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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 157140, 4 pages
Affine-Periodic Solutions for Dissipative Systems
1College of Mathematics, Jilin University, Changchun 130012, China
2School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
Received 4 November 2013; Accepted 10 December 2013
Academic Editor: Bingwen Liu
Copyright © 2013 Yu Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
As generalizations of Yoshