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Periodic solutions for the p-Laplacian neutral functional differential system
Advances in Difference Equations volume 2013, Article number: 367 (2013)
By using the generalized Borsuk theorem in coincidence degree theory, we prove the existence of periodic solutions for the p-Laplacian neutral functional differential system.
In recent years, the existence of periodic solutions for the Rayleigh equation and the Liénard equation has been studied (see [1–9]). By using topological degree theory, some results on the existence of periodic solutions are obtained.
Motivated by the works in [1–9], we consider the existence of periodic solutions of the following system:
where , , are periodic functions with period T; is an symmetric matrix of constants, is a constant. is given by
The is a homeomorphism of with the inverse . By using the theory of coincidence degree, we obtain some results to guarantee the existence of periodic solutions. Even for , the results in this paper are also new.
In what follows, we use to denote the Euclidean inner product in and to denote the -norm in , i.e., .
The norm in is defined by .
The corresponding -norm in is defined by
and the -norm in is
Let be the Sobolev space.
Lemma 1.1 (See )
Suppose and , then
In order to use coincidence degree theory to study the existence of T-periodic solutions for (1.1), we rewrite (1.1) in the following form:
If is a T-periodic solution of (1.2), must be a T-periodic solution of (1.1). Thus, the problem of finding a T-periodic solution for (1.1) reduces to finding one for (1.2).
Let with the norm , with the norm . Clearly, X and Z are Banach spaces.
Denote the operator A by
It is easy to see that , . So, L is a Fredholm operator with index zero. Let and be defined by
and let denote the inverse of .
where , , .
From (1.3), one can easily see that N is L-compact on , where Ω is an open bounded subset of X.
Lemma 1.2 (See )
Suppose that are eigenvalues of the matrix C. If , , then A has a continuous bounded inverse with the following relationships:
, , , where
where q is a constant with ;
In the proof of our results on the existence of periodic solutions, we use the following generalized Borsuk theorem in coincidence degree theory of Gaines and Mawhin .
Lemma 1.3 Let X and Z be real normed vector spaces. Let L be a Fredholm mapping of index zero. Ω is an open bounded subset of X and Ω is symmetric with respect to the origin and contains it. Let be L-compact and such that
Then, for every , the equation has at least one solution in Ω.
2 Main results
Theorem 2.1 Suppose that the matrix C satisfies the conditions of Lemma 1.2 and that there exist constants , , and such that
(H1) or , ;
(H2) , .
Then equation (1.1) has at least one T-periodic solution for .
Proof For any , let
Consider the following parameter equation:
Let be a possible T-periodic solution of (2.3) for some , then is a T-periodic solution of the following system:
Noticing that is a T-periodic solution, we have
Multiplying the two sides of (2.2) by and integrating them on the interval , by (2.3) and (H1)-(H2), we obtain
On the other hand,
So, multiplying the two sides of (2.2) by and integrating them on the interval , by (H1)-(H2), we get
Furthermore, we have
It is obvious that there exist and such that
From (2.4) and (2.5), we can see
from which it follows that there exists a positive number such that
By using Lemma 1.2, we get
From (2.6), there exists such that , and
Therefore and .
Since , , there exist and such that
From (2.4), we have
Clearly, for each , there exists such that . Thus, for any , we have
Choose a number , and let , then for any , . It is easy to see that is L-compact on , is (2.1) and . From Lemma 1.3, (2.1) has at least one T-periodic solution , is a T-periodic solution of (1.1). □
Theorem 2.2 Let , where are eigenvalues of the matrix C with , . Suppose that there exist constants , and such that
(H3) there is a constant such that ;
(H4) , ;
(H5) , either or for , where .
Then (1.1) has at least one T-periodic solution for .
Proof Let be a possible T-periodic solution of (2.1). From assumption (H3), there exists a constant such that
From (H3) and (2.2), we have
Integrating both sides of (2.2) over , we get
So, there exist such that
From (H4), one can see . Let , , then . By Lemma 1.1, one can obtain
Noticing the periodicity of , we have
From Minkovski’s inequality, we have
In view of (2.7) and Lemma 1.2, we get
Since , from (2.8), there exists a constant such that
From (2.9) and (2.10), we know that the rest of the proof of the theorem is similar to that of Theorem 2.1. □
Remark 2.1 If , system (1.1) can be reduced to the system in .
If and , system (1.1) can be reduced to the system in .
Example 2.1 Consider the following system:
where , , are periodic functions with period T; . Clearly, .
then, by Theorem 2.1, (2.11) has at least one T-periodic solution for .
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The authors thank the referees for helpful suggestions. This work was supported by Grants Nos. 10871052 and 109010600 from NNSF of China, and by Grant No. S2011010005029 from NSF of Guangdong.
The authors declare that they have no competing interests.
The first author carried out the studies, the second author participated in the studies and drafted the manuscript. All authors read and approved the final manuscript.
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Wang, Z., Song, C. Periodic solutions for the p-Laplacian neutral functional differential system. Adv Differ Equ 2013, 367 (2013). https://doi.org/10.1186/1687-1847-2013-367
- periodic solutions
- coincidence degree