In this section, an existence result of antiperiodic solutions for (1.4) will be given.

Theorem 3.1.

Assume that

there exists a nonnegative function

such that

Then (1.4) has at least one antiperiodic solution.

Remark 3.2.

When
,
is equal to 1. It is easy to see that condition (
) in [15] is stronger than condition (
) of Theorem 3.1.

For making use of Leray-Schauder degree theory to prove the existence of antiperiodic solutions for (1.4), we consider the homotopic equation of (1.4) as follows:

Define the operator

by

Let

be the Nemytski operator

Obviously, the operator

is invertible and the antiperiodic problem of (3.3) is equivalent to the operator equation

We begin with some lemmas below.

Lemma 3.3.

Suppose that the assumption

is true. Then the antiperiodic solution

of (3.3) satisfies

where
is a positive constant only dependent of
and
.

Proof.

Multiplying the both sides of (3.3) with

and integrating it over

, we get

and

, we have

By hypothesis

, there exists a nonnegative constant

such that

Thus, from (3.11), we have

where
.

For each

, we get

Similarly, we obtain that

Basing on Lemma 2.1, it can be shown from (3.17) and (3.14) that

Let

, then

The proof is complete.

Lemma 3.4.

Suppose that the assumption

is true. Then, for the possible antiperiodic solution

of (3.3), there exists a prior bounds in

, that is,

satisfies

where
is a positive constant independent of
.

Proof.

By (3.15), there exists

such that

. Hence, (3.8) yields that

From (3.16), there exists

such that

, which implies that

. Therefore, integrating the both sides of (3.3) over

, we have

Thus, we get from (3.8) that

Noting that

, we obtain that

Combining (3.21) with (3.25), we have

where
. The proof is complete.

Now we give the proof of Theorem 3.1.

Proof of Theorem 3.1..

Obviously, the set
is an open-bounded set in
and zero element
.

From the definition of operator

, it is easy to see that

Hence, the operator

sends

into

. Let us define the operator

by

Obviously, the operator
is completely continuous in
and the fixed points of operator
are the antiperiodic solutions of (1.4).

With this in mind, let us define the completely continuous field

by

By (3.20), we get that zero element

for all

. So that, the following Leray-Schauder degrees are well defined and

Consequently, the operator
has at least one fixed point in
by using Lemma 2.2. Namely, (1.4) has at least one antiperiodic solution. The proof is complete.