Admissible integral manifolds for partial neutral functional-differential equations
UDC 517.9 We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space $X$  of the form \begin{align*}&...
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| author | Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan |
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UDC 517.9
We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space $X$  of the form \begin{align*}& \dfrac{\partial}{\partial t}Fu_t= A(t)Fu_t +f(t,u_t),\quad t \ge s,\quad t,s\in\mathbb{R},\\& u_s=\phi\in\mathcal{C}:= C([-r, 0], X)\end{align*} under the conditions that the family of linear partial differential operators $\left(A(t)\right)_{t\in\mathbb{R}}$ generates the evolution family $\left(U(t,s)\right)_{t\geq s}$ with an exponential dichotomy on the whole line $\mathbb{R};$  the difference operator  $ F\colon\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator $f$ satisfies the $\varphi$-Lipschitz condition, i.e., $ \|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in\mathcal{C},$ where $\varphi(\cdot)$ belongs to an admissible function space defined on $\mathbb{R}.$  We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates.  Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces.  We  apply our results to the finite-delayed heat equation for a material with memory.  |
| doi_str_mv | 10.37863/umzh.v74i10.6257 |
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DOI: 10.37863/umzh.v74i10.6257
UDC 517.9
Nguyen Thieu Huy1, Vu Thi Ngoc Ha (School Appl. Math. and Informatics, Hanoi Univ. Sci. and Technology,
Vietnam),
Trinh Xuan Yen (Hung Yen Univ. Technology and Education, Vietnam)
ADMISSIBLE INTEGRAL MANIFOLDS
FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL EQUATIONS
ДОПУСТИМI IНТЕГРАЛЬНI МНОГОВИДИ ДЛЯ НЕЙТРАЛЬНИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible
classes of solutions to the partial neutral functional-differential equation in Banach space X of the form
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \geq s, t, s \in \BbbR ,
us = \phi \in \scrC := C([ - r, 0], X)
under the conditions that the family of linear partial differential operators (A(t))t\in \BbbR generates the evolution family
(U(t, s))t\geq s with an exponential dichotomy on the whole line \BbbR ; the difference operator F : \scrC \rightarrow X is bounded and
linear, and the nonlinear delay operator f satisfies the \varphi -Lipschitz condition, i.e., \| f(t, \phi ) - f(t, \psi )\| \leq \varphi (t)\| \phi - \psi \| \scrC for
\phi , \psi \in \scrC , where \varphi (\cdot ) belongs to an admissible function space defined on \BbbR . We also prove that an unstable manifold of
the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron
equation combined with the admissibility of function spaces. We apply our results to the finite-delayed heat equation for a
material with memory.
Доведено iснування та властивiсть притягання для допустимих iнварiантних нестiйких та центрально-нестiйких
многовидiв допустимих класiв розв’язкiв нейтрального функцiонально-диференцiального рiвняння з частинними
похiдними в банаховому просторi X вигляду
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \geq s, t, s \in \BbbR ,
us = \phi \in \scrC := C([ - r, 0], X)
за умови, що множина лiнiйних операторiв частинного диференцiювання (A(t))t\in \BbbR породжує еволюцiйну множину
(U(t, s))t\geq s, що має експоненцiальну дихотомiю на всiй прямiй \BbbR ; рiзницевий оператор F : \scrC \rightarrow X є обмеженим i
лiнiйним, а нелiнiйний оператор затримки f задовольняє умову \varphi -Лiпшиця, тобто \| f(t, \phi ) - f(t, \psi )\| \leq \varphi (t)\| \phi - \psi \| \scrC
для \phi , \psi \in \scrC , де \varphi (\cdot ) належить допустимому функцiональному простору, визначеному на \BbbR . Ми також доводимо,
що нестiйкий многовид з допустимого класу притягує всi iншi розв’язки з експоненцiальною швидкiстю. Наш
основний метод базується на рiвняннi Ляпунова – Перрона в поєднаннi з допустимiстю функцiональних просторiв.
Отриманi результати застосовано до рiвняння теплопровiдностi зi скiнченною затримкою для матерiалу з пам’яттю.
1. Introduction and preliminaries. The main concern of this paper is the existence and attraction
property of an unstable manifold of \scrE -class for solutions to the partial neutral functional-differential
equation (PNFDE)
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \in \BbbR , (1.1)
1 Corresponding author, e-mail: huy.nguyenthieu@hust.edu.vn.
c\bigcirc NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN, 2022
1364 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1365
where (A(t))t\in \BbbR is a family of (possibly unbounded) linear operators on a Banach X; F : \scrC \rightarrow X
is a bounded linear operator called a difference operator, f : \BbbR \times \scrC \rightarrow X is a continuous nonlinear
operator called a delay operator, where \scrC = C([ - r, 0], X), and ut is the history function defined by
ut(\theta ) := u(t+ \theta ) for \theta \in [ - r, 0].
The investigation for existence of an invariant manifold for solutions to (1.1) is of great impor-
tance since, on the one hand, it describes the behavior of solutions around a steady state or near
some specific solution, and on the other hand, it attracts all other solutions of the equation so that
the research of properties of all solutions can be deduced to studying the solutions on that manifold
using the reduction principle. The classical conditions for the presence of such a manifold are two
folds, firstly, the exponential dichotomy of the solution operators of corresponding linear homoge-
neous equations, and secondly the uniform Lipschitz continuity of the nonlinear term f(t, ut) with a
sufficiently small Lipschitz constant, i.e., \| f(t, \phi ) - f(t, \psi )\| \leq q\| \phi - \psi \| \scrC for sufficiently small q
(see, e.g., [9, 12, 13] and the references therein).
Huy [2] showed such results for general semilinear evolution equations with nonlinear terms
being \varphi -Lipschitz and suitable for complicated diffusion processes. Moreover, in [1], Huy has
proved the existence of a new type of invariant manifolds, called the invariant stable manifolds of
admissible classes. Such manifolds have been constituted by trajectories belonging to the admissible
Banach space E which can be Lp-space, Lorentz spaces Lp,q or some interpolation space.
The purpose of the present paper is to prove the existence of unstable manifolds of admissible
classes and their attraction property. We prove the existence of such manifolds for Eq. (1.1), when
its linear part (B(t))t\geq 0 generates the evolution family having an exponential dichotomy on \BbbR , and
its nonlinear term is \varphi -Lipschitz, i.e., \| f(t, \phi ) - f(t, \psi )\| \leq q\| \phi - \psi \| \scrC , where \phi , \psi \in \scrC and \varphi (t) is
a real and positive function which belong admissible function space.
As mentioned in [4], when handling with PNFDE we face a difficult fact that the differential
operators do not apply directly to u(t) but to Fut, and hence the variation-of-constant formula
is available only for Fut. Therefore, we write F in the form F = \delta 0 - (\delta 0 - F ), with Dirac
distribution \delta 0 concentrated at 0. Then we need certain “smallness” of \Psi := \delta 0 - F. It can be
proved that, using a renorming procedure, the smallness of \Psi can be substituted by the fact that
\Psi has “no mass in 0”, and, in case that \Psi is written as an operator integral with a kernel \eta of
bounded variation, the condition “having no mass in 0” of \Phi is equivalent to the fact that \eta is
non-atomic at 0 (see the details in [3]). Furthermore, another difficulty is lying in the fact that the
admissibly inertial manifold is constituted by trajectories of the solutions belonging to (rescaledly)
general admissible function spaces which are not necessary L\infty -spaces. Therefore, the techniques
and methodology used in the paper [4] cannot directly be applied here. Instead, we use the duality
arguments together with generalized Hölder inequalities to obtain necessary estimates correspon-
ding to the dichotomy of the evolution family. Then we apply our techniques and results in [1]
(see also [5]) of using admissibility of function spaces to construct the solutions of Lyapunov –
Perron equation which will be used to derive the existence of invariant unstable manifolds of \scrE -
class and center-invariant unstable of \scrE -class. Our main results are contained in Theorems 2.2, 2.3
and 3.1.
Next, we recall notions and concepts for latter use.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1366 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
For a Banach spaces X (with norm \| \cdot \| ) and a given r > 0, we denote by \scrC := C([ - r, 0], X)
the Banach space of all continuous functions from [ - r, 0] into X, equipped the norm \| \phi \| \scrC =
= \mathrm{s}\mathrm{u}\mathrm{p}\theta \in [ - r,0] \| \phi (\theta )\| for \phi \in \scrC . For a continuous function v : \BbbR - \rightarrow X and each t \in \BbbR , the history
function vt \in \scrC is defined by vt(\theta ) = v(t+ \theta ) for all \theta \in [ - r, 0].
Definition 1.1. A family of bounded linear operators \scrU = (U(t, s))t\geq s on a Banach space X
is a (strongly continuous, exponentially bounded) evolution family on the line if
(i) U(t, t) = Id and U(t, r)U(r, s) = U(t, s) for t \geq r \geq s,
(ii) the map (t, s) \mapsto \rightarrow U(t, s)x is continuous for every x \in X,
(iii) there are constants K \geq 1 and \alpha \in \BbbR such that \| U(t, s)\| \leq Ke\alpha (t - s) for t \geq s.
This notion has been invented to represent the solutions to Cauchy problem
du(t)
dt
= A(t)u(t), t \geq s,
u(s) = xs \in X,
(1.2)
where (A(t))t\in \BbbR is a family of (unbounded) linear operators on X, which generates the evolution
family \scrU = (U(t, s))t\geq s. That is to say, under some appropriate conditions, the solutions to Cauchy
problem (1.2) can be represented by that evolution family as u(t) := U(t, s)u(s). We refer the reader
to Pazy [11] (see also [10]) for a detailed treatment of the matter.
We then briefly recall some notions on function spaces taken from Massera and Schäffer [7] and
Huy et al. [1, 5, 6].
Let E be admissible function spaces and E\prime be its associate space defined as in [5, 6]. Then we
set
\scrE := \scrE (\BbbR , \scrC ) := \{ g : \BbbR \rightarrow \scrC : g is strongly measurable and \| g(\cdot )\| \scrC \in E\}
endowed with the norm
\| g\| \scrE :=
\bigm\| \bigm\| \| g(\cdot )\| \scrC \bigm\| \bigm\| E .
Then clearly \scrE is a Banach space, called the Banach space corresponding to the admissible function
space E. Moreover, the following hypothesis is needed in our strategy.
Standing Hypothesis 1.1. We will consider the Banach function space E and its associate space
E\prime such that both are admissible spaces. Furthermore, we suppose that E\prime contains an exponentially
E -invariant function \varphi \geq 0 satisfying that, for any fixed \nu > 0, the function h\nu (\cdot ), defined by
h\nu (t) := \| e - \nu | t - \cdot | \varphi (\cdot )\| E\prime for t \in \BbbR ,
belongs to E.
