y)(t)−N3(y)(t)| =
∣∣∣∣∣
0∫
t
AT (t− s)[g(s, ys)− g(s, ys)]ds+
+
t∫
0
T (t− s)[(Buny )(s)− (Buny )](s)ds+
+ g(t, yt)− g(t, yt) +
t∫
0
T (t− s)[f(s, ys)− f(s, ys)]ds
∣∣∣∣∣ ≤
≤M2
0∫
t
l̃n(s)‖ys − ys‖ds+ l̃n(t)‖yt − yt‖+
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SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 305
+ M‖B‖
t∫
0
|(uny )(s)− (uny )(s)|ds+M
t∫
0
ln(s)‖ys − ys‖ds ≤
≤ 1
τ
eτLn(t)‖y − y‖n + eτLn(t) l̂n‖y − y‖n +
+ MM‖W−1‖
t∫
0
[nM2‖l̃n(s)‖‖y − y‖∞ds+
+ M2‖ln‖L([0,n])‖y − y‖∞]ds+ eτLn(t) 1
τ
‖y − y‖n ≤
≤ 1
τ
eτLn(t)‖y − y‖n +
1
τ
eτLn(t)‖y − y‖n +
1
τ
eτLn(t)‖y − y‖n +
+
1
τ
eτLn(t)‖y − y‖n + sup
t∈[0,n]
|l̃(t)|‖y − y‖n.
Therefore,
‖N3(y)−N3(y)‖n ≤
(
4
τ
+ sup
t∈[0,n]
|l̃(t)
)
‖y − y‖n,
showing that N3 is a contraction for all n ∈ N. From the choice of Y1 there is no y ∈ ∂Y1 such
that y = λN3(y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative [1] we
deduce that N3 has a unique fixed point which is a mild solution to (5), (6).
6. Controllability for second order semilinear neutral FDEs. In this section we study the
initial value problem (7), (8).
Definition 6.1. A function y ∈ C([−r,∞), E) is said to be a mild solution of (7), (8) if
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η, and
y(t) = C(t)φ(0) + S(t)[η − g(0, φ)] +
0∫
t
C(t− s)g(s, ys)ds+
+
t∫
0
S(t− s)(Bu)(s)ds+
t∫
0
S(t− s)f(s, ys)ds.
Theorem 6.1. Assume that hypotheses H1), H2), H5), H6) and A1), A2) hold. Then the IVP
(7), (8) has a unique mild solution.
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306 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
Proof. Define the control
uny (t) = W−1
[
x1 − C(n)φ(0) + [η − g(0, φ)]S(n) +
+
n∫
0
C(n− s)g(s, ys)ds+
n∫
0
S(n− s)f(s, ys)ds
]
.
Transform the problem (7), (8) into a fixed point problem. Consider the operator N3 :
C([−r,∞), E) → C([−r,∞), E) defined by:
N3(y)(t) =
φ(t), if t ∈ [−r, 0];
C(t)φ(0) + [η − g(0, φ)]S(t)+
+
0∫
t
C(t− s)g(s, ys)ds+
+
t∫
0
S(t− s)(Buny )(s)ds+
+
t∫
0
S(t− s)f(s, ys)ds, if t ∈ [0,∞).
We can easily show (as in the previous theorems with minor appropriate modifications) that the
operator n4 is a contraction. The details are left to the reader.
7. An example. As an application of our results we consider the following partial neutral
functional differential equation of the form
∂
∂t
[z(t, x)− p(t, z(t− r, x))] =
∂2
∂x2
z(t, x) +
+ Q(t, z(t− r, x), zx(t− r, x)) +Bu(t), 0 ≤ x ≤ π, t ∈ [0,∞), (9)
z(t, 0) = z(t, π), t ≥ 0,
(10)
z(t, x) = φ(t, x), −r ≤ t ≤ 0,
where φ is continuous. Let
g(t, wt)(x) = p(t, w(t− x)), 0 ≤ x ≤ π,
and
f(t, wt)(x) = Q
(
t, w(t− x),
∂
∂x
w(t− x)
)
, 0 ≤ x ≤ π.
