The Existence and Stability of Synchronizing Solution of Non-Autonomous Equations with Multiple Delays ()
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1. Introduction
In this paper, we consider the following non-autonomous evolution equation with multiple delays in a Hilbert space H:
(1.1)
where
is a positive definite selfadjoint operator with compact resolvent,
is a nonlinear mapping,
are nonnegative constants, and
is bounded.
In this paper, our aim is to study the existence and stability of synchronizing solution of Equation (1.1). Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. The result be of most interest when we choose
be translation compact (resp. recurrent or almost periodic or quasiperiodic or periosdic), then we can obtain the synchronizing solution of Equation (1.1) is also translation compact (resp. recurrent or almost periodic or quasiperiodic or periosdic). This result provides a general approach for guaranteeing the existence and stability of periodic, quasiperiodic, almost periodic or recurrent solution of the equation.
The rest of the paper is organized as follows. In Section 2, we provide some preliminaries. In Section 3, we establish the existence and stability of synchronizing solutions under some convergence condition.
2. Preliminaries
This section consists of some preliminary work.
2.1. Analytic Semigroups
Let H be a Hilbert space with the inner product
. We will use
to denote the norm of H and use
to denote the norm of bounded linear operators on H. Let
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be a positive definite selfadjoint operator with compact resolvent, and let
![](//html.scirp.org/file/68925x19.png)
Be the eigenvalues of A (counting with multiplicity) with the corresponding eigenvectors
which form a canonical basis of H.
For
, define the powers
as follows:
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Let
![]()
Then,
is a Hilbert space with the inner product
and norm
defined as
![]()
respectively. We also know that for any
, the embedding
is compact; moreover, it holds that
(2.1)
2.2. Pullback Attractors
We recall some basic definitions and facts in the theory of non-autonomous dynamical systems for skew-product flows on complete metric spaces.
Let
be a complete metric space,
be a metric space which will be called the base space (or symbol space).
is a mapping,
form a group, that is,
satisfies
1)
;
2)
.
Definition 2.1 A mapping
is said to be a continuous cocycle on X with respect to group
, if
1)
;
2)
and
;
3)
is continuous.
The mapping
defined by
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forms a semigroup on
and is called a skew-product flow.
Definition 2.2 A family
of nonempty compact sets of X is called a global pullback attractor of the cocycle
if it is
-invariant, that is,
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and pullback attracting, that is, for any bounded subset B of X,
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and is the minimal family of compact sets that is both invariant and pullback attracting.
2.3. Global Pullback Attractor of (1.1)
We present essential conditions on the nonlinearity F to guarantee the dissipation and the existence of pullback attractor of (1.1).
We first discuss the well-posedness of the initial value problem of the equation.
Let
, where
.
is endowed with the norm
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For
and
, we define
by
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Consider the initial value problem of the evolution equation with delays
(2.2)
where
is continuous and there exist positive constants
and
such that F satisfies the following conditions:
(H1) For all
and ![]()
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(H2) ![]()
(H3) For any
and bounded interval J, there exists
such that
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for all
and
, where (and hereafter)
denotes the ball in
centered at 0 with radius R;
and
is bounded, that is, there exists a positive constant
such that for all ![]()
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Theorem 2.3 Assume that
and F satisfies (H1)-(H3). Then, the problem (2.2) has a unique global mild solution
which depends on
continuously, and
![]()
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Proof. The proof can be obtained by Theorem 5 in [3].
Remark 2.4
satisfies the following integral equation:
. (2.3)
Let the space
be equipped with the compact-open topology:
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It is well known that this topology is metrizable and
is a complete metric space.
Give
is bounded, we define the base space
as
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So the shift operator
defined for each
by
,
forms a continuous dynamical system on the base space
.
Define
as follows:
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where
is the unique solution of the problem (2.2) with
. Then
is a cocycle system on
with the base space
and driving system
.
Since Theorem 12 in [3], we have the following existence result concerning the pullback attractors.
Theorem 2.5 Let
. If F satisfies conditions (H1)-(H3), then
has a unique global pullback attractor
.
3. Synchronizing Solutions
In this section, we establish some results on synchronizing solutions for (1.1), by developing some techniques inspired by works [2] and [1]. It is known that if g has some special structure, i.e., periodic, quasiperiodic, almost periodic etc., then we can obtain a compact base space with same structure. Combined with the theory of uniform pullback attractors for dynamical systems in [6], we will prove that under some convergence condition, Equation (1.1) have some entire solution
that synchronize with the motion of the driving system. We call
synchronizing solutions for (1.1).
Now, we consider that
is translation compact, then the base space
is compact.
If furthermore, the Lipschitz coefficients
of F in the set
satisfy:
, (H4)
then we have the following results about synchronizing solutions for (1.1).
Theorem 3.1 Assume
, and F satisfies (H1)-(H4). Let
is translation compact in
. Then:
1) There exists a
such that for each
,
is the unique bounded entire solution of (1.1) on
;
2) For any
, there exists a unique solution
of (1.1) on
with initial value
that satisfies
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Proof. By Theorem 2.5, we have proved that the cocycle mapping
has a pullback attractor
, and we know that
is bounded. So
is given as the union all bounded entire solution.
As Definition 2.2, it is
-invariant, that is,
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One can also write the non-autonomous invariance property as
(3.1)
In what follows we show that for each
,
is in fact a singleton, i.e.,
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for some
.
Let
. By invariance property (3.1), for any
there exist
such that
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We know that
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where
is the solution of (2.2) with initial value
, and
![]()
where
is the solution of (2.2) with initial value
.
Let
, we have
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Taking inner product with
and using Hӧlder’s inequality,
, Poincáre’s inequality and Young’s inequality, we have
(3.2)
which yields that
(3.3)
where
. Integrating from 0 to t, we obtain
(3.4)
where
.
Let
. We obain
(3.5)
Since
and (H4), we have
. Then by Gronwall’s lemma we have
(3.6)
Then, we can obtain that
(3.7)
which implies
as
Hence,
.
Now define
as
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We infer from Corollary 2.8 in [6] that
is upper semi-continuous in
. This reduces to the continuity of
in
when the
are single point sets. Hence,
is continuous. For each
, set
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By invariance property of
one trivially checks that
is precisely the unique solution of (1.1) on
. Since Theorem 4.3 in [5],
is a uniform pullback attractor. That is, for any
, we have
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where
denotes the semi-Hausdorff distance in
. Then, it is uniformly forwards attracting,
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Thus we can deduce that
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The proof is complete.
Corollary 3.2 Let
is periodic (resp. quasiperiodic, almost periodic, recurrent), then under con- ditions of Theorem 3.1 the non-autonomous Equation (1.1) admits a unique periodic (resp. quasiperiodic, almost periodic, recurrent) solution
and every other solution of this equation are asymptotically periodic (resp. asymptotically quasiperiodic, asymptotically almost periodic, asymptotically recurrent).
Proof. Let
be a function from Theorem 3.1, then according to this theorem we conclude that
is the unique bounded solution of (1.1) synchronizing with the motion
of the driving system
. In particular, if
is periodic (resp. quasiperiodic, almost periodic, recurrent), then so is
By (2) of Theorem 3.1, we know that every other solution of this equation is asymptotically periodic (resp. asymptotically quasiperiodic, asymptotically almost periodic, asymptotically recurrent). The proof is complete.
Acknowledgements
This work was supported by NNSF (11261027), NNSF (11161026) and the Research Funds of Lanzhou City University (LZCU-BS2015-01).