We refer the readers to [5] for various examples of admissible spaces and their applications to
invariant manifolds of admissible classes. Typical examples of admissible spaces satisfying the above
hypothesis are Lp-spaces with one type of exponentially Lp-invariant functions of the form \beta e - \alpha | t|
for t \in \BbbR and any fixed \beta , \alpha > 0.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1367
2. Existence and attractiveness of admissible unstable manifolds. In this section, we prove
the existence of an admissible unstable manifold of \scrE -class for the mild solutions of Eq. (1.1).
Throughout this section we assume that the evolution family \{ U(t, s)\} t\geq s has an exponential di-
chotomy on \BbbR . We recall now the notion of exponential dichotomies on the whole line.
Definition 2.1. An evolution family (U(t, s))t\geq s on the Banach space X is said to have an
exponential dichotomy on \BbbR if there exist bounded linear projections P (t), t \in \BbbR , on X and
positive constants N, \nu such that:
(a) U(t, s)P (s) = P (t)U(t, s), t \geq s,
(b) the restriction U(t, s)| : \mathrm{K}\mathrm{e}\mathrm{r}P (s) \rightarrow \mathrm{K}\mathrm{e}\mathrm{r}P (t), t \geq s, is an isomorphism
\bigl(
and we denote
its inverse by (U(t, s)| )
- 1 = U(s, t)| for t \geq s
\bigr)
,
(c) \| U(t, s)x\| \leq Ne - \nu (t - s)\| x\| for x \in P (s)X, t \geq s,
(d) \| U(s, t)| x\| \leq Ne - \nu (t - s)\| x\| for x \in \mathrm{K}\mathrm{e}\mathrm{r}P (t), t \geq s.
The projections P (t), t \in \BbbR , are called the dichotomy projections, and the constants N, \nu are
the dichotomy constants.
For an evolution family (U(t, s))t\geq s having an exponential dichotomy on the whole line, we can
define the Green function on \BbbR as follows:
\scrG (t, \tau ) =
\left\{ P (t)U(t, \tau ) for t \geq \tau ,
- U(t, \tau )| (I - P (\tau )) for t < \tau .
(2.1)
Thus, we have
\| \scrG (t, \tau )\| \leq N(1 +H)e - \nu | t - \tau | for all t \not = \tau ,
where H := \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| P (t)\| < \infty . Note that the exponential dichotomy of (U(t, s))t\geq s implies that
H := \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| P (t)\| < \infty and the map t \mapsto \rightarrow P (t) is strongly continuous (see [8], Lemma 4.2, for
the same discussion).
We give next the notion of the \varphi -Lipschitz of the nonlinear term f.
Definition 2.2. Let E be an admissible Banach function space and \varphi be a positive function
belonging to E. A function f : \BbbR \times \scrC \rightarrow X is said to be \varphi -Lipschitz if f satisfies:
(i) \| f(t, 0)\| = 0 for all t \in \BbbR ,
(ii) \| f(t, \phi 1) - f(t, \phi 2)\| \leq \varphi (t)\| \phi 1 - \phi 2\| \scrC for all t \in \BbbR and all \phi 1, \phi 2 \in \scrC .
Note that if f(t, \phi ) is \varphi -Lipschitz, then \| f(t, \phi )\| \leq \varphi (t)\| \phi \| \scrC for all \phi \in \scrC and t \in \BbbR . Note
also that \varphi is locally integrable (because it belongs to an admissible space), it follows that f(t, ut)
is locally integrable.
To prove the existence of an unstable manifold, instead of (1.1), we consider the following
integral equations:
Fut = U(t, s)F\phi +
t\int
s
U(t, \xi )f(\xi , u\xi )d\xi for t \geq s,
us = \phi \in \scrC .
(2.2)
We note that if the evolution family (U(t, s))t\geq s arising from the well-posed Cauchy prob-
lem (1.2), then the function u : \BbbR - \rightarrow X, which satisfies (2.2) for some given function f, is called a
mild solution of the semilinear problems
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1368 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \geq s,
us = \phi \in \scrC .
The following lemma gives the form of bounded solutions to Eq. (2.2).
Lemma 2.1. Let the evolution family (U(t, s))t\geq s have an exponential dichotomy with the cor-
responding projections P (t), t \in \BbbR , and the dichotomy constants N, \nu > 0. Assume Standing
Hypothesis 1.1 and let \varphi be an exponentially E -invariant function defined as in that Standing Hy-
pothesis 1.1. Let F : \scrC \rightarrow X and f : \BbbR \times \scrC \rightarrow X be respectively the difference and delay operators.
Suppose that the difference operator F is of the form F = \delta 0 - \Psi for \Psi \in \scrL (\scrC , X) with \| \Psi \| \leq 1,
and \delta 0 being the Dirac function concentrated at 0. Suppose that f is \varphi -Lipschitz and u(t) is a
solution to Eq. (2.2) on ( - \infty , t0] such that the function x(t) =
\Biggl\{
ut for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs
to \scrE .
Then, for t \leq t0, the function u(t) satisfies
Fut = U(t, t0)| \nu 1 +
t0\int
- \infty
\scrG (t, \tau )f(\tau , u\tau )d\tau (2.3)
for some \nu 1 \in X1(t0) = (I - P (t0))X, where \scrG (t, \tau ) is the Green function defined as in (2.1).
Proof. Put z(t) =
\int t0
- \infty
\scrG (t, \tau )f(\tau , u\tau )d\tau for all t \leq t0. We have
\| z(t)\| \leq
t0\int
- \infty
N(1 +H)e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau \leq
\leq N(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau .
Since t+ \theta \in [ - r + t, t] for fixed t \in ( - \infty , t0] and \theta \in [ - r, 0], we have
\| zt\| \scrC = \mathrm{s}\mathrm{u}\mathrm{p}
- r\leq \theta \leq 0
\| y(t+ \theta )\| \leq N(1 +H)e\nu r
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau for t \leq t0.
Since e - \nu | t - \cdot | \varphi (\cdot ) \in E\prime , \| u\cdot \| \scrC \in E using the “Hölder inequality” [6] (inequality (15)), we obtain
\| zt\| \scrC \leq N(1 +H)e\nu r\| e - \nu | t - \cdot | \varphi (\cdot )\| E\prime \| \| u\cdot \| \scrC \| E = N(1 +H)e\nu rh\nu (t)\| u(\cdot )\| \scrE for t \leq t0.
Therefore, by Banach lattice properties we have that z(\cdot ) \in \scrE and
\| z(\cdot )\| \scrE \leq N(1 +H)e\nu r\| h\nu (\cdot )\| E\| u(\cdot )\| \scrE .
By straightforward calculations, we get
z(t0) = U(t0, t)z(t) +
t0\int
t
U(t0, \tau )f(\tau , u\tau )d\tau for t \leq t0.
Indeed,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1369
U(t0, t)z(t) +
t0\int
t
U(t0, \tau )f(\tau , u\tau )d\tau =
=
t0\int
t
U(t0, \tau )f(\tau , u\tau )d\tau + U(t0, t)
t0\int
- \infty
\scrG (t, \tau )f(\tau , u\tau )d\tau =
=
t0\int
t
U(t0, \tau )f(\tau , u\tau )d\tau +
t\int
- \infty
U(t0, \tau )P (\tau )f(\tau , u\tau )d\tau -
t0\int
t
U(t0, \tau )(I - P (\tau ))f(\tau , u\tau )d\tau =
=
t0\int
- \infty
U(t0, \tau )P (\tau )f(\tau , u\tau )d\tau =
t0\int
- \infty
\scrG (t0, \tau )f(\tau , u\tau )d\tau = z(t0).
On the other hand,
Fut0 = U(t0, t)Fut +
t0\int
t
U(t0, \tau )f(\tau , u\tau )d\tau for t \leq t0.
Hence, Fut0 - z(t0) = U(t0, t)(Fut - z(t)). For \xi \leq t, we have
P (t)Fut = P (t)U(t, \xi )Fu\xi + P (t)
t\int
\xi
U(t, \tau )f(\tau , u\tau )d\tau =
= U(t, \xi )P (\xi )Fu\xi +
t\int
\xi
U(t, \tau )P (\tau )f(\tau , u\tau )d\tau .
Therefore, letting \xi \rightarrow - \infty , we obtain
P (t)[Fut - z(t)] = P (t)Fut -
t\int
- \infty
U(t, \tau )P (\tau )f(\tau , u\tau )d\tau = \mathrm{l}\mathrm{i}\mathrm{m}
\xi \rightarrow - \infty
U(t, \xi )P (\xi )Fu\xi .
We assume that \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow - \infty U(t, \xi )P (\xi )Fu\xi = m \not = 0. On the other hand,\bigm\| \bigm\| U(t, \xi )P (\xi )Fu\xi
\bigm\| \bigm\| \leq Ne - \nu (t - \xi )\| P (\xi )Fu\xi \| \leq Ne - \nu (t - \xi )H(1 + \| \Psi \| )\| u\xi \| \scrC .
So, e - \nu \xi \| U(t, \xi )P (\xi )Fu\xi \| \leq Ne - \nu tH(1+\| \Psi \| )\| u\xi \| \scrC for all \xi \leq t. By Banach lattice property, we
have e - \nu \xi \| U(t, \xi )P (\xi )Fu\xi \| \in E. Moreover, we also obtain \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow - \infty e - \nu \xi \| U(t, \xi )P (\xi )Fu\xi \| =
= \infty . Therefore,
\mathrm{s}\mathrm{u}\mathrm{p}
\xi \leq t
\xi \int
\xi - 1
e - \nu \tau \| U(t, \tau )P (\tau )Fu\tau \| d\tau = \infty .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1370 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
This contradict to E \lhook \rightarrow M(\BbbR ) (see [6], Remark 1.5). So, \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow - \infty U(t, \xi )P (\xi )Fu\xi = 0.
Thus, Fut - z(t) \in \mathrm{K}\mathrm{e}\mathrm{r}(P (t)). This leads to Fut0 - z(t0) = U(t0, t)(Fut - z(t)) \in \mathrm{K}\mathrm{e}\mathrm{r}(P (t0)).
Putting \nu 1 = Fut0 - z(t0), we have that Fut = U(t, t0)| \nu 1 + z(t) for all t \leq t0.
Lemma 2.1 is proved.
Remark 2.1. We call Eq. (2.3) the Lyapunov – Perron equation. By computing directly, we can
see that the converse of Lemma 2.1 is also true in the sense that all solutions of Eq. (2.3) on ( - \infty , t0]
satisfy Eq. (2.2) for all s \leq t \leq t0.
In case the evolution (U(t, s))t\geq s have an exponential dichotomy, using the projections (P (t))t\in \BbbR
on X, we can define the operators
\bigl( \widetilde P (t)\bigr)
t\in \BbbR on \scrC as follows.
For each t \in \BbbR , we set that
\widetilde P (t) : \scrC - \rightarrow \scrC ,
( \widetilde P (t)\phi )(\theta ) = U(t+ \theta , t)| (I - P (t))\phi (0) for all \theta \in [ - r, 0]. (2.4)
We easily see that
\bigl( \widetilde P (t)\bigr) 2 = \widetilde P (t), so the operators
\bigl( \widetilde P (t)\bigr)
t\in \BbbR are projections on \scrC . Moreover,
\mathrm{I}\mathrm{m} \widetilde P (t) = \Bigl\{ \phi \in \scrC : \phi (\theta ) = U(t+ \theta , t)| \nu 1 for all \theta \in [ - r, 0] for some \nu 1 \in \mathrm{K}\mathrm{e}\mathrm{r}P (t)
\Bigr\}
. (2.5)
We then come to our first result on the existence, uniqueness and exponential stability of solution to
(2.3) with initial function belonging to \mathrm{I}\mathrm{m} \widetilde P (t). To do this, we first recall the notion of the integral
translation operators \Lambda 1 (see [6], Definition 1.3, Proposition 1.6) as follows: for \varphi \in E, \Lambda 1\varphi is
defined by \Lambda 1\varphi (t) :=
\int t+1
t
\varphi (\tau )d\tau belong to E for all t \in \BbbR .
Theorem 2.1. Let the evolution family \{ U(t, s)\} t\geq s have an exponential dichotomy with the
dichotomy projections P (t), t \in \BbbR , and constants N, \nu > 0. Consider the projections \widetilde P (t) defined
as in (2.4), and function h\nu defined as in Standing Hypothesis 1.1. Let the difference operator F :
\scrC \rightarrow X be of the form F = \delta 0 - \Psi for \Psi \in \scrL (\scrC , X) with \| \Psi \| \leq 1, and \delta 0 being the Dirac function
concentrated at 0. Suppose that the delay operator f : \BbbR \times \scrC \rightarrow X is \varphi -Lipschitz for \varphi \in E\prime being an
exponentially E -invariant function as in Standing Hypothesis 1.1, and set k = N(1 +H)e\nu r\| h\nu \| E .
Then, if
k
1 - \| \Psi \|
< 1, there corresponding to each \phi \in \mathrm{I}\mathrm{m} \widetilde P (t0) one and only one solution u(t) of
(2.3) on ( - \infty , t0] satisfying the conditions that \widetilde P (t0)\widetilde ut0 = \phi and x(t) =
\Biggl\{
ut for t \leq t0,
0 for t > t0,
t \in \BbbR ,
belongs to \scrE , where the function \widetilde ut0 is defined by \widetilde ut0(\theta ) = Fut0+\theta for all - r \leq \theta \leq 0. Moreover,
if
N(1 +H)e\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - \| \Psi \|
< 1, then the following estimate is valid for any two solutions
u(\cdot ), v(\cdot ) corresponding to different initial function \phi , \psi \in \mathrm{I}\mathrm{m} \widetilde P (t0):
\| ut - vt\| \scrC \leq C\mu e
- \mu (t0 - t)\| \phi (0) - \psi (0)\| for all t \leq t0, (2.6)
where \mu is a positive number satisfying
0 < \mu < \nu + \mathrm{l}\mathrm{n}
\biggl(
1 - N(1 +H)e\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - \| \Psi \|
\biggr)
and
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1371
C\mu =
Ne\nu r
1 - \| \Psi \| - N(1 +H)e\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
(1 - \| \Psi \| )(1 - e - (\nu - \mu ))
.
Proof. Firstly, we prove that there corresponding to each \phi \in \mathrm{I}\mathrm{m} \widetilde P (t0) one and only one
solution u(t) in \scrE of Eq. (2.3) on ( - \infty , t0]. Since \phi \in \mathrm{I}\mathrm{m} \widetilde P (t0), by (2.5), there exists \nu 1 \in
\in \mathrm{K}\mathrm{e}\mathrm{r}P (t0) such that \phi (\theta ) = U(t0 + \theta , t0)| \nu 1 for all - r \leq \theta \leq 0. Clearly, \nu 1 = \phi (0). Denote
by Cb(( - \infty , t0], X) the Banach space of bounded, continuous and X -valued functions defined on
( - \infty , t0]. For \nu 1 = \phi (0) \in \mathrm{K}\mathrm{e}\mathrm{r}P (t0) as above, we define a mapping
\widetilde F\phi : Cb(( - \infty , t0], X) \rightarrow Cb(( - \infty , t0], X)
by
( \widetilde F\phi u)(t) = U(t, t0)| \nu 1 +
t0\int
- \infty
\scrG (t, \tau )f(\tau , u\tau )d\tau .
We define the operator \widetilde \Psi : Cb(( - \infty , t0], X) \rightarrow Cb(( - \infty , t0], X) by
(\widetilde \Psi u)(t) = \Psi ut for t \leq t0.
Since \| \Psi \| < 1, we have \| \widetilde \Psi \| \leq \| \Psi \| < 1. Therefore, the operator (I - \widetilde \Psi ) is invertible.
We now put T := (I - \widetilde \Psi ) - 1 \widetilde F\phi . Then we have
\| ( \widetilde F\phi u)(t)\| \leq Ne - \nu (t0 - t)\| \nu 1\| +N(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau =
= NT+
t0
e\nu (t)\| \nu 1\| +N(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau .
Since t+ \theta \in [ - r + t, t] for fixed t \in ( - \infty , t0] and all \theta \in [ - r, 0], we obtain
\| (Tu)(t)\| \scrC \leq
\infty \sum
n=0
\| \Psi \| n
\left( NT+
t0
e\nu (t)e
\nu r\| \nu 1\| +N(1 +H)e\nu r
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau \| \scrC d\tau
\right) .
According to the “Hölder inequality”, we get
\| (Tu)(t)\| \scrC \leq 1
1 - \| \Psi \|
\biggl(
NT+
t0
e\nu (t)e
\nu r\| \nu 1\| +N(1 +H)e\nu rh\nu (t)\| u(\cdot )\| \scrE
\biggr)
.
Therefore, by Banach lattice properties we have (Tu)(\cdot ) \in \scrE and
\| (Tu)(\cdot )\| \scrE \leq 1
1 - \| \Psi \|
\biggl(
NN1\| e\nu \| Ee\nu r\| \nu 1\| +N(1 +H)e\nu r\| h\nu (\cdot )\| E\| u(\cdot )\| \scrE
\biggr)
.
Hence, the transformation T acts from \scrE into \scrE . Next, we will prove T is a contraction mapping.
Using the Neumann series, we obtain
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1372 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
(Tu)(t) - (Tv)(t) =
\Biggl[ \Biggl( \infty \sum
n=0
\widetilde \Psi n
\Biggr) \widetilde F\phi u
\Biggr]
(t) -
\Biggl[ \Biggl( \infty \sum
n=0
\widetilde \Psi n
\Biggr) \widetilde F\phi v
\Biggr]
(t) =
=
\Bigl[ \Bigl( \widetilde F\phi u
\Bigr)
(t) -
\Bigl( \widetilde F\phi v
\Bigr)
(t)
\Bigr]
+
\Bigl[ \Bigl( \widetilde \Psi \widetilde F\phi u
\Bigr)
(t) -
\Bigl( \widetilde \Psi \widetilde F\phi v
\Bigr)
(t)
\Bigr]
+ . . . .
We then estimate
\bigm\| \bigm\| \bigm\| \Bigl( \widetilde F\phi u
\Bigr)
(t) -
\Bigl( \widetilde F\phi v
\Bigr)
(t)
\bigm\| \bigm\| \bigm\| \leq
t0\int
- \infty
\| \scrG (t, \tau )(f(\tau , u\tau ) - f(\tau , v\tau ))\| d\tau \leq
\leq N(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau - v\tau \| \scrC d\tau for t \leq t0.
Next, by induction we can easily see that
\bigm\| \bigm\| \bigm\| \Bigl( \widetilde \Psi n \widetilde F\phi u
\Bigr)
(t) -
\Bigl( \widetilde \Psi n \widetilde F\phi v
\Bigr)
(t)
\bigm\| \bigm\| \bigm\| \leq \| \Psi \| nN(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau - v\tau \| \scrC d\tau for t \leq t0.
From the above claim it follow that
\| (Tu)(t) - (Tv)(t)\| \leq
\infty \sum
n=0
\| \Psi \| nN(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau - v\tau \| \scrC d\tau =
=
1
1 - \| \Psi \|
N(1 +H)
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau - v\tau \| \scrC d\tau for t \leq t0.
Since t+ \theta \in [ - r + t, t] for fixed t \in ( - \infty , t0] and all \theta \in [ - r, 0], we have
\| (Tu)(t) - (Tv)(t)\| \scrC = \mathrm{s}\mathrm{u}\mathrm{p}
- r\leq \theta \leq 0
\| (Tu)(t+ \theta ) - (Tv)(t+ \theta )\| \leq
\leq 1
1 - \| \Psi \|
N(1 +H)e\nu r
t0\int
- \infty
e - \nu | t - \tau | \varphi (\tau )\| u\tau - v\tau \| \scrC d\tau .
Since e - \nu | t - \cdot | \varphi (\cdot ) \in E\prime , \| u\tau - v\tau \| \scrC \in E, and using the “Hölder inequality” [6] (inequality (15)) it
follows from the above inequality that
\| (Tu)(t) - (Tv)(t)\| \scrC \leq 1
1 - \| \Psi \|
N(1 +H)e\nu r\| e - \nu | t - \cdot | \varphi (\cdot )\| E\prime \| \| u(\cdot ) - v(\cdot )\| \scrC \| E \leq
\leq 1
1 - \| \Psi \|
N(1 +H)e\nu rh\nu (t)\| u(\cdot ) - v(\cdot )\| \scrE for t \leq t0.
By the Banach lattice property of E and the fact that h\nu (\cdot ) \in E it follows that \| Tu(\cdot )\| \scrC \in E. Thus,
Tu \in \scrE , and we have
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1373
\| Tu - Tv\| \scrE \leq 1
1 - \| \Psi \|
N(1 +H)e\nu r\| h\nu \| E .\| u - v\| \scrE =
k
1 - \| \Psi \|
\| u - v\| \scrE .
Next, if
k
1 - \| \Psi \|
< 1, the transformation T is a contraction mapping from \scrE to it self. Hence, there
exists a unique u(\cdot ) \in \scrE such that Tu = u. This yield that u(t), t \leq t0, is the unique solution of
(2.3) with
( \widetilde F\phi ut0)(\theta ) = U(t0 + \theta , t0)| \nu 1 +
t0\int
- \infty
\scrG (t0 + \theta , \tau )f(\tau , u\tau )d\tau for all \theta \in [ - r, 0],
and (I - P (t0))Fut0 = \nu 1 = \phi (0). Therefore, \widetilde P (t0)\widetilde ut0 = \phi by the definition of \widetilde P (t0) (see (2.4)).
Secondly, by using the similar arguments as in [4] (Theorem 3.5) combined with the “Hölder
inequality” in the admissible function spaces E and E\prime , we obtain (2.6).
Theorem 2.1 is proved.
Definition 2.3. A set U \subset \BbbR \times \scrC is said to be unstable manifold of \scrE -class for the solution
to Eq. (2.2) if for every t \in \BbbR the phase space splits into a direct sum \scrC = \widetilde X0(t) \oplus \widetilde X1(t) with
corresponding projections \widetilde P (t), t \in \BbbR (i.e., \widetilde X0(t) = \mathrm{I}\mathrm{m} \widetilde P (t), \widetilde X1(t) = \mathrm{K}\mathrm{e}\mathrm{r} \widetilde P (t)) such that
\mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| \widetilde P (t)\| <\infty , and there exists a family of Lipschitz continuous mapping
\widetilde yt : \widetilde X0(t) \rightarrow \widetilde X1(t), t \in \BbbR ,
with the Lipschitz constants being independent of t such that
(i) U =
\Bigl\{
(t, \psi + \widetilde yt(\psi )) \in \BbbR \times ( \widetilde X0(t)\oplus \widetilde X1(t))| t \in \BbbR , \psi \in \widetilde X0(t)
\Bigr\}
, and we denote by
Ut = \{ \psi + \widetilde yt(\psi ) : (t, \psi + \widetilde yt(\psi )) \in U, t \in \BbbR \} ;
(ii) Ut is homeomorphic to \widetilde X0(t) for all t \in \BbbR ;
(iii) to each t0 \in \BbbR , \phi \in Ut0 there corresponds one and only one solution u(\cdot ) of Eq. (2.2) on
( - \infty , t0] satisfying the conditions that \widetilde ut0 = \phi and
x(t) =
\left\{ ut for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs to \scrE ,
where the functions \widetilde ut0 is defined as in Theorem 2.1. Moreover, any two solutions u(\cdot ) and v(\cdot ) of
(2.2) corresponding to different initial functions \phi 1, \phi 2 \in Ut0 backwardly and exponentially attract
each other in the sense that there exist positive constants \mu and C\mu independent of t0 such that
\| ut - vt\| \scrC \leq C\mu e
- \mu (t0 - t)
\bigm\| \bigm\| \bigm\| \Bigl( \widetilde P (t0)\phi 1\Bigr) (0) - \Bigl( \widetilde P (t0)\phi 2\Bigr) (0)\bigm\| \bigm\| \bigm\| for t \leq t0;
(iv) U is positively F -invariant under (2.2), i.e., if u(t), t \in \BbbR , is a solution to (2.2) satisfying
conditions that \widetilde ut0 \in Ut0 and function x(t) =
\Biggl\{
ut for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs to \scrE for some
t0 \in \BbbR , then we have \widetilde ut \in Ut for all t \in \BbbR , where the function \widetilde ut is defined as in Theorem 2.1 with
t0 being replaced by t, i.e., \widetilde ut(\theta ) = Fut+\theta for all - r \leq \theta \leq 0 and t \in \BbbR .
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1374 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
We now prove the existence of an unstable manifold of \scrE -class.
Theorem 2.2. Under the hypotheses of Theorem 2.1 and function e\nu (t) = e - \nu | t| for all t \in \BbbR .
Then if the f is \varphi -Lipschitz with \varphi satisfying
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
N(1 +H)e\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - \| \Psi \|
,
N2N1(1 +H)e\nu r\| e\nu \| E\| \varphi \| E\prime
1 - k - \| \Psi \|
\biggr\}
< 1,
then there exists an invariant unstable manifold U of \scrE -class for the solutions of Eq. (2.2), where k
is defined as in Theorem 2.1.
Proof. The proof of this theorem can be done by the similar way as in [4] (Theorem 3.7) and
using the structures of bounded solution as in Lemma 2.1. We just note that the family of Lipschitz
mapping (\widetilde yt)t\in \BbbR determining the unstable manifold of \scrE -class in Definition 2.3 by
\widetilde yt : \widetilde X0(t) \rightarrow \widetilde X1(t), t \in \BbbR ,
\widetilde yt(\phi )(\theta ) = t\int
- \infty
\scrG (t+ \theta , \tau )f(\tau , u\tau )d\tau for all \theta \in [ - r, 0].
Here, u(\cdot ) is the unique solution of Eq. (2.2) on ( - \infty , t] satisfying \widetilde P (t)\widetilde ut = \phi and x(t) =
=
\Biggl\{
ut for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs to \scrE (note that the existence and uniqueness of u(\cdot ) is guaran-
teed by Theorem 2.1). Using the “Hölder inequality”, we obtain \widetilde yt is Lipschitz continuous with the
Lipschitz constant
k1 =
N2N1(1 +H)e\nu r\| e\nu \| E\| \varphi \| E\prime
1 - k - \| \Psi \|
(2.7)
independent of t.
Theorem 2.2 is proved.
The next we will prove the attraction property of an invariant unstable manifold of \scrE -class
for solutions of Eq. (2.2). Concretely, we will show that the unstable manifold of \scrE -class U =
= \{ (t,Ut)\} t\in \BbbR F -exponentially attracts all solutions to Eq. (2.2) in the sense that any solution u(\cdot )
to (2.2) is exponentially attracted to some F -induced trajectory u\ast (\cdot ) lying in the unstable manifold
of \scrE -class (i.e., \widetilde u\ast t \in Ut for all t \in \BbbR ). Precisely, we will prove the following theorem.
Theorem 2.3. Assume that conditions of Theorem 2.2 are satisfied. For each fixed 0 < \alpha < \nu ,
we define the functions e\nu - \alpha (t) = e - (\nu - \alpha )| t| and h\nu - \alpha (t) = \| e - (\nu - \alpha )| t - \cdot | \varphi (\cdot )\| E\prime for t \in \BbbR .
Suppose that
l
1 - \| \Psi \|
< 1, where
l = N(1 +H)e2\nu r \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
Nk1 +
(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - e - (\nu - \alpha )
, NN1k1\| e\nu - \alpha \| E + \| h\nu - \alpha \| E
\biggr\}
,
k1 is defined in (2.7). Then the unstable manifold of \scrE -class U = \{ (t,Ut)\} t\in \BbbR F -exponentially
attracts all solutions to Eq. (2.2) in the sense that for any solution u(\cdot ) to (2.2) with initial function
u\xi there exists a solution u\ast (\cdot ) such that \widetilde u\ast t \in Ut for all t \in \BbbR such that
\| ut - u\ast t \| \scrC \leq Ce - \alpha (t - \xi )\| u\xi - u\ast \xi \| \scrC for t \geq \xi ,
where \widetilde u\ast t (\theta ) = Fu\ast t+\theta for all \theta \in [ - r, 0], t \in \BbbR .
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1375
Proof. For any fixed \xi \in \BbbR , we introduce the space
C\xi ,\alpha =
\Bigl\{
x(\cdot ) \in \scrE such that x(t) = 0 for t < \xi and e\alpha (\cdot - \xi )\| x(\cdot )\| \scrC \in E \cap L\infty (\BbbR )
\Bigr\}
,
which is a Banach space endowed with the norm
\| x(\cdot )\| \alpha = \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{
\| e\alpha (\cdot - \xi )\| x(\cdot )\| \scrC \| E , \| e\alpha (\cdot - \xi )\| x(\cdot )\| \scrC \| \infty
\Bigr\}
.
We will find u\ast (\cdot ) in the form u\ast (t) = u(t) + \omega (t) such that z(t) =
\Biggl\{
\omega t for t \geq \xi ,
0 for t < \xi ,
belongs
to C\xi ,\alpha .
We see that u\ast (\cdot ) is a solution to (2.2) if and only if \omega (\cdot ) is a solution of the equation
F\omega t = U(t, \xi )F\omega \xi +
t\int
\xi
U(t, \tau )
\bigl[
f(\tau , u\tau + \omega \tau ) - f(\tau , u\tau )
\bigr]
d\tau .
To simplify the representation, we put g(t, \omega t) = f(t, ut + \omega t) - f(t, ut). Then g : \BbbR \times \scrC \rightarrow X is
also \varphi -Lipschitz and g(t, 0) = 0. The equation for \omega (t) can be rewritten as
F\omega t = U(t, \xi )F\omega \xi +
t\int
\xi
U(t, \tau )g(\tau , \omega \tau )d\tau . (2.8)
In the same way as in the proof of Lemma 2.1 and Remark 2.1, we observe that the solution \omega (t) of
(2.8) defines on [\xi - r,\infty ) (here \omega (t) = 0 for t < \xi - r) such that z(t) belongs to \scrE if and only if
satisfies
F\omega t = U(t, \xi )\nu 0 +
\infty \int
\xi
\scrG (t, \tau )g(\tau , \omega \tau )d\tau for some \nu 0 \in \mathrm{I}\mathrm{m}P (\xi ) and t \geq \xi (2.9)
and
F\omega t = U(2\xi - t, \xi )\nu 0 +
\infty \int
\xi
\scrG (2\xi - t, \tau )g(\tau , \omega \tau )d\tau for some \nu 0 \in \mathrm{I}\mathrm{m}P (\xi ) and t \in [\xi - r, \xi ].
(2.10)
We will choose \nu 0 \in \mathrm{I}\mathrm{m}P (\xi ) such that u\ast \xi = u\xi + \omega \xi \in U\xi . This means\Bigl(
I - \widetilde P (\xi )\Bigr) (u\xi + \omega \xi )(0) = \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(\theta ) for \theta \in [ - r, 0].
Hence,
\nu 0 =
\Bigl(
\omega \xi - \widetilde P (\xi )\omega \xi
\Bigr)
(0) = -
\Bigl(
u\xi - \widetilde P (\xi )u\xi \Bigr) (0) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0) =
= - P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0). (2.11)
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1376 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
Substituting (2.11) into (2.9) and (2.10), we obtain
F\omega t =
\left\{
U(t, \xi )
\Bigl[
- P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0)
\Bigr]
+
+
\int \infty
\xi
\scrG (t, \tau )g(\tau , \omega \tau )d\tau for t \geq \xi ,
U(2\xi - t, \xi )
\Bigl[
- P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0)
\Bigr]
+
+
\int \infty
\xi
\scrG (2\xi - t, \tau )g(\tau , \omega \tau )d\tau for t \in [\xi - r, \xi ].
(2.12)
Thus, u\ast (t) is a solution to (2.2) and satisfies u\ast \xi \in U\xi if and only if \omega (t) satisfies (2.12).
Next, in order to prove the existence of u\ast (t) satisfying assertions of the theorem, we will find
solution \omega (t) of Eq. (2.12) in the Banach space C\xi ,\alpha . To do this, we define a mapping
\widetilde F\phi : C([\xi - r,\infty ), X) \rightarrow C([\xi - r,\infty ), X)
by
\Bigl( \widetilde F\phi \omega
\Bigr)
(t) =
\left\{
U(t, \xi )
\Bigl[
- P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0)
\Bigr]
+
+
\int \infty
\xi
\scrG (t, \tau )g(\tau , \omega \tau )d\tau for t \geq \xi ,
U(2\xi - t, \xi )
\Bigl[
- P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0)
\Bigr]
+
+
\int \infty
\xi
\scrG (2\xi - t, \tau )g(\tau , \omega \tau )d\tau for t \in [\xi - r, \xi ].
We also define the operator \widetilde \Psi : C([\xi - r,\infty ), X) \rightarrow C([\xi - r,\infty ), X) by
\Bigl( \widetilde \Psi u\Bigr) (t) =
\left\{ \Psi (ut) for t \geq \xi ,
\Psi (u\xi ) for \xi - r \leq t < \xi .
Since \| \Psi \| < 1, we have \| \widetilde \Psi \| \leq \| \Psi \| < 1. Therefore, the operator I - \widetilde \Psi is invertible and we now
put T = (I - \widetilde \Psi ) - 1 \widetilde F\phi . We will prove that transformation T as above acts from C\xi ,\alpha into C\xi ,\alpha is a
contraction mapping. Firstly, we show that T\omega \in C\xi ,\alpha . Indeed, for t \geq \xi - r, using the Neumann
series, we have
(T\omega )(t) =
\Biggl[ \Biggl( \infty \sum
n=0
\widetilde \Psi n
\Biggr) \widetilde F\phi \omega
\Biggr]
(t).
Then we estimate\bigm\| \bigm\| \bigm\| \Bigl( \widetilde F\phi \omega
\Bigr)
(t)
\bigm\| \bigm\| \bigm\| \leq Ne - \nu (t - \xi )\| \nu 0\| +N(1 +H)
\infty \int
\xi
e - \nu | t - \tau | \varphi (\tau )\| \omega \tau \| \scrC d\tau for t \geq \xi
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1377
and similarly
\bigm\| \bigm\| \bigm\| \Bigl( \widetilde F\phi \omega
\Bigr)
(t)
\bigm\| \bigm\| \bigm\| \leq Ne - \nu (\xi - t)\| \nu 0\| +N(1 +H)
\infty \int
\xi
e - \nu | 2\xi - t - \tau | \varphi (\tau )\| \omega \tau \| \scrC d\tau for t \in [\xi - r, \xi ].
From the inequality \| \widetilde \Psi \| \leq \| \Psi \| , it follows that
\| (T\omega )(t)\| \leq
\infty \sum
n=0
\| \Psi \| n
\left[ Ne - \nu (t - \xi )\| \nu 0\| +N(1 +H)
\infty \int
\xi
e - \nu | t - \tau | \varphi (\tau )\| \omega \tau \| \scrC d\tau
\right] for t \geq \xi
and
\bigm\| \bigm\| (T\omega )(t)\bigm\| \bigm\| \leq
\infty \sum
n=0
\| \Psi \| n
\left[ Ne - \nu (\xi - t)\| \nu 0\| +
+N(1 +H)
\infty \int
\xi
e - \nu | 2\xi - t - \tau | \varphi (\tau )\| \omega \tau \| \scrC d\tau
\right] for t \in [\xi - r, \xi ].
Therefore, for t \geq \xi ,
\bigm\| \bigm\| (T\omega )(t)\bigm\| \bigm\| \scrC \leq 1
1 - \| \Psi \|
\left[ Ne\nu re - \nu (t - \xi )\| \nu 0\| +N(1 +H)e\nu r
\infty \int
\xi
e - \nu | t - \tau | \varphi (\tau )\| \omega \tau \| \scrC d\tau
\right] .
Thus,
e\alpha (t - \xi )\| (T\omega )(t)\| \scrC \leq
1
1 - \| \Psi \|
\left[ Ne\nu r\| \nu 0\| +N(1+H)e\nu r
\infty \int
\xi
e - (\nu - \alpha )| t - \tau | \varphi (\tau )e\alpha (\tau - \xi )\| \omega \tau \| \scrC d\tau
\right] \leq
\leq 1
1 - \| \Psi \|
\biggl[
Ne\nu r\| \nu 0\| +
N(1 +H)e\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - e - (\nu - \alpha )
\| e\alpha (t - \xi )\| \omega t\| \scrC \| \infty
\biggr]
.
So, e\alpha (t - \xi )\| (T\omega )(t)\| \scrC \in L\infty (\BbbR ).
On the other hand, we also have
e\alpha (t - \xi )\| (T\omega )(t)\| \scrC \leq
\leq 1
1 - \| \Psi \|
\left[ Ne\nu re - (\nu - \alpha )(t - \xi )\| \nu 0\| +N(1 +H)e\nu r
\infty \int
\xi
e - (\nu - \alpha )| t - \tau | \varphi (\tau )e\alpha (\tau - \xi )\| \omega \tau \| \scrC d\tau
\right] \leq
\leq 1
1 - \| \Psi \|
\Bigl[
Ne\nu r
\Bigl(
T+
\xi e\nu - \alpha
\Bigr)
(t)\| \nu 0\| +N(1 +H)e\nu rh\nu - \alpha (t)\| e\alpha (\tau - \xi )\| \omega \tau \| \scrC \| E
\Bigr]
.
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1378 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
By the Banach lattice property of E the e\alpha (t - \xi )\| (T\omega )(t)\| \scrC \in E\cap L\infty (\BbbR ). This leads to T\omega \in C\xi ,\alpha .
Next, by the Lipschitz continuity of \widetilde y\xi , we obtain
\| \nu 0\| =
\bigm\| \bigm\| \bigm\| - P (\xi )u(\xi ) + \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0)
\bigm\| \bigm\| \bigm\| \leq
\leq
\bigm\| \bigm\| \bigm\| \widetilde y\xi \Bigl( \widetilde P (\xi )u(\xi )\Bigr) (0) - P (\xi )u(\xi )
\bigm\| \bigm\| \bigm\| + \bigm\| \bigm\| \bigm\| \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0) - \widetilde y\xi \Bigl( \widetilde P (\xi )u(\xi )\Bigr) (0)\bigm\| \bigm\| \bigm\| \leq
\leq
\bigm\| \bigm\| \bigm\| \widetilde y\xi \Bigl( \widetilde P (\xi )u\xi \Bigr) - (I - \widetilde P (\xi ))u\xi \bigm\| \bigm\| \bigm\|
\scrC
+ k1
\bigm\| \bigm\| \bigm\| \widetilde P (\xi )\omega \xi
\bigm\| \bigm\| \bigm\|
\scrC
\leq
\leq m(\xi ) + k1N(1 +H)e\nu r\| \omega \xi \| \scrC
for
m(\xi ) =
\bigm\| \bigm\| \bigm\| \widetilde y\xi \Bigl( \widetilde P (\xi )u\xi \Bigr) - (I - \widetilde P (\xi ))u\xi \bigm\| \bigm\| \bigm\|
\scrC
\leq m(\xi ) + k1N(1 +H)e\nu r\| \omega \| \alpha .
So,
\| T\omega \| \alpha \leq \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, N1\| e\nu - \alpha \| E\}
Ne\nu rm(\xi )
1 - \| \Psi \|
+
l
1 - \| \Psi \|
\| \omega \| \alpha . (2.13)
We then prove that T is a contraction mapping. Indeed, for \omega , v belongs to C\xi ,\alpha . Then, for
\nu 0 = \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + \omega \xi )
\Bigr)
(0), \mu 0 = \widetilde y\xi \Bigl( \widetilde P (\xi )(u\xi + v\xi )
\Bigr)
(0), we have
e\alpha (t - \xi )\| (T\omega )(t) - (Tv)(t)\| \scrC \leq 1
1 - \| \Psi \|
\left[ Ne\nu re - (\nu - \alpha )(t - \xi )\| \nu 0 - \mu 0\| +
+N(1 +H)e\nu r
\infty \int
\xi
e - (\nu - \alpha )| t - \tau | \varphi (\tau )e\alpha (\tau - \xi )\| \omega \tau - v\tau \| \scrC d\tau
\right] .
On the other hand, \| \nu 0 - \mu 0\| \leq k1N(1 +H)e\nu r\| \omega - v\| \alpha .
Thus,
\| e\alpha (t - \xi )\| T\omega - Tv\| \scrC \| \infty \leq
\leq 1
1 - \| \Psi \|
\biggl[
k1N
2(1 +H)e2\nu r\| \omega - v\| \alpha +
N(1 +H)e2\nu r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - e - (\nu - \alpha )
\| \omega - v\| \alpha
\biggr]
and
\| e\alpha (t - \xi )\| T\omega - Tv\| \scrC \| E \leq
\leq 1
1 - \| \Psi \|
\bigl[
k1N
2N1(1 +H)e2\nu r\| e\nu - \alpha \| E\| \omega - v\| \alpha +N(1 +H)e\nu r\| h\nu - \alpha \| E\| \omega - v\| \alpha
\bigr]
.
Therefore,
\| T\omega - Tv\| \alpha \leq l
1 - \| \Psi \|
\| \omega - v\| \alpha .
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1379
Since
l
1 - \| \Psi \|
< 1, we obtain that T is a contraction on the Banach space C\xi ,\alpha . Hence, the equation
T\omega = \omega has a unique solution \omega \in C\xi ,\alpha . From (2.13) we get
\| \omega \| \alpha \leq \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, N1\| e\nu - \alpha \| E\} Ne\nu rm(\xi )
1 - \| \Psi \| - l
.
We have therefore completed the proof of the existence of the solution u\ast = u + \omega of Eq. (2.2)
satisfying \widetilde u\ast t \in Ut for t \geq \xi and
\| ut - u\ast t \| \scrC = \| \omega t\| \scrC \leq e\nu re - \alpha (t - \xi )\| \omega \| \alpha \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, N1\| e\nu - \alpha \| E\} Ne\nu rm(\xi )e - \alpha (t - \xi )
1 - \| \Psi \| - l
=
=
\mathrm{m}\mathrm{a}\mathrm{x}\{ 1, N1\| e\nu - \alpha \| E\} Ne\nu re - \alpha (t - \xi )
1 - \| \Psi \| - l
\bigm\| \bigm\| \bigm\| \widetilde y\xi \Bigl( \widetilde P (\xi )u\xi \Bigr) - (I - \widetilde P (\xi ))u\xi \bigm\| \bigm\| \bigm\|
\scrC
=
= C\| u\xi - u\ast \xi \| \scrC for all t \geq \xi .
Theorem 2.3 is proved.
3. Exponential trichotomy and center-invariant unstable manifolds on \BbbR . In this section,
we will generalize Theorem 2.2 to the case that the evolution family (U(t, s))t\geq s has an exponen-
tial trichotomy on \BbbR and the nonlinear forcing term f is \varphi -Lipschitz. In this case, under similar
conditions as in above section we will prove that there exists a center-invariant unstable manifold of
\scrE -class for the solutions to Eq. (2.2). We now recall the definition of an exponential trichotomy and
a center-invariant unstable manifold of \scrE -class.
