- Open Access
Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian
© Yang and Du; licensee Springer 2012
- Received: 24 April 2012
- Accepted: 8 November 2012
- Published: 27 November 2012
By means of the generalized Mawhin’s continuation theorem, we present some sufficient conditions which guarantee the existence of at least one T-periodic solution for a generalized Liénard neutral functional differential system with p-Laplacian.
- periodic solutions
- neutral equations
- generalized Mawhin’s continuation theorem
with ; with ; τ is a given constant; is a real matrix with .
However, when B of (1.1) is a matrix, there are few existence results of periodic solutions for neutral differential systems. In , when B is a symmetric matrix, the authors studied a second-order p-Laplacian neutral functional differential system and obtained the existence of periodic solutions. In , when B is a general matrix, the authors studied a second-order neutral differential system. But for p-Laplacian functional differential system, to the best our knowledge, there are no results on the existence of periodic solutions. Hence, in this paper, we will study system (1.1) and obtain the existence of periodic solutions by using the generalization of Mawhin’s continuation theorem.
Furthermore, we suppose that with , . It is obvious that the function has a unique inverse denoted by .
Lemma 2.1 ()
- (2)For all , , , where
, , for all , .
Definition 2.1 ()
is a closed subset of Z;
is linearly homeomorphic to , .
Definition 2.2 ()
, , ,
where is the complement space of KerM in X, i.e., ; P, Q are two projectors satisfying , , , .
Lemma 2.2 ()
is quasi-linear and , is M-compact in . In addition, if the following conditions hold:
(A1) , ;
(A2) , ;
(A3) , is a homeomorphism.
Then the abstract equation has at least one solution in .
Lemma 2.3 ()
Let with and . Suppose that the function has a unique inverse , . Then .
Lemma 2.4 ()
- (1)For any fixed , there must be a unique such that the equation
The function defined as above is continuous and sends bounded sets into bounded sets.
Lemma 2.5 ()
Since , ImM is a closed set in Z, then we have the following.
Lemma 3.1 Let M be as defined by (3.2), then M is a quasi-linear operator.
Lemma 3.2 If f, g, e, γ satisfy the above conditions, then is M-compact.
where F is defined by (3.4) and is a constant vector in which depends on x. By Lemma 2.4, we know that exists uniquely. Hence, is well defined.
Hence, is equicontinuous on . By using the Arzelà-Ascoli theorem, we have is completely continuous on .
Secondly, we show that is M-compact in four steps, i.e., the conditions of Definition 2.2 are all satisfied.
Step 2. We show that , , . Because , we get , i.e., . The inverse is true.
Hence, is M-compact in . □
Theorem 3.3 Suppose that , are eigenvalues of the matrix B with , , and there exist positive constants , and such that
(H1) , , , for each ,
(H2) , , for each ,
(H3) , , for each .
where is defined by (3.14).
Proof We complete the proof in three steps.
Step 2. Let , we shall prove that is a bounded set. , then , , we have for each . By assumption (H1) we have and . So, is a bounded set.
Applying Lemma 2.2, we complete the proof. □
Remark Assumption (H1) guarantees that condition (A2) of Lemma 2.2 is satisfied. Furthermore, using assumptions (H1)-(H3), we can easily estimate prior boud of the solution to Eq. (1.1).
, , , , .
By using Theorem 3.3, we know that Eq. (3.25) has at least one 2π-periodic solution.
This work was supported by NSF of Jiangsu education office (11KJB110002), Postdoctoral Fundation of Jiangsu (1102096C), Postdoctoral Fundation of China (2012M511296) and Jiangsu province fund (BK2011407).
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