- Research Article
- Open Access
© Taiyong Chen et al. 2010
Received: 2 March 2010
Accepted: 19 August 2010
Published: 25 August 2010
Antiperiodic problems arise naturally from the mathematical models of various of physical processes (see [1, 2]), and also appear in the study of partial differential equations and abstract differential equations (see [3–5]). For instance, electron beam focusing system in travelling-wave tube's theories is an antiperiodic problem (see ).
During the past twenty years, antiperiodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera's type criterion was presented in  and the validity of the monotone iterative technique was shown in . Moreover, for higher-order ordinary differential equations, the existence of antiperiodic solutions was considered in [9–12]. Recently, existence results were extended to antiperiodic boundary value problems for impulsive differential equations (see ), and antiperiodic wavelets were discussed in .
using of the main assumption as follows:
In the past few decades, many important results relative to (1.3) with certain boundary conditions have been obtained. We refer the reader to [17–20] and the references cited therein. However, to the best of our knowledge, there exist relatively few results for the existence of antiperiodic solutions of (1.3). Moreover, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see ). Thus, it is worthwhile to continue to investigate the existence of antiperiodic solutions for (1.3).
Note that, is also a -periodic solution of (1.4) or (1.5) if is a -antiperiodic solution of (1.4) or (1.5). Hence, from the arguments in this paper, we can also obtain the existence results of periodic solutions for above equations.
The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3, we establish some sufficient conditions for the existence of antiperiodic solutions of (1.4), basing on Leray-Schauder principle. Then, in Section 4, we obtain two existence results of antiperiodic solutions with symmetry for (1.5). Finally, in Section 5, some explicit examples are given to illustrate the main results. Our results are different from those of bibliographies listed above.
Lemma 2.1 (Wirtinger inequality).
Lemma 2.2 (Continuation theorem).
3. Antiperiodic Solutions for (1.4)
In this section, an existence result of antiperiodic solutions for (1.4) will be given.
Then (1.4) has at least one antiperiodic solution.
When , is equal to 1. It is easy to see that condition ( ) in  is stronger than condition ( ) of Theorem 3.1.
We begin with some lemmas below.
The proof is complete.
Now we give the proof of Theorem 3.1.
Proof of Theorem 3.1..
4. Antiperiodic Solutions with Symmetry for (1.5)
In this section, we will prove the existence of even antiperiodic solutions or odd antiperiodic solutions for (1.5).
where is a positive constant independent of . So that, our problem is reduced to construct one completely continuous operator in which sends into , such that the fixed points of operator in some open-bounded set are the even antiperiodic solutions of (1.5).
From the similar arguments in the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . Thus, (1.5) has at least one even antiperiodic solution. The proof is complete.
Based on the proof of Theorem 3.1, for the possible odd antiperiodic solutions of (4.3), there exists a prior bounds in . Hence, our problem is reduced to construct one completely continuous operator in which sends into , such that the fixed points of operator in some open-bounded set are the odd antiperiodic solutions of (1.5).
By a similar way as the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . So that, (1.5) has at least one odd antiperiodic solution. The proof is complete.
In this section, we will give some examples to illustrate our main results.
Moreover, the conditions of Theorem 4.1 are also satisfied. Thus (5.1) has at least one even antiperiodic solution.
The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China (10771212), the Fundamental Research Funds for the Central Universities, and the Science Foundation of China University of Mining and Technology (2008A037).
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