Definition 3.1. A given evolution family (U(t, s))t\geq s is said to have an exponential trichotomy
on \BbbR if there are three families of projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3, and positive constants N,
\alpha , \beta with \alpha < \beta such that the following conditions are fulfilled:
(i) \mathrm{s}\mathrm{u}\mathrm{p}t\in \BbbR \| Pj(t)\| <\infty , j = 1, 2, 3;
(ii) P1(t) + P2(t) + P3(t) = Id for t \in \BbbR and Pj(t)Pi(t) = 0 for all j \not = i;
(iii) Pj(t)U(t, s) = U(t, s)Pj(s) for t \geq s and j = 1, 2, 3;
(iv) U(t, s)| \mathrm{I}\mathrm{m}Pj(s) are homeomorphisms from \mathrm{I}\mathrm{m}Pj(s) onto \mathrm{I}\mathrm{m}Pj(t) for all t \geq s and j =
= 2, 3, respectively; we also denote the inverse of U(t, s)| \mathrm{I}\mathrm{m}P2(s) by U(s, t)| , s \leq t;
(v) for all t \geq s and x \in X, the following estimates hold:
\| U(t, s)P1(s)x\| \leq Ne - \beta (t - s)\| P1(s)x\| ,
\| U(s, t)| P2(t)x\| \leq Ne - \beta (t - s)\| P2(t)x\| ,
\| U(t, s)P3(s)x\| \leq Ne\alpha (t - s)\| P3(s)x\| .
The projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3, are called the trichotomy projections, and the constants
N, \alpha , \beta are the trichotomy constants.
Using the trichotomy projections we can now construct three families of projections
\bigl\{ \widetilde Pj(t)
\bigr\}
,
t \in \BbbR , j = 1, 2, 3, on \scrC as follows:
( \widetilde Pj(t)\phi )(\theta ) = U(t+ \theta , t)| (I - Pj(t))\phi (0) for all \theta \in [ - r, 0] and \phi \in \scrC . (3.1)
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1380 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
Definition 3.2. Let the evolution family (U(t, s))t\geq s have an exponential trichotomy with the
trichotomy projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3, and constants N, \alpha , \beta given as in Definition 3.1.
A set C \subset \BbbR \times \scrC is said to be a center-invariant unstable manifold of \scrE -class for the solutions
to Eq. (2.2) if there exists a family of Lipschitz continuous mappings
ft : \mathrm{I}\mathrm{m} \widetilde P1(t) \rightarrow \mathrm{I}\mathrm{m}
\Bigl( \widetilde P2(t) + \widetilde P3(t)
\Bigr)
with projections
\bigl\{ \widetilde Pj(t)
\bigr\}
, t \in \BbbR , j = 1, 2, 3, defined as in Eq. (3.1), and Lipschitz constants being
independent of t such that Ct = \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}(ft) has the following properties:
(i) Ct is homeomorphic to \mathrm{I}\mathrm{m} \widetilde P1(t) for all t \in \BbbR .
(ii) To each t0 \in \BbbR , \phi \in Ct0 there corresponds one and only one solution u(t) to Eq. (2.2) on
( - \infty , t0] satisfying e - \gamma (t0+\theta )Fut0+\theta = \phi (\theta ) for \theta \in [ - r, 0] and
z(t) =
\left\{ e
- \gamma (t+\cdot )ut(\cdot ) for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs to \scrE , where \gamma =
\beta + \alpha
2
.
Moreover, for any two solutions u(t) and v(t) to Eq. (2.2) corresponding to different functions \phi ,
\psi \in Ct0 , we have the estimate
\| ut - vt\| \scrC \leq C\mu e
(\gamma - \mu )(t0 - t)
\bigm\| \bigm\| \bigm\| ( \widetilde P1(t0)\phi )(0) - ( \widetilde P1(t0)\psi )(0)
\bigm\| \bigm\| \bigm\| for t \leq t0,
where \mu , C\mu are positive constants independent of t0, u(\cdot ), and v(\cdot ).
(iii) C is positively F -invariant under Eq. (2.2) in the sense that, if u(t), t \leq t0, is the solution
to Eq. (2.2) satisfying the conditions that the function e - \gamma (t0+\cdot )\~ut0(\cdot ) \in Ct0 and
z(t) =
\left\{ e
- \gamma (t+\cdot )ut(\cdot ) for t \leq t0,
0 for t > t0,
t \in \BbbR , belongs to \scrE ,
then the function e - \gamma (t+\cdot )\~ut(\cdot ) \in Ct for all t \leq t0, where \~ut(\theta ) = Fut0+\theta for all - r \leq \theta \leq 0.
We come to our second main result. It proves the existence of a center-unstable manifold of
\scrE -class for solutions of Eq. (2.2).
Theorem 3.1. Let the evolution family (U(t, s))t\geq s have an exponential trichotomy with the
trichotomy projections \{ Pj(t)\} , t \in \BbbR , j = 1, 2, 3, and constants N, \alpha , \beta given as in Defi-
nition 3.1. Assume Standing Hypothesis 1.1 and let the functions \varphi , h\nu , e\nu , and the operators
F, f be determined as in Theorem 2.1 and e\nu = e - \nu | t| for all t \in \BbbR . Set q := \mathrm{s}\mathrm{u}\mathrm{p}\{ \| Pj(t)\| :
t \in \BbbR , j = 1, 3\} , N0 := \mathrm{m}\mathrm{a}\mathrm{x}\{ N, 2Nq\} , \nu 0 =
\beta - \alpha
2
and
\widetilde k := (1 +H)N0e
\nu 0r\| h\nu 0\| E .
Then, if
\mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
N0(1 +H)e\nu 0r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - \| \Psi \|
,
N2
0N1(1 +H)e\nu 0r\| e\nu 0\| E\| \varphi \| E\prime
1 - \widetilde k - \| \Psi \|
\Biggr\}
< 1
for each fixed \beta > \alpha , there exists a center-invariant unstable manifold of \scrE -class for the solutions
to Eq. (2.2).
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1381
Proof. Set P (t) := P1(t) and Q(t) := P2(t) + P3(t) = Id - P (t) for t \in \BbbR . We have
that P (t) and Q(t) are projections complemented to each other on X. Then we define the families
of projections \{ \widetilde Pj(t)\} , t \in \BbbR , j = 1, 2, 3, on \scrC as in Eq. (3.1). Setting \widetilde P (t) = \widetilde P1(t) and\widetilde Q(t) = \widetilde P2(t) + \widetilde P3(t), t \in \BbbR , we obtain that \widetilde P (t) and \widetilde Q(t) are complemented projections on \scrC for
each t \in \BbbR . We consider the following rescaling evolution family:
\widetilde U(t, s) = e - \gamma (t - s)U(t, s) for all t \geq s, where \gamma :=
\beta + \alpha
2
.
We now prove that the evolution family \widetilde U(t, s) has an exponential dichotomy with dichotomy
projections P (t), t \in \BbbR . Indeed,
P (t)\widetilde U(t, s) = e - \gamma (t - s)P1(t)U(t, s) = e - \gamma (t - s)U(t, s)P1(s) = \widetilde U(t, s)P (s).
Since U(t, s)| \mathrm{I}\mathrm{m}Pj(s) is a homeomorphism from \mathrm{I}\mathrm{m}Pj(s) onto \mathrm{I}\mathrm{m}Pj(t) for t \geq s, j = 2, 3, and
\mathrm{I}\mathrm{m}(P2(t) +P3(t)) = \mathrm{K}\mathrm{e}\mathrm{r}P (t) for all t \in \BbbR , we have that \widetilde U(t, s)| \mathrm{K}\mathrm{e}\mathrm{r}P (s) is also a homeomorphism
from KerP (s) onto KerP (t), and we denote \widetilde U(s, t)| := (\widetilde U(t, s)| \mathrm{K}\mathrm{e}\mathrm{r}P (s))
- 1 for s \leq t. By the
definition of exponential trichotomy we obtain\bigm\| \bigm\| \bigm\| \widetilde U(t, s)P (s)x
\bigm\| \bigm\| \bigm\| \leq e - (\beta +\gamma )(t - s)\| P (s)x\| for all t \geq s.
On the other hand, \bigm\| \bigm\| \bigm\| \widetilde U(s, t)| Q(t)x
\bigm\| \bigm\| \bigm\| = e - \gamma (t - s)
\bigm\| \bigm\| U(s, t)| (P2(t) + P3(t))x
\bigm\| \bigm\| \leq
\leq Ne - \gamma (t - s)(e - \beta (t - s)\| P2(t)x\| + e\alpha (t - s)\| P3(t)x\| ) =
= Ne - \gamma (t - s)(e - \beta (t - s)\| P2(t)Q(t)x\| + e\alpha (t - s)\| P3(t)Q(t)x\| )
for all t \geq s and x \in X.
Putting q := \mathrm{s}\mathrm{u}\mathrm{p}\{ \| Pj(t)\| , t \in \BbbR , j = 2, 3\} , we finally get the following estimate:\bigm\| \bigm\| \bigm\| \widetilde U(s, t)| Q(t)x
\bigm\| \bigm\| \bigm\| \leq 2Nqe -
\beta - \alpha
2
(t - s)\| Q(t)x\| .
Therefore, \widetilde U(t, s) has an exponential dichotomy with the dichotomy projections P (t), t \geq 0, and
constants N0 := \mathrm{m}\mathrm{a}\mathrm{x}\{ N, 2Nq\} , \nu 0 :=
\beta - \alpha
2
.
Put \widehat u(t) = e - \gamma tu(t), and define the mapping \widetilde f : \BbbR \times \scrC \rightarrow X as follows:
\widetilde f(t, \phi ) = e - \gamma tf(t, e\gamma (t+\cdot )\phi (\cdot )) for (t, \phi ) \in \BbbR \times \scrC .
Obviously, \widetilde f is also \varphi -Lipschitz. Thus, we can rewrite Eq. (2.2) in the new form
F \widehat ut = \widetilde U(t, s)F \widehat us + t\int
s
\widetilde U(t, \xi ) \widetilde f(\xi , \widehat u\xi )d\xi for all t \geq s,
\widehat us(\cdot ) = e - \gamma (s+\cdot )\phi (\cdot ) \in \scrC .
(3.2)
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1382 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
Hence, by Theorem 2.2, we obtain that if
\mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
N0(1 +H)e\nu 0r(N1 +N2)\| \Lambda 1\varphi \| \infty
1 - \| \Psi \|
,
N2
0N1(1 +H)e\nu 0r\| e\nu 0\| E\| \varphi \| E\prime
1 - \widetilde k - \| \Psi \|
\Biggr\}
< 1,
then there exists an invariant unstable manifold of \scrE -class U for the solutions to Eq. (3.2). Returning
to Eq. (2.2) by using the relation u(t) := e\gamma t\widehat u(t) and Theorems 2.1, 2.2, we can easily verify the
properties of C which are stated in (i), (ii) and (iii) in Definition 3.2. Thus, C is a center-invariant
unstable manifold of \scrE -class for the solutions of Eq. (2.2).
