In this section, we will show our main results and prove them.

Theorem 3.1.

Assume that (H1) and (H2) hold. Moreover,
and the impulsive functions
are odd about
, then IBVP (1.1) has infinitely many classical solutions.

Proof.

Obviously,
is an even functional and
. We divide our proof into three parts in order to show Theorem 3.1.

Firstly, We will show that

satisfies the Palais-Smale condition. Let

be a bounded sequence such that

. Then there exists constants

such that

By (2.8), (2.9), (3.1), and (H1), we have

It follows that

is bounded in

. From the reflexivity of

, we may extract a weakly convergent subsequence that, for simplicity, we call

in

. In the following we will verify that

strongly converges to

in

. By (2.9) we have

By

in

, we see that

uniformly converges to

in

. So

By (3.3), (3.4), we obtain
as
. That is,
strongly converges to
in
, which means the that P. S. condition holds for
.

Secondly, we verify the condition (A1) in Theorem 2.3. Let

, then

, where

. In view of (H2), take

, there exists an

such that for every

with

,

Hence, for any

with

, by (2.8) and (3.5) , we have

Take
, then

Finally, we verify condition (A2) in Theorem 2.3. According to (H1), for any

and

we have that

for all

and

. This implies that

for all

and

. Similarly, we can prove that there is a constant

such that

for all

and

. Since

is continuous on

, there exists

such that

on

. Thus, we have

where
.

Similarly, there exist constants

such that

For every

and

, by (2.8), (3.9), and (3.10), we have that the following inequality:

holds. Take
such that
, since
, (3.11) implies that there exists
such that
and
for
. Since
is a finite dimensional subspace, there exists
such that
on
. By Theorem 2.3,
possesses infinite many critical points; that is, IBVP (1.1) has infinite many classical solutions.

Theorem 3.2.

Assume that (H1) and the first equality in (H2) hold. Moreover,
is odd about
and the impulsive functions
are odd and nonincreasing. Then IBVP (1.1) has infinitely many classical solutions.

Proof.

We only verify (A1) in Theorem 2.3. Since
are odd and nonincreasing continuous functions, then for any
,
. So we have
. Take
, like in (3.6) we can obtain the result.

Theorem 3.3.

Suppose that the first inequalities in (H1), (H3), and (H4) hold. Furthermore, one assumes that
and the impulsive functions
are odd about
and we have the following.

(H7)There exists
such that

Then IBVP (1.1) has infinitely many classical solutions.

Proof.

Obviously,

is an even functional and

. Firstly, we will show that

satisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the first inequalities in (H1) and (H4), we have

It follows that
is bounded in
. In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.

Secondly, as in Theorem 3.1, we can obtain that condition (A2) in Theorem 2.1 is satisfied.

Take the same direct sum decomposition

as in Theorem 3.1. For any

, by (2.8), (H3), and (H4), we obtain

In view of (H7), set

, then we have

Therefore,
By Theorem 2.3,
possesses infinite many critical points, that is, IBVP (1.1) has infinite many classical solutions.

Theorem 3.4.

Assume that the second inequalities in (H1), (H5), and (H6) hold, moreover, one assumes the following.

(H8) There exists

such that

Then IBVP (1.1) has at least two classical solutions.

Proof.

We will use Theorems 2.1 and 2.2 to prove the main results. Firstly, we will show that

satisfies the Palais-Smale condition. Similarly, as in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the second inequalities in (H1) and (H5), we have

It follows that
is bounded in
. In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.

Let

, which will be determined later. Set

, then

is a closed ball. From the reflexivity of

, we can easily obtain that

is bounded and weakly sequentially closed. We will show that

is weakly lower semicontinuous on

. Let

Then

. By

on

we see that

uniformly converges to

in

. So

is weakly continuous. Clearly,

is continuous, which, together with the convexity of

, implies that

is weakly lower semicontinuous. Therefore,

is weakly lower semi-continuous on

. So by Theorem 2.1, without loss of generality, we assume that

. Now we will show that

For any

, by (H5) and (H6), we have

In view of (H8), take
, we have
, for any
. So
.

Next we will verify that there exists a

with

such that

. Let

. Then by (3.10) and (H5), we have

Since
, we have
. Therefore, there exists a sufficiently large
with
such that
. Set
, then
. So by Theorem 2.2, there exists
such that
. Therefore,
and
are two critical points of
, and they are classical solutions of IBVP (1.1).

Remark 3.5.

Obviously, if
is a bounded function, in view of Theorem 3.4, we can obtain the same result.

Theorem 3.6.

Suppose that (H4) and (H5) hold. Then IBVP (1.1) has at least one solution.

Proof.

The proof is similar to that in [19], and we omit it here.

Corollary 3.7.

Suppose that
and impulsive functions
are bounded, then IBVP (1.1) has at least one solution.