Let
be a Banach space,
denotes the space of linear operators on
;
denotes the space of bounded linear operators on
.
is the Banach space with the usual supremum norm. Denote
and define
is continuous at
,
is continuous from left and has right-hand limits at
and
.

It can be seen that endowed with the norm
,
is a Banach space.

In order to investigate periodic solutions, we introduce the following two spaces:

It can be seen that endowed with the norm
,
is a Banach space.

We introduce assumption [H1].

[H1.1]:
is the infinitesimal generator of a
-semigroup
on
with domain
.

[H1.2]:There exists
such that
where
.

[H1.3]:For each
,
,
.

Under the assumption [H1], consider

and the associated Cauchy problem

For every

,

is an invariant subspace of

, using ([

38, Theorem 5.2.2, page 144]), step by step, one can verify that the Cauchy problem (2.5) has a unique classical solution

represented by

, where

given by

The operator
is called impulsive evolution operator associated with
and
.

The following lemma on the properties of the impulsive evolution operator
associated with
and
is widely used in this paper.

Lemma 2.1.

*Let assumption [H1]*
hold. The impulsive evolution operator
has the following properties.

(1)For
,
, there exists a
such that

(2)For
,
,
.

(3)For
,
,
.

(4)For
,
,
.

(5)For

, there exits

,

such that

Proof.

- (1)
By assumption [H1.1], there exists a constant

such that

. Using assumption [H1.3], it is obvious that

, for

. (2) By the definition of

-semigroup and the construction of

, one can verify the result immediately. (3) By assumptions [H1.2], [H1.3], and elementary computation, it is easy to obtain the result. (4) For

,

, by virtue of (3) again and again, we arrive at

- (5)
Without loss of generality, for

,

This completes the proof.

In order to study the asymptotical properties of periodic solutions, it is necessary to discuss the exponential stability of the impulsive evolution operator
. We first give the definition of exponential stable for
.

Definition 2.2.

,

is called exponentially stable if there exist

and

such that

Assumption [H2]:

is exponentially stable, that is, there exist

and

such that

An important criteria for exponential stability of a
-semigroup is collected here.

Lemma 2.3 (see [38, Lemma 7.2.1]).

*Let*
be a
-semigroup on
, and let
be its infinitesimal generator. Then the following assertions are equivalent:

(1)
is exponentially stable.

(2)For every

there exits a positive constants

such that

Next, four sufficient conditions that guarantee the exponential stability of impulsive evolution operator
are given.

Lemma 2.4.

Assumptions [H1]

and [H2]

hold. There exists

such that

Then,
is exponentially stable.

Proof.

Without loss of generality, for

, we have

Suppose

and let

Then,

Let
and
, then we obtain

Lemma 2.5.

*Assume that assumption [H1]*
holds. Suppose

If there exists

such that

for

where

*Then*,
is exponentially stable.

Proof.

It comes from (2.17) that

where
is denoted the number of impulsive points in
.

For

, by (2.16), we obtain the following two inequalities:

By (5) of Lemma 2.1, let
,
,

Lemma 2.6.

*Assume that assumption [H1]*
holds. The limit

*Suppose there exists*
such that

*Then*,
is exponentially stable.

Proof.

Let

with

. It comes from

that there exits a

enough small such that

From (2.27), we know that

Here, we only need to choose
small enough such that
, by (5) of Lemma 2.1 again, let
,
, we have

Lemma 2.7.

*Assume that assumption [H1]*
holds. For some

,

,

*Imply the exponential stability of*
.

Proof.

It comes from the continuity of

, the inequality

and the boundedness of
,
are convergent, that
for every
and fixed
. This shows that
is bounded for each
and fixed
and hence, by virtue of uniform boundedness principle, there exists a constant
such that
for all
. Let
denote the operator given by
,
and
is fixed. Clearly,
is defined every where on
and by assumption it maps
and it is a closed operator. Hence, by closed graph theorem, it is a bounded linear operator from
to
. Thus, there exits a constant
such that
for all
and
,
is fixed.

Thus, for

,

where

. Fix

. Then, for any

we can write

for some

and

and we have

where
and
. Since
, this shows that our result.