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SOME UNIQUENESS RESULTS ON CONTROLLABILITY FOR FUNCTIONAL SEMILINEAR DIFFERENTIAL . . . 307
Take E = L2[0, π] and define A : D(A) ⊂ E → E by Aw = w′′ with domain
D(A) = {w ∈ E,w,w′ are absolutely continuous, w′′ ∈ E,w(0) = w(π) = 0}.
Then
Aw =
∞∑
n=1
n2(w,wn)wn, w ∈ D(A),
where ( , ) is the inner product in L2 and wn(s) =
√
2
π
sin ns, n = 1, 2, . . . , is the orthogonal
set of eigenvectors of A. It is well known (see [10]) that A is the infinitesimal generator of an
analytic semigroup T (t), t ≥ 0, in E and is given by
T (t)w =
∞∑
n=1
exp(−n2t)(w,wn)wn, w ∈ E.
Since the analytic semigroup T (t) is compact there exist constants m1 ≥ 1 and m2 > 0 such
that
‖T (t)‖ ≤ m1 and ‖AT (t)‖ ≤ m2.
Assume that the operator B : U → Y, U ⊂ [0,∞), is a bounded linear operator and for each
b > 0 the operator
Wu =
b∫
0
T (b− s)Bu(s)ds
has a bounded inverseW−1 which takes values inL2([0,∞), U)\kerW.Assume that there exists
a constant L > 0 such that
|p(t, w(t− x)| ≤ L.
Also assume that there exists an integrable function σ : J → [0,∞) such that
|q(t, w(t− x))| ≤ σ(t)Ω(‖w‖)
where Ω : [0,∞) → (0,∞) is continuous and nondecreasing with
∞∫
1
ds
s+ Ω(s)
= +∞.
Assume that for each R > 0, there exists a function l̃R ∈ L1
loc(J,E) such that
|q(t, w(t− x))− q(t, w(t− x))| ≤ l̃R(t)‖w − w‖,
t ∈ [0,∞), w, w ∈ E, with ‖w‖, ‖w‖ ≤ R.
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308 A. ARARA, M. BENCHOHRA, AND A. OUAHAB
We can show that problem (5), (6) is an abstract formulation of problem (9), (10). Since all
the conditions of Theorem 5.1 are satisfied, the problem (9), (10) has a unique solution z on
[−r,∞)× [0, π].
1. Frigon M., Granas A. Résultats de type Leray – Schauder pour des contractions sur des espaces de Fréchet //
Ann. Sci. Math. Québec. — 1998. — 22, № 2. — P. 161 – 168.
2. Balachandran K., Dauer J.P. Controllability of nonlinear systems in Banach spaces. A survey // J. Optimiz.
Theory and Appl. — 2002. — 115. — P. 7 – 28.
3. Benchohra M., Ntouyas S.K. Controllability on infinite time horizon of nonlinear differential equations in
Banach spaces with nonlinear conditions // An. şti. Univ. Iaşi. Mat. (N. S.) — 2001. — 47. — P. 277 – 286.
4. Yosida K. Functional analysis. — 6-th edn. — Berlin: Springer, 1980.
5. Fattorini H. O. Second order linear differential equations in Banach spaces // Math. Stud. — Amsterdam:
North-Holland, 1985. — 108.
6. Goldstein J. A. Semigroups of linear operators and applications. — Oxford; New York, 1985.
7. Travis C. C., Webb G. F. Second order differential equations in Banach spaces // Proc. Int. Symp. Nonlinear
Equations in Abstract Spaces. — New York: Acad. Press, 1978 — P. 331 – 361.
8. Travis C. C., Webb G. F. Cosine families and abstract nonlinear second order differential equations // Acta
Math. hung. — 1978. — 32. — P. 75 – 96.
9. Engel K.J., Nagel R. One-parameter semigroups for linear evolution equations. — New York: Springer, 2000.
10. Pazy A. Semigroups of linear operators and applications to partial differential equation. — New York: Spri-
nger, 1983.
Received 15.05.2003
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