- Research Article
- Open Access

# Existence and Lyapunov Stability of Periodic Solutions for Generalized Higher-Order Neutral Differential Equations

- Jingli Ren
^{1}, - Wing-Sum Cheung
^{2}Email author and - Zhibo Cheng
^{1}

**Received:**17 May 2010**Accepted:**23 June 2010**Published:**19 July 2010

## Abstract

Existence and Lyapunov stability of periodic solutions for a generalized higher-order neutral differential equation are established.

## Keywords

- Periodic Solution
- Delay Differential Equation
- Lyapunov Stability
- Fredholm Operator
- Degree Theory

## 1. Introduction

and by using topological degree theory and some analysis skills, existence results of periodic solutions for (1.2) have been presented.

where is given by with being a constant, is a continuous function defined on and is periodic with respect to with period , that is, for all , and , are constants.

Since the neutral operator is divided into two cases and , it is natural to study the neutral differential equation separately according to these two cases. The case has been studied in [5]. Now we consider (1.3) for the case . So throughout this paper, we always assume that , and the paper is organized as follows. We first transform (1.3) into a system of first-order differential equations, and then by applying Mawhin's continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions for (1.3). The Lyapunov stability of periodic solutions for the equation will then be established. Finally, an example is given to illustrate our results.

## 2. Preparation

First, we recall two lemmas. Let and be real Banach spaces and let be a Fredholm operator with index zero; here denotes the domain of . This means that is closed in and . Consider supplementary subspaces , of , , respectively, such that , . Let and denote the natural projections. Clearly, and so the restriction is invertible. Let denote the inverse of .

Let be an open bounded subset of with . A map is said to be -compact in if is bounded and the operator is compact.

Lemma 2.1 (see [12]).

Suppose that and are two Banach spaces, and suppose that is a Fredholm operator with index zero. Let be an open bounded set and let be -compact on . Assume that the following conditions hold:

(1)

(2)

(3) , where is an isomorphism.

Then, the equation has a solution in .

Lemma 2.2 (see [13]).

Lemma 2.3.

Remark 2.4.

This lemma is basically proved in [3, 10]. For the convenience of the readers, we present a detailed proof here as follows.

Proof.

We split it into the following two cases.

Case 1 ( ).

Case 2 ( ).

This proves (1) and (2) of Lemma 2.3. Finally, (3) is easily verified.

Lemma 2.5 (see [4]).

It is clear that, if is a -periodic solution to (2.25), then must be a -periodic solution to (1.3). Thus, the problem of finding a -periodic solution for (1.3) reduces to finding one for (2.25).

From (2.30) and (2.33), it is clear that and are continuous, and is bounded, and so is compact for any open bounded . Hence, is -compact on . For the function defined as (2.24), we have the following.

Lemma 2.6.

where

Proof.

This completes the proof of Lemma 2.6.

Remark 2.7.

## 3. Main Results

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel.

for all .

Theorem 3.1.

If and hold, then (1.3) has at least one nonconstant -periodic solution.

Proof.

Let . Then, obviously , and .

Now take . By the analysis above, it is easy to see that , , and conditions (1) and (2) of Lemma 2.1 are satisfied.

which means that condition (3) of Lemma 2.1 is also satisfied. By applying Lemma 2.1, we conclude that equation has a solution on ; that is, (1.3) has a -periodic solution with .

Finally, observe that is not constant. For, if (constant), then from (1.3) we have , which contradicts the assumption that . The proof is complete.

Theorem 3.2.

If and hold, then (1.3) has at least one nonconstant -periodic solution if one of the following conditions holds:

(1) ,

(2) and

Proof.

We claim that is bounded.

Case 1.

Case 2.

This proves the claim, and the rest of the proof of the theorem is identical to that of Theorem 3.1.

Remark 3.3.

where and , then the results of Theorems 3.1 and 3.2 still hold.

Remark 3.4.

and the results of Theorems 3.1 and 3.2 still hold.

Next, we study the Lyapunov stability of the periodic solutions of (3.32).

Theorem 3.5.

Assume that holds. Then every -periodic solution of (3.32) is Lyapunov stable.

Proof.

Hence, is a Lyapunov function for nonautonomous (3.32) (see [15, page 50]), and so the -periodic solution of (3.32) is Lyapunov stable.

