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Periodic solutions for a kind of higher-order neutral functional differential equation with variable parameter
Advances in Difference Equations volume 2014, Article number: 187 (2014)
In this paper, we consider a kind of higher-order neutral equation with distributed delay and variable parameter: . By using the classical coincidence degree theory of Mawhin, sufficient conditions for the existence of periodic solutions are established. Recent results in the literature are generalized and significantly improved. Furthermore, two examples are given to illustrate that the results are almost sharp.
MSC:34K10, 30D05, 34B45.
This paper is devoted to the application of Mawhin’s continuation theorem to investigate the existence of periodic solutions for the following equation:
where p, q are continuous periodic functions with period , with , , , n is a positive integer, , is a bounded variation function, and , where is the total variation of over .
In recent years, the existence of periodic solution for functional differential equations has been studied extensively (see [1–7]). For example, in , the authors studied the following equation with a deviating argument:
Du et al.  studied the second-order neutral equation with variable parameter:
In , the authors proved for the first time the lemma (Lemma 2.1) for the existence of with and some properties of when is not a constant. Then they established sufficient conditions for the existence of periodic solutions of (1.3) by using Mawhin’s theorem.
In , Wang and Lu discussed a kind of high-order neutral functional differential equation with distributed delay:
with a constant. However, there are several errors in the proof of Theorems 3.1 and 3.2 of . The main purpose of this paper is to improve the results of  and modify the errors. Meanwhile, the problem considered in paper  is generalized to the higher-order case in our work. Moreover, two examples are given to demonstrate our results.
2 Related lemmas
For convenience we denote , , , , and define the spaces
with the norm and
with the norm . Clearly, and are all Banach spaces.
Define a linear operator by
Lemma 2.1 
If , then A has a continuous inverse on satisfying
Let X and Y be real Banach spaces and be a Fredholm operator with index zero, i.e., ImL is closed and . If L is a Fredholm operator with index zero, then there exist continuous projections and such that , and is invertible. Denote by the inverse of . Define , where is an open and bounded set; N is L-compact on , if QN is continuous and bounded and is compact on .
Lemma 2.2 
Let X and Y be two Banach spaces with norms and , respectively, and an open and bounded set. Suppose is a Fredholm operator of index zero and is L-compact. In addition, if
for , ;
, where is a homeomorphism.
Then the abstract equation has at least one solution in .
Define the linear operator by with
and the nonlinear operator
From , , we can obtain
where , . From , we have , . Let be a solution of , from Lemma 2.1, , then
Clearly, . Define continuous projections
It is easy to see that and . So . Notice that ImL is closed, then L is a Fredholm operator of index zero.
where , , are defined by the equation ; here
Whereafter, since , can be determined by
It follows from Lemma 2.1, the definition of N, Q, , and the continuity of f, g, q that N is L-compact.
3 Main results
Theorem 3.1 Suppose n is an even integer and with . In addition, if there exist constants and such that
(H1) (or <0), whenever , where ,
then (1.1) has at least one periodic solution provided
Proof Without loss of generality, we may assume that , when . Now, we will complete the proof by three steps.
Step 1. Let , , , i.e.,
Integrating both sides of (3.2) on , we have
By the integral mean value theorem, there exists a constant such that . So (H1) implies that . Since and on , by the properties of the Riemann-Stieltjes integral, there exists a constant such that . Because is a T-periodic function, there exists an integer such that , . Then . Hence,
Multiplying both sides of (3.2) by and integrating them over , one gets
Since , there exist , , such that . Thus,
By the Hölder inequality and (3.7), we have
In view of (H2) and the properties of the bounded variation function, , there exists a constant such that
when . Let
where . By (3.4) and (3.6), we get
In view of , there exists a constant such that . Then
Step 2. Let . Then for , , , and , that is,
Repeating the process of Step 1, we see that is bounded, that is, there exists such that for .
Step 3. Let , where , then . In view of Step 1 and Step 2, conditions (i) and (ii) in Lemma 2.2 are all satisfied. Next, we will show that (iii) of Lemma 2.2 holds.
Set the isomorphism by . For , , define the homotopy
Since for , . So for . Hence, by the homotopy invariance of the Brouwer degree, we obtain
Applying Lemma 2.2, we reach the conclusion.
For the case , when , a similar argument can complete the proof. Here we omit it. □
Remark 3.1 In , the calculation of formula (3.5) is wrong. So the bound of is not evaluated correctly. The same error appeared again in the next theorem. We modified those errors in this paper.
Theorem 3.2 Suppose that n is an odd integer and , where and k is odd. Moreover, if , and conditions (H1), (H2) and (3.1) are satisfied. Then (1.1) has at least one periodic solution.
Proof Without loss of generality, we assume that , when . As in the proof of Theorem 3.1, define the set , for , , (3.2) can be obtained. Multiplying both sides of (3.2) by and integrating them over , we obtain
In view of
by the Hölder inequality and (3.6), (3.7) in the proof of Theorem 3.1, one gets
Since the condition (3.1) holds, is bounded. The remainder can be proved in the same way as in Theorem 3.1. □
Remark 3.2 In , the first inequality in (3.17) cannot be obtained without assuming that is odd. In our paper, we correct this error by adding such a condition.
Remark 3.3 In our paper, we consider the n-order period equation (1.1); in this sense, we generalize the model in  to higher order under the equivalent conditions. Moreover, in view of the variable parameter in (1.1), we develop the results in  with the constant coefficient c.
As an application, we list the following examples.
Example 4.1 Consider
Corresponding to (1.1), we have , , , , , , , , . Obviously, , ; we have
Therefore, Theorem 3.1 implies that (4.1) has at least one 2π-periodic solution. In fact, is such a solution.
Example 4.2 Consider
Corresponding to (1.1), we have , , , , , , , , . Obviously, , ; we have
Therefore, Theorem 3.2 implies that (4.2) has at least one 2π-periodic solution. It is easy to see that is such a solution.
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The work is supported by the National Natural Science Foundation of China (No. 11226148 and No. 61273016) and the Natural Science Foundation of Zhejiang Province (LY12F05006).
The authors declare that they have no competing interests.
AY performed the theory analysis and carried out the computations. HW participated in the design of the study and helped to draft the manuscript. All authors have read and approved the final manuscript.