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Periodic solutions for a kind of neutral functional differential systems
Boundary Value Problems volume 2014, Article number: 151 (2014)
In this paper, we analyze some properties of the linear difference operator , , and then, by using the coincidence degree theory of Mawhin, a kind of neutral differential systems with non-constant matrix is studied. Some new results on the existence of periodicity are obtained. It is worth noting that is no longer a constant matrix, which is different from the corresponding ones of past work.
The field of neutral functional equations (in short NFDEs) is making significant breakthroughs in its practice; it is no longer only a specialist’s field. In many practical systems, models of systems are described by NFDEs in which the models depend on the delays of state and state derivatives. Practical examples for neutral systems include population ecology, heat exchanges, mechanics, and economics; see –. In particular, qualitative analysis such as periodicity and stability of solutions of NFDEs has been studied extensively by many authors. We refer to – for some recent work on the subject of periodicity and stability of neutral equations.
In the last few years, the stability of neutral systems of various classes with time delays has received an ever-growing interest from many authors. Many sufficient conditions have been proposed to guarantee the asymptotic stability for neutral time delay systems. We only mention the work of some authors –. It is well known that the existence of periodic solutions of neutral equations and neutral systems is a very basic and important problem, which plays a role similar to stability. Thus, it is reasonable to seek conditions under which the resulting periodic neutral system would have a periodic solution. Much progress has been seen in this direction and many criteria are established based on different approaches. However, there is no paper for investigating the existence of periodic solutions of neutral system with non-constant matrix. In addition, to the best of our knowledge, most of the existing results deal with scalar NFEDs or neutral systems with a constant matrix. For example, in papers –, based on Mawhin’s continuation theorem, several types of scalar neutral equations have been studied:
For a neutral system, we note that Lu and Ge  studied the following system:
But C is a constant symmetric matrix. The purpose of this paper is to investigate the existence of periodic solutions to the nonlinear neutral system with non-constant matrix of the form
where , , ; , ; , ; , ; T, and τ are given constants with .
Throughout this paper, we use some notation:
; , ;
with the norm
with the norm
Clearly, and are Banach spaces.
2 Main lemmas
If, then operatorhas a continuous inverseon, satisfying
where, is a real symmetric matrix.
We will give some properties of .
Suppose thatare eigenvalues of. Then the operatorhas continuous inverseon, satisfying
Since is a real symmetric matrix, there exists an orthogonal matrix such that
Consider the system
where we have equivalence to
where , . On the other hand, a component of the vector in system (2.1) is
From Lemma 2.1, we have
Thus, exists and
When , by (2.2) we get
Thus, by (2.3) we have
Similar to the above proof, when , we get
Let X and Y be two Banach spaces and let be a linear operator, a Fredholm operator with index zero (meaning that ImL is closed in Y and ). If L is a Fredholm operator with index zero, then there exist continuous projectors , such that , , and is invertible. Denote by the inverse of .
Let Ω be an open bounded subset of X, a map is said to be L-compact in if is bounded and the operator is relatively compact. We first give the famous Mawhin continuation theorem.
Suppose that X and Y are two Banach spaces andis a Fredholm operator with index zero. Furthermore, is an open bounded set andis L-compact on. If all the following conditions hold:
, , ,
then the equationhas a solution on.
3 Main results
Suppose that, is a nonzero periodic solution of (3.1) and there exists a constantsuch that
(H1):, is bounded in the set (or), where
(H2): (or <0), whenever, .
(H3):Suppose thatare eigenvalues of, , and there exists a constantsuch that
Then system (1.1) has at least one T-periodic solution, if (or), , and, , where
where . Then system (1.1) obeys the operator equation . We have , . Then
where . Since , we have . Let be a nonzero periodic solution of
then and , where I is an unit matrix. We get
Obviously, ImL is closed in and . So L is a Fredholm operator with index zero. Define continuous projectors P, Q:
Since and , is an embedding operator. Hence is a completely continuous operator in ImL. By the definitions of Q and N, one knows that is bounded on . Hence the nonlinear operator N is L-compact on . We complete the proof by three steps.
Step 1. Let . We show that is a bounded set. We have , i.e.,
Integrating both sides of (3.2) over , we have
Let be bounded in and
By assumption (H1), if , there exists a constant such that . From (3.3) and (3.4), we get
We claim that there exists a point such that
In fact, for , we have , and by (3.4), we have , which is a contradiction; see (3.3). So there must be a point such that
Similar to the above proof, there must be a point such that
If , by (3.7) we have
Let . This proves (3.6).
If , from (3.8) and the fact that is continuous on ℝ, there is a point between and such that . This also proves (3.6). Let , , . Then . Hence we get
Multiplying the two sides of system (3.2) by and integrating them over , combining with , by (3.9) we have
From (3.9) and , there is a constant such that
In view of (3.9) and (3.10), we get
From Lemma 2.2, and (3.2), if , we have
From assumption (H3) and (3.10)-(3.12), we get
So there exists a constant such that
Since , , there is a constant vector such that ; then by (3.13) we get
Step 2. Let , we shall prove that is a bounded set. We have , ; then
When , , we have
Then we have
Otherwise, if, , , then from assumption (H2), we have
which is a contradiction; see (3.14). When , , we have
Then we have
Otherwise, if , , then from assumption (H2), we have
which is a contradiction; see (3.14). Hence is a bounded set.
Step 3. Let , then , , and from the above proof, is satisfied. Obviously, condition (2) of Lemma 2.3 is also satisfied. Now we prove that condition (3) of Lemma 2.3 is satisfied. We have , , . There at least exists a such that . When , take the homotopy
Then, by using assumption (H2), we have . When , take the homotopy
We also have . Then by degree theory,
Applying Lemma 2.3, we reach the conclusion. □
When or , there are no existence results for periodic solutions for system (1.1). We hope that there is interest in doing further research on this issue.
As an application, we consider the following system:
Clearly, system (3.15) is a particular case of system (1.1). Obviously,
Here assumptions (H1)-(H2) are all satisfied. In addition,
By using Theorem 3.1, when , we know that system (3.15) has at least one 2π-periodic solution.
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The authors wish to thank the anonymous referee for his/her valuable suggestions to this paper. This work is supported by the NSFC of China (11171085).
The authors declare that they have no competing interests.
ZM performed all the steps of proof in this research and also wrote the paper. BD suggested many good ideas that made this paper possible and helped to improve the manuscript. All authors read and approved the final manuscript.
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He, Z.M., Du, B. Periodic solutions for a kind of neutral functional differential systems. Bound Value Probl 2014, 151 (2014). https://doi.org/10.1186/s13661-014-0151-1
- periodic solutions
- Mawhin’s continuation theorem