Periodic solutions for a kind of higher-order neutral functional differential equation with variable parameter
© Yang and Wang; licensee Springer. 2014
Received: 21 March 2014
Accepted: 26 June 2014
Published: 22 July 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.
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 , 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.
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
with the norm . Clearly, and are all Banach spaces.
Lemma 2.1 
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 
for , ;
, where is a homeomorphism.
Then the abstract equation has at least one solution in .
It is easy to see that and . So . Notice that ImL is closed, then L is a Fredholm operator of index zero.
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 ,
Proof Without loss of generality, we may assume that , when . Now, we will complete the proof by three steps.
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.
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.
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.
Therefore, Theorem 3.1 implies that (4.1) has at least one 2π-periodic solution. In fact, is such a solution.
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.
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).
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