- Research Article
- Open Access
Existence and Lyapunov Stability of Periodic Solutions for Generalized Higher-Order Neutral Differential Equations
© Jingli Ren et al. 2011
- Received: 17 May 2010
- Accepted: 23 June 2010
- Published: 19 July 2010
Existence and Lyapunov stability of periodic solutions for a generalized higher-order neutral differential equation are established.
- Periodic Solution
- Delay Differential Equation
- Lyapunov Stability
- Fredholm Operator
- Degree Theory
and by using topological degree theory and some analysis skills, existence results of periodic solutions for (1.2) have been presented.
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 . 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.
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 .
Lemma 2.1 (see ).
Lemma 2.2 (see ).
We split it into the following two cases.
This proves (1) and (2) of Lemma 2.3. Finally, (3) is easily verified.
Lemma 2.5 (see ).
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.
This completes the proof of Lemma 2.6.
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel.
This proves the claim, and the rest of the proof of the theorem is identical to that of Theorem 3.1.
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).
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.
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).
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