Finally, we present an example to illustrate our result.

Example 3.6.

assumption holds with .

Case 1.

If , then by (1) of Theorem 3.2, (3.40) has at least one nonconstant -periodic solution.

Case 2.

So by (2) of Theorem 3.2, (3.40) has at least one nonconstant -periodic solution.

## Declarations

### Acknowledgments

This paper is partially supported by the National Natural Science Foundation of China (10971202), and the Research Grant Council of Hong Kong SAR, China (project no. HKU7016/07P).

## Authors’ Affiliations

## References

- Lu S, Ge W: On the existence of periodic solutions for neutral functional differential equation.
*Nonlinear Analysis: Theory, Methods & Applications*2003, 54(7):1285-1306. 10.1016/S0362-546X(03)00187-1View ArticleMathSciNetMATHGoogle Scholar - Lu S:Existence of periodic solutions for a
-Laplacian neutral functional differential equation.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 70(1):231-243. 10.1016/j.na.2007.11.053View ArticleMathSciNetGoogle Scholar - Lu S, Ge W, Zheng Z: Periodic solutions to neutral differential equation with deviating arguments.
*Applied Mathematics and Computation*2004, 152(1):17-27. 10.1016/S0096-3003(03)00530-7View ArticleMathSciNetMATHGoogle Scholar - Peng S:Periodic solutions for
-Laplacian neutral Rayleigh equation with a deviating argument.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(5-6):1675-1685. 10.1016/j.na.2007.07.007View ArticleMATHGoogle Scholar - Ren J, Cheng Z: Periodic solutions for generalized high-order neutral differential equation in the critical case.
*Nonlinear Analysis: Theory, Methods & Applications*2009, 71(12):6182-6193. 10.1016/j.na.2009.06.011View ArticleMathSciNetMATHGoogle Scholar - Shen J, Liang R: Periodic solutions for a kind of second order neutral functional differential equations.
*Applied Mathematics and Computation*2007, 190(2):1394-1401. 10.1016/j.amc.2007.02.137View ArticleMathSciNetMATHGoogle Scholar - Wang Q, Dai B: Three periodic solutions of nonlinear neutral functional differential equations.
*Nonlinear Analysis: Real World Applications*2008, 9(3):977-984. 10.1016/j.nonrwa.2007.01.014View ArticleMathSciNetMATHGoogle Scholar - Wang K, Lu S: On the existence of periodic solutions for a kind of high-order neutral functional differential equation.
*Journal of Mathematical Analysis and Applications*2007, 326(2):1161-1173. 10.1016/j.jmaa.2006.03.078View ArticleMathSciNetMATHGoogle Scholar - Wu J, Wang Z: Two periodic solutions of second-order neutral functional differential equations.
*Journal of Mathematical Analysis and Applications*2007, 329(1):677-689. 10.1016/j.jmaa.2006.06.084View ArticleMathSciNetMATHGoogle Scholar - Zhang MR: Periodic solutions of linear and quasilinear neutral functional-differential equations.
*Journal of Mathematical Analysis and Applications*1995, 189(2):378-392. 10.1006/jmaa.1995.1025View ArticleMathSciNetMATHGoogle Scholar - Zhu Y, Lu S:Periodic solutions for
-Laplacian neutral functional differential equation with deviating arguments.
*Journal of Mathematical Analysis and Applications*2007, 325(1):377-385. 10.1016/j.jmaa.2005.10.084View ArticleMathSciNetMATHGoogle Scholar - Gaines RE, Mawhin JL:
*Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics, vol. 568*. Springer, Berlin, Germany; 1977:i+262.Google Scholar - Zhang M:Nonuniform nonresonance at the first eigenvalue of the
-Laplacian.
*Nonlinear Analysis: Theory, Methods & Applications*1997, 29(1):41-51. 10.1016/S0362-546X(96)00037-5View ArticleMathSciNetMATHGoogle Scholar - Hale JK:
*Theory of Functional Differential Equations*. Springer, New York, NY, USA; 1977.View ArticleMATHGoogle Scholar - Glendinning P:
*Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge Texts in Applied Mathematics*. Cambridge University Press, Cambridge, UK; 1994:xiv+388.View ArticleGoogle Scholar

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