4. Examples.
Example 4.1. Consider the finite delayed heat equation for a material with memory which has
formula
\partial
\partial t
u(t, x) = m(t)
\partial 2
\partial x2
\left[ u(t, x) + t\int
t - 1
(t - s)(t - s - 1)u(s, x)ds
\right] +
+
t\int
t - 1
[ - 2(t - s) + 1]u(s, x)ds+ a(t)
t\int
t - 1
\mathrm{l}\mathrm{n}(1 + | u(s, x)| )ds,
(4.1)
u(t, 0) = u(t, \pi ) = 0, t \geq s,
us(\theta , x) = u(s+ \theta , x) = \psi (\theta , x), x \in [0, \pi ], \theta \in [ - 1, 0],
where a(t) is defined by a(t) = | l| e - \eta | t| , \eta > 1 and l \not = 0, the given function \psi is continuous. The
function m(\cdot ) \in L1,\mathrm{l}\mathrm{o}\mathrm{c}(\BbbR ) and satisfies the condition m1 \geq m(t) \geq m0 > 0 for fixed constants m0,
m1 and a.e. t \in \BbbR .
We choose the Hilbert space X = L2[0, \pi ], and let A : X \rightarrow X be defined by
A(v) = v\prime \prime
with the domain D(A) =
\bigl\{
v \in W 2,2[0, \pi ] : v(0) = v(\pi ) = 0
\bigr\}
.
Also, for \scrC = C([ - 1, 0], X), we define the difference and delay operators F and f as
F : \scrC \rightarrow X, F (v) = v(0) +
0\int
- 1
b( - \theta )v(\theta )d\theta ,
f : \BbbR \times \scrC \rightarrow X, f(t, \phi ) = | l| e - \eta | t|
0\int
- 1
\mathrm{l}\mathrm{n}(1 + | (\phi (\theta ))(x)| )d\theta , t \in \BbbR , \theta \in [ - 1, 0]. (4.2)
It is obvious that
b(t) = t(t - 1) satisfies b(0) = b(1) = 0,
F = \delta 0 +\Psi , here \Psi (\cdot ) = -
\int 0
- 1
b( - \theta ) \cdot (\theta )d\theta with \| \Psi \| =
\int 0
- 1
| \theta (\theta - 1)| d\theta = 5
6
< 1.
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ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1383
Note that the fact that f takes value in X = L2[0, \pi ] can be easily seen by using the Minkowskii
inequality.
Putting now A(t) = m(t)A, u(t) = u(t, \cdot ), t \in \BbbR and \phi (\theta ) = \psi (\theta , \cdot ); \theta \in [ - 1, 0] the Eq. (4.1)
can now be rewritten as
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \geq s, t, s \in \BbbR ,
us = \phi \in \scrC := C([ - 1, 0], X).
From the definition of A, it can easily seen that A is the generator of an analytic semigroup
(T (t))t\geq 0 =
\bigl(
etA
\bigr)
t\geq 0
with \sigma (A) =
\bigl\{
- 1, - 4, . . . , - n2, - (n+ 1)2, . . .
\bigr\}
and \sigma (A) \cap i\BbbR = \varnothing .
Applying the spectral mapping theorem for analytic semigroups, we get
\sigma (T (t)) = et\sigma (A) =
\Bigl\{
e - t, e - 4t, . . . , e - n2t, . . .
\Bigr\}
and \sigma (T (t)) \cap \{ z \in \BbbC : | z| = 1\} = \varnothing for all t > 0. Therefore, for fixed t0 > 0, the spec-
trum of operator T (t0) splits into two disjoint sets \sigma 0, \sigma 1, where \sigma 0 \subset \{ z \in \BbbC : | z| < 1\} , \sigma 1 \subset
\subset \{ z \in \BbbC : | z| > 1\} .
Next, we choose P = P (t0) to be the Riesz projections corresponding to spectral set \sigma 0, and
Q = Id - P. Clearly, P and Q commute with T (t) for all t \geq 0. We denote by TQ(t) = T (t)Q the
restriction of T (t) on \mathrm{I}\mathrm{m}Q. As known Semigroup Theory, in this case, the semigroup (T (t))t\geq 0 is
called hyperbolic (or having an exponential dichotomy) and restriction TQ(t) is invertible. Moreover,
there are positive constants N, \gamma such that
\| T (t)| PX\| \leq Ne - \gamma t, (4.3)
\| TQ( - t)\| \leq \| TQ(t) - 1\| \leq Ne - \gamma t (4.4)
for all t \geq 0.
Clearly, the family (A(t))t\in \BbbR = (m(t)A)t\in \BbbR generates the evolution family (U(t, s))t\geq s defined
by the formula
U(t, s) = T
\left( t\int
s
m(\tau )d\tau
\right) for all t \geq s.
From the dichotomy estimates of (T (t))t\geq 0 in (4.3), it is straightforward to check that evolution
family (U(t, s))t\geq s has an exponential dichotomy with projection P and constants N, \nu = \gamma m0 by
the following estimates:
\| U(t, s)| PX\| = \| T (t - s)| PX\| \leq Ne - \nu (t - s),
\| U(s, t)| \| = \|
\bigl(
U(t, s)| \mathrm{K}\mathrm{e}\mathrm{r}P
\bigr) - 1\| = \| TQ( - (t - s))| \| \leq Ne - \nu (t - s)
for all t \geq s.
We now take E = Lp(\BbbR ), 1 \leq p \leq +\infty , the delay operator f : \BbbR \times \scrC \rightarrow X defined as in (4.2)
and check that f is \varphi -Lipschitz with \varphi (t) = | l| e - \eta | t| \in E\prime = Lq(\BbbR ) for
1
p
+
1
q
= 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1384 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
Indeed, the condition (i) in Definition 2.2 is evident. To verify the condition (ii) in that definition
we use Minkowskii inequality and the fact that \mathrm{l}\mathrm{n}(1 + h) \leq h for all h \geq 0. Then
\bigm\| \bigm\| f(t, \phi 1)(x) - f(t, \phi 2)(x)
\bigm\| \bigm\|
2
= | l| e - \eta | t|
\left( \pi \int
0
\left( 0\int
- 1
\mathrm{l}\mathrm{n}
1 + | (\phi 1(\theta ))(x)|
1 + | (\phi 2(\theta ))(x)|
d\theta
\right) 2dx
\right)
1
2
\leq
\leq | l| e - \eta | t|
0\int
- 1
\left( \pi \int
0
\mathrm{l}\mathrm{n}2
1 + | (\phi 1(\theta ))(x)|
1 + | (\phi 2(\theta ))(x)|
dx
\right) 1
2
d\theta =
= | l| e - \eta | t|
0\int
- 1
\left( \pi \int
0
\mathrm{l}\mathrm{n}2
\biggl(
1 +
| (\phi 1(\theta ))(x)| - | (\phi 2(\theta ))(x)|
1 + | (\phi 2(\theta ))(x)|
\biggr)
dx
\right) 1
2
d\theta \leq
\leq | l| e - \eta | t|
0\int
- 1
\left( \pi \int
0
| (\phi 1(\theta ))(x) - (\phi 2(\theta ))(x)| 2dx
\right) 1
2
d\theta =
= | l| e - \eta | t|
0\int
- 1
\| \phi 1(\theta ) - \phi 2(\theta )\| 2 d\theta \leq
\leq | l| e - \eta | t| \mathrm{s}\mathrm{u}\mathrm{p}
\theta \in [ - 1,0]
\| \phi 1(\theta ) - \phi 2(\theta )\| 2.
Hence, f is \varphi -Lipschitz. In the space Lp(\BbbR ), the constants N1, N2 are defined by N1 = N2 = 1.
We have
\| \varphi \| E\prime = | l|
\left( +\infty \int
- \infty
e - \eta q| t| dt
\right)
1
q
= | l|
\biggl(
2
\eta q
\biggr) 1
q
.
Also, the function h\nu (\cdot ) can be computed by
h\nu (t) =
\bigm\| \bigm\| e - \nu | t - \cdot | \varphi (\cdot )
\bigm\| \bigm\|
Lq
= | l|
\Biggl(
e - \nu q| t| - e - \eta q| t|
(\eta - \nu )q
+
e - \eta q| t| + e - \nu q| t|
(\eta + \nu )q
\Biggr) 1
q
for t \in \BbbR .
Therefore, h\nu \in Lp for
1
p
+
1
q
= 1 and
\| h\nu \| Lp \leq | l|
\biggl(
2\eta
q(\nu + \eta )(\eta - \nu )
\biggr) 1
q
\biggl(
4
\nu p
\biggr) 1
p
.
We have \| e\nu \| Lp =
\biggl(
2
\nu p
\biggr) 1
p
and \Lambda 1\varphi (t) =
\int t+1
t
\varphi (\tau )d\tau . Thus, \| \Lambda 1\varphi \| \infty \leq | l| (e\eta - 1)
\eta
.
By Theorem 2.2, we obtain that if
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1385
12N(1 +H)e\nu r\times
\times \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
e\eta - 1
\eta
,
N
(\nu p)
1
p (\eta q)
1
q
\Biggl(
1 - 6N(1 +H)| l|
\biggl(
2\eta
q(\nu + \eta )(\nu - \eta )
\biggr) 1
q
\biggl(
4
\nu p
\biggr) 1
p
\Biggr)
\right\} < 1,
then there exists an unstable manifold of \scrE -class U for the mild solutions to problem (4.1), and this
manifold has the attraction property given in Theorem 2.3.
Example 4.2. Consider the above Example 4.1, in Eq. (4.1) we replace the boundary condition
by
u\prime x(t, 0) = u\prime x(t, \pi ) = 0, t \geq s.
Then we choose the Hilbert space X = L2[0, \pi ], and let A : X \rightarrow X be defined by
A(v) = v\prime \prime
with the domain D(A) =
\bigl\{
v \in W 2,2[0, \pi ] : v\prime (0) = v\prime (\pi ) = 0
\bigr\}
.
Putting now A(t) = m(t)A, u(t) = u(t, \cdot ), t \in \BbbR , and \phi (\theta ) = \psi (\theta , \cdot ), \theta \in [ - 1, 0], the Eq. (4.1)
can now be rewritten as
\partial
\partial t
Fut = A(t)Fut + f(t, ut), t \geq s, t, s \in \BbbR ,
us = \phi \in \scrC := C([ - 1, 0], X).
