In this section, some existence results on anti-periodic solutions with symmetry of (1.3) and (1.4) will be given.
Theorem 3.1
Assume that
(H1) the functions and are odd in t, i.e.,
(H2) there exist non-negative functions such that
(H3) .
Then (1.3) has at least one even anti-periodic solution , i.e., satisfies
Proof For making use of the Leray-Schauder degree theory to prove the existence of even anti-periodic solutions for (1.3), we consider the following homotopic equation of (1.3):
(3.1)
Define the operator by
Obviously, the operator is invertible. Let be the Nemytskii operator
By hypothesis (H1), it is easy to see that
Thus, the operator sends into . Hence, the problem of even anti-periodic solutions for (3.1) is equivalent to the operator equation
From hypotheses (H2), (H3) and (5) in [10], for the possible even anti-periodic solution of (3.1), there exists a prior bounds in , i.e., satisfies
(3.2)
where is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the even anti-periodic solutions of (1.3).
With this in mind, let us define the set as follows:
Obviously, the set is a open bounded set in and zero element . Define the completely continuous operator by
Let us define the completely continuous field by
By (3.2), we get that zero element for all . So, the following Leray-Schauder degrees are well defined and
Consequently, the operator has at least one fixed point in by using Lemma 2.1. Namely, (1.3) has at least one even anti-periodic solution. The proof is complete. □
Theorem 3.2
Assume that
(H4) the function is even in t, x and is even in t, i.e.,
and the assumptions (H2), (H3) are true.
Then (1.3) has at least one odd anti-periodic solution , i.e., satisfies
Proof We consider the homotopic equation (3.1) of (1.3). Define the operator by
Let be the Nemytskii operator
By hypothesis (H4), it is easy to see that
Thus, the operator sends into . Hence, the problem of odd anti-periodic solutions for (3.1) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the odd anti-periodic solutions of (1.3). With this in mind, let us define the following set:
Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 3.3
Assume that
(H5) the functions and are even in t, i.e.,
and the assumptions (H2), (H3) are true.
Then (1.4) has at least one even anti-periodic solution.
Proof We consider the homotopic equation of (1.4) as follows:
(3.3)
Define the operator by
Let be the Nemytskii operator
By hypothesis (H5), it is easy to see that
Thus, the operator sends into . Hence, the problem of even anti-periodic solutions for (3.3) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the even anti-periodic solutions of (1.4). With this in mind, let us define the following set:
where is a positive constant independent of λ. Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 3.4
Assume that
(H6) the function is odd in t, x and is odd in t, i.e.,
and the assumptions (H2), (H3) are true.
Then (1.4) has at least one odd anti-periodic solution.
Proof We consider the homotopic equation (3.3) of (1.4). Define the operator by
Let be the Nemytskii operator
By hypothesis (H6), it is easy to see that
Thus, the operator sends into . Hence, the problem of odd anti-periodic solutions for (3.3) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator which sends into , such that the fixed points of operator in some open bounded set are the odd anti-periodic solutions of (1.4). With this in mind, let us define the set as follows:
Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
When , we can remove the assumption (H2) in Theorem 3.1, Theorem 3.2 and obtain the following results.
Theorem 3.5
Assume that
(H7) and the assumption (H1) is true.
Then (1.3) () has at least one even anti-periodic solution.
Theorem 3.6 Suppose that the assumptions (H4), (H7) are true. Then (1.3) () has at least one odd anti-periodic solution.
Basing on the proof of Theorem 2 in [10], for the possible anti-periodic solution of (3.1) (), the hypothesis (H7) yields that there exists a prior bounds in , i.e., satisfies
where is a positive constant independent of λ. The remainder of the proof work of Theorem 3.5 and Theorem 3.6 is quite similar to the proof of Theorem 3.1 and Theorem 3.2, so we omit the details.