^{1}

^{1}

^{2}

In this paper, we consider an abstract non-autonomous evolution equation with multiple delays in a Hilbert space H: *u'*(*t*) + *Au*(*t*) = *F*(*u*(*t-r _{1}*

_{}),...,

*u*((

*t-r*

_{n}_{}))

*+*

*g*(

*t*), where

*A*:

*D*(

*A*)?

*H*→

*H*is a positive definite selfadjoint operator,

*F*:

*H*

^{n}

_{a}→

*H*is a nonlinear mapping, r

_{1},...,r

_{n}are nonnegative constants, and

*g*(

*t*)∈

*C*(□;

*H*) is bounded. Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.

In this paper, we consider the following non-autonomous evolution equation with multiple delays in a Hilbert space H:

where

This partial differential equations with delays (1.1) has extensive physical background and realistic mathematical model, hence it has been considerably developed and the numerous properties of their solutions have been studied, see [

In this paper, our aim is to study the existence and stability of synchronizing solution of Equation (1.1). Motivated by [

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.

This section consists of some preliminary work.

Let H be a Hilbert space with the inner product

be a positive definite selfadjoint operator with compact resolvent, and let

Be the eigenvalues of A (counting with multiplicity) with the corresponding eigenvectors

For

Let

Then,

respectively. We also know that for any

We recall some basic definitions and facts in the theory of non-autonomous dynamical systems for skew-product flows on complete metric spaces.

Let

1)

2)

Definition 2.1 A mapping

1)

2)

3)

The mapping

forms a semigroup on

Definition 2.2 A family

and pullback attracting, that is, for any bounded subset B of X,

and is the minimal family of compact sets that is both invariant and pullback attracting.

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

For

Consider the initial value problem of the evolution equation with delays

where

(H1) For all

(H3) For any

for all

and

Theorem 2.3 Assume that

Proof. The proof can be obtained by Theorem 5 in [

Remark 2.4

Let the space

It is well known that this topology is metrizable and

Give

So the shift operator

forms a continuous dynamical system on the base space

Define

where

Since Theorem 12 in [

Theorem 2.5 Let

In this section, we establish some results on synchronizing solutions for (1.1), by developing some techniques inspired by works [

Now, we consider that

If furthermore, the Lipschitz coefficients

then we have the following results about synchronizing solutions for (1.1).

Theorem 3.1 Assume

1) There exists a

2) For any

Proof. By Theorem 2.5, we have proved that the cocycle mapping

As Definition 2.2, it is

One can also write the non-autonomous invariance property as

In what follows we show that for each

for some

Let

We know that

where

where

Let

Taking inner product with

which yields that

where

where

Let

Since

Then, we can obtain that

which implies

Now define

We infer from Corollary 2.8 in [

By invariance property of

where

Thus we can deduce that

The proof is complete.

Corollary 3.2 Let

Proof. Let

This work was supported by NNSF (11261027), NNSF (11161026) and the Research Funds of Lanzhou City University (LZCU-BS2015-01).

Jinying Wei,Yongjun Li,Xiaohua Zhuo, (2016) The Existence and Stability of Synchronizing Solution of Non-Autonomous Equations with Multiple Delays. Journal of Applied Mathematics and Physics,04,1294-1299. doi: 10.4236/jamp.2016.47136