(4.5)
From the definition of A, it can easily seen that A is the generator of an analytic semigroup
(T (t))t\geq 0 =
\bigl(
etA
\bigr)
t\geq 0
with \sigma (A) =
\bigl\{
0, - 1, - 4, . . . , - n2, - (n+ 1)2, . . .
\bigr\}
. Applying the spectral
mapping theorem for analytic semigroups we get
\sigma (T (t)) = et\sigma (A) =
\Bigl\{
e - t, e - 4t, . . . , e - n2t, . . .
\Bigr\}
\cup \{ 1\} .
Therefore, for fixed t0 > 0, the spectrum of operator T (t0) splits into three disjoint sets \sigma 1, \sigma 2, \sigma 3,
where \sigma 1 \subset \{ z \in \BbbC : | z| < 1\} , \sigma 2 \subset \{ z \in \BbbC : | z| > 1\} , \sigma 3 \subset \{ z \in \BbbC : | z| = 1\} .
Next, we choose P1 = P1(t0), P2 = P2(t0), P3 = P3(t0) to be the Riesz projections correspon-
ding to spectral set \sigma 1, \sigma 2, \sigma 3. Clearly, P1, P2, P3 commute with T (t) for all t \geq 0. We can see that
P1 + P2 + P3 = Id and PiPj = 0 \forall i \not = j. Moreover, there exist are positive constants M, \delta such
that
\| T (t)| P1X\| \leq Me - \delta t \forall t \geq 0.
We denote Q := P2 +P3 = Id - P1 and consider the semigroup on \mathrm{I}\mathrm{m}Q such that TQ(t) = T (t)Q
the restriction of T (t) on \mathrm{I}\mathrm{m}Q. Because \sigma 2 \cup \sigma 3 = \sigma (TQ(t0)) implies (TQ(t))t\geq 0 can be extended
into group (TQ(t))t\in \BbbR in \mathrm{I}\mathrm{m}Q. Moreover, there exist positive constants K, \epsilon 0, and \epsilon 1, \epsilon 0 < \epsilon 1,
such that
\| TQ( - t)| P2X\| = \|
\bigl(
TQ(t)| P2X
\bigr) - 1\| \leq Ke - \epsilon 1t \forall t \geq 0,
\| TQ(t)| P3X\| \leq Ke\epsilon 0| t| \forall t \in \BbbR .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1386 NGUYEN THIEU HUY, VU THI NGOC HA, TRINH XUAN YEN
Thus, the semigroup (T (t))t\geq 0 having an exponential trichotomy with the trichotomy projections
\{ Pj\} , j = 1, 2, 3, and constants N, \epsilon 0, \beta 0 satisfies
\| T (t)| P1X\| \leq Ne - \beta 0t,
\| T ( - t)| P2X\| = \|
\bigl(
T (t)| P2X
\bigr) - 1\| \leq Ne - \beta 0t, (4.6)
\| T (t)| P3X\| \leq e\epsilon 0t,
where N := max\{ K,M\} , \beta 0 := min\{ \delta , \epsilon 1\} .
Clearly, the family (A(t))t\in \BbbR = (m(t)A)t\in \BbbR generates the evolution family (U(t, s))t\geq s defined
by the formula
U(t, s) = T
\left( t\int
s
m(\tau )d\tau
\right) for all t \geq s.
From the trichotomy estimates of (T (t))t\geq 0 in (4.2), it is straightforward to check that evolution
family (U(t, s))t\geq s has an exponential trichotomy with projections Pk, k = 1, 2, 3, and trichotomy
constants N, \beta := \epsilon 1m0, \alpha := \epsilon 0m0 by the following estimates:
\| U(t, s)| P1X\| = \| T (t - s)| P1X\| \leq Ne - \beta (t - s),
\| U(s, t)| \| = \|
\bigl(
U(t, s)| P2X
\bigr) - 1\| \leq Ne - \beta (t - s),
\| U(t, s)| P3X\| = \| T (t - s)| P3X\| \leq Ne\alpha (t - s)
for all t \geq s.
Set q := \mathrm{s}\mathrm{u}\mathrm{p}\{ \| Pj(t)\| : t \in \BbbR , j = 1, 3\} , N0 := \mathrm{m}\mathrm{a}\mathrm{x}\{ N, 2Nq\} , \nu 0 =
\beta - \alpha
2
. By Theorem 3.1
and result in the Example 4.1, we obtain that if
12N0(1 +H)e\nu 0r\times
\times \mathrm{m}\mathrm{a}\mathrm{x}
\left\{
e\eta - 1
\eta
,
N0
(\nu 0p)
1
p (\eta q)
1
q
\Biggl(
1 - 6N0(1 +H)| l|
\biggl(
2\eta
q(\nu 0 + \eta )(\nu 0 - \eta )
\biggr) 1
q
\biggl(
4
\nu 0p
\biggr) 1
p
\Biggr)
\right\} < 1,
then there exists a center-invariant unstable manifold of \scrE -class C for the mild solutions to prob-
lem (4.5).
References
1. Nguyen Thieu Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, J. Different. Equat.,
246, 1820 – 1844 (2009).
2. Nguyen Thieu Huy, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a
half-line, J. Math. Anal. and Appl., 354, 372 – 386 (2009).
3. Nguyen Thieu Huy, Pham Van Bang, Hyperbolicity of solution semigroups for linear neutral differential equations,
Semigroup Forum, 84, 216 – 228 (2012).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL-DIFFERENTIAL . . . 1387
4. Nguyen Thieu Huy, Pham Van Bang, Unstable manifolds for partial neutral differential equations and admissibility
of function spaces, Acta Math. Vietnam, 42, 187 – 207 (2017).
5. N. T. Huy, V. T. N. Ha, Admissible integral manifolds for semi-linear evolution equations, Ann. Polon. Math., 112,
127 – 163 (2014).
6. N. T. Huy, T. V. Duoc, D. X. Khanh, Attraction property of admissible integral manifolds and applications to
Fisher – Kolmogorov model, Taiwanese J. Math., 20, 365 – 385 (2016).
7. J. J. Massera, J. J. Schäffer, Linear differential equations and function spaces, Acad. Press, New York (1966).
8. N. V. Minh, F. Räbiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy
of evolution equations on the half line, Integral Equations and Operator Theory, 32, 332 – 353 (1998).
9. N. V. Minh, J. Wu, Invariant manifolds of partial functional differential equations, J. Different. Equat., 198, 381 – 421
(2004).
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13. J. Wu, Theory and applications of partial functional differential equations, Springer-Verlag (1996).
Received 04.08.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
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| id | umjimathkievua-article-6257 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:46Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/80/79b2d1a2481f0e3606fa995dad314e80.pdf |
| spelling | umjimathkievua-article-62572022-12-17T13:00:55Z Admissible integral manifolds for partial neutral functional-differential equations ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS Admissible integral manifolds for partial neutral functional-differential equations Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan ADMISSIBLE INTEGRAL UDC 517.9 We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space $X$&nbsp; of the form&nbsp;\begin{align*}&amp; \dfrac{\partial}{\partial t}Fu_t= A(t)Fu_t +f(t,u_t),\quad t \ge s,\quad t,s\in\mathbb{R},\\&amp; u_s=\phi\in\mathcal{C}:= C([-r, 0], X)\end{align*}&nbsp;under the conditions that the family of linear partial differential operators $\left(A(t)\right)_{t\in\mathbb{R}}$ generates the evolution family $\left(U(t,s)\right)_{t\geq s}$ with an exponential dichotomy on the whole line $\mathbb{R};$&nbsp;&nbsp;the difference operator&nbsp; $ F\colon\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator $f$ satisfies the $\varphi$-Lipschitz condition, i.e.,&nbsp;$ \|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$ for $\phi,\psi \in\mathcal{C},$ where $\varphi(\cdot)$ belongs to an admissible function space defined on $\mathbb{R}.$&nbsp;&nbsp;We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates.&nbsp;&nbsp;Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces.&nbsp;&nbsp;We&nbsp; apply our results to the finite-delayed heat equation for a material with memory.&nbsp; УДК 517.9 Допустимі інтегральні многовиди для&nbsp; нейтральних функціонально-диференціальних рівнянь Доведено існування та властивість притягання для допустимих інваріантних нестійких та центрально-нестійких многовидів допустимих класів розв’язків нейтрального функціонально-диференціального рівняння з частинними похідними&nbsp; в банаховому просторі $X$ вигляду&nbsp;\begin{align*}&amp; \dfrac{\partial}{\partial t}Fu_t= A(t)Fu_t +f(t,u_t),\quad t \ge s,\quad t,s\in\mathbb{R},\\&amp; u_s=\phi\in\mathcal{C}:= C([-r, 0], X)\end{align*} &nbsp;за умови, що множина лінійних операторів частинного&nbsp;&nbsp;диференціювання&nbsp; $\left(A(t)\right)_{t\in\mathbb{R}}$ породжує еволюційну множину $\left(U(t,s)\right)_{ t\geq s},$ що має експоненціальну дихотомію на всій прямій $\mathbb{R};$ різницевий оператор&nbsp;$ F\colon\mathcal{C}\to X$ є обмеженим і лінійним, а нелінійний оператор затримки $f$ задовольняє умову $\varphi$-Ліпшиця, тобто&nbsp;$ \|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$ для $\phi,\psi \in\mathcal{C},$ де $\varphi(\cdot)$ належить допустимому функціональному простору, визначеному на $\mathbb{R}.$&nbsp;&nbsp;Ми також доводимо, що&nbsp; нестійкий многовид з допустимого класу притягує всі інші розв'язки з експоненціальною швидкістю.&nbsp;&nbsp;Наш основний метод базується на рівнянні Ляпунова – Перрона в поєднанні з допустимістю функціональних просторів.&nbsp;&nbsp; Отримані результати застосовано до&nbsp; рівняння теплопровідності зі скінченною затримкою для матеріалу з пам’яттю. Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6257 10.37863/umzh.v74i10.6257 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1364 - 1387 Український математичний журнал; Том 74 № 10 (2022); 1364 - 1387 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6257/9313 Copyright (c) 2022 Thieu Huy Nguyen |
| spellingShingle | Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan Nguyen, Thieu Huy Ha, Vu Thi Ngoc Yen, Trinh Xuan Admissible integral manifolds for partial neutral functional-differential equations |
| title | Admissible integral manifolds for partial neutral functional-differential equations |
| title_alt | ADMISSIBLE INTEGRAL MANIFOLDS FOR PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS Admissible integral manifolds for partial neutral functional-differential equations |
| title_full | Admissible integral manifolds for partial neutral functional-differential equations |
| title_fullStr | Admissible integral manifolds for partial neutral functional-differential equations |
| title_full_unstemmed | Admissible integral manifolds for partial neutral functional-differential equations |
| title_short | Admissible integral manifolds for partial neutral functional-differential equations |
| title_sort | admissible integral manifolds for partial neutral functional-differential equations |
| topic_facet | ADMISSIBLE INTEGRAL |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6257 |
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