Abstract
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.
MSC:34B15, 34L30.
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1 Introduction
This paper is devoted to investigating the following p-Laplacian Liénard neutral differential system:
where ;
with ; with ; τ is a given constant; is a real matrix with .
When the matrix B is a constant, Zhang [1] studied the properties of a difference operator A and obtained the following results: define the operator A on
where , c is a constant. If , then A has a unique continuous bounded inverse satisfying
On the basis of Zhang’s work, Lu [2] further studied the properties of the difference operator A and gave the following inequality properties for A:
-
(1)
;
-
(2)
, ;
-
(3)
, .
After that, by using the above results, many researchers studied the existence of periodic solutions for some kinds of differential equations; see [3–7]. In a recent paper [8], when the constant c of (1.2) is a variable , we generalized the results of [1] and obtained the following results. If , then the operator A has continuous inverse on , satisfying
-
(1)
-
(2)
Using the above results, we have obtained some existence results of periodic solutions for first-order, second-order and p-Laplacian neutral equations with a variable parameter; see [9–11].
However, when B of (1.1) is a matrix, there are few existence results of periodic solutions for neutral differential systems. In [12], 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 [13], 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.
2 Main lemmas
In this section, we give some notations and lemmas which will be used in this paper. Let
with the norm , with the norm
U is a complex such that
is a Jordan’s normal matrix, where
with , is the set of eigenvalues of matrix B. Let
Furthermore, we suppose that with , . It is obvious that the function has a unique inverse denoted by .
Lemma 2.1 ([13])
Suppose that the matrix U and the operator are defined by (2.1) and (2.2), respectively, and for all , . Then has its inverse with the following properties:
-
(1)
, .
-
(2)
For all , , , where
and is a constant with .
-
(3)
, , for all , .
Definition 2.1 ([14])
Let X and Z be two Banach spaces with norms , , respectively. A continuous operator
is said to be quasi-linear if
-
(i)
is a closed subset of Z;
-
(ii)
is linearly homeomorphic to , .
Definition 2.2 ([14])
Let be an open and bounded set with the origin , , is said to be M-compact in if there exists a subset of Z satisfying and an operator being continuous and compact such that for ,
-
(a)
,
-
(b)
, , ,
-
(c)
and ,
-
(d)
, ,
where is the complement space of KerM in X, i.e., ; P, Q are two projectors satisfying , , , .
Lemma 2.2 ([14])
Let X and Z be two Banach spaces with norms , , respectively and be an open and bounded nonempty set. Suppose
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 ([15])
Let with and . Suppose that the function has a unique inverse , . Then .
For fixed and , define
Lemma 2.4 ([16])
The function has the following properties:
-
(1)
For any fixed , there must be a unique such that the equation
-
(2)
The function defined as above is continuous and sends bounded sets into bounded sets.
Lemma 2.5 ([17])
Let be a constant, such that , . Then
3 Main results
For convenience of applying Lemma 2.2, the operators A, M, are defined by
where . For convenience of the proof, let
then . By (3.1)-(3.3), Eq. (1.1) is equivalent to the operator equation , where . Then we have
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.
Let
Lemma 3.2 If f, g, e, γ satisfy the above conditions, then is M-compact.
Proof Let . For any bounded set , define ,
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.
We first show that is completely continuous on . Let
we have
From the properties of f, g, e, γ, obviously, , . Then by Lemma 2.1 is uniformly bounded on . Now, we show is equicontinuous. , is sufficiently small, then there exists , for , by , we have
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 1. By , we have , so , here θ is an n-dimension zero vector. On the other hand, . Clearly, , so , then . So, we have
Step 2. We show that , , . Because , we get , i.e., . The inverse is true.
Step 3. When , from the above proof, we have . So, we get . , we have and . In this case, when , we have
Hence,
Step 4. , we have
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 .
Then Eq. (1.1) has at least one T-periodic solution if one of the following two conditions is satisfied:
where is defined by (3.14).
Proof We complete the proof in three steps.
Step 1. Let . We show that is a bounded set. If , then , i.e.,
Integrating both sides of (3.5) over , we have
which together with assumption (H1) leads to the fact that there exists a point such that
Let , , . Then
Thus,
By (3.6), we have
and
On the other hand, multiplying the two sides of Eq. (3.5) by from the left side and integrating them over , we have
Let , then
By (3.9), we have
By assumption (H3), we have
From (3.10) and (3.11), we have
From , assumption (H2), Lemma 2.5 and (3.12), we have
Since , by (3.13), there exists a positive constant such that
Then by (3.8),
By (3.5), we have
where , . Take , then
and . Because there exists a such that , , so by (3.15), we get
and
By (3.16) and Lemma 2.1, we have
Now, we consider . In the formal case, we get
By classical elementary inequalities, we see that there is a constant , which is dependent on p only, such that
Case 2.1. If , then
Case 2.2. If , by (3.18) and (3.19), we have
From (3.17) and (3.21), we have
When , from , we know that there exists a constant such that
When , there must be a constant such that
Hence, from (3.14), (3.20), (3.23) and (3.24), we have
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.
Step 3. Let , then , . From the above proof, is satisfied. Obviously, condition (A2) of Lemma 2.2 is also satisfied. Now, we prove that condition (A3) of Lemma 2.2 is satisfied. Take the homotopy
where is a homeomorphism with , . , we have , , then
then we have
By using assumption (H1), we have . And then, by the degree theory,
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).
As an application, we consider the following example:
where
, , , , .
Obviously, , ,
Since
so assumption (H1) is satisfied. Take , then
and assumption (H2) is satisfied. Take , then
and assumption (H3) is satisfied. Hence, assumptions (H1)-(H3) are all satisfied. Take
such that
Take , then
By using Theorem 3.3, we know that Eq. (3.25) has at least one 2π-periodic solution.
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Acknowledgements
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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The first author QY gave an example for verifying the paper’s results. The corresponding author BD gave the proof for all the theorems. QY and BD read and approved the final manuscript.
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Yang, Q., Du, B. Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian. J Inequal Appl 2012, 270 (2012). https://doi.org/10.1186/1029-242X-2012-270
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DOI: https://doi.org/10.1186/1029-242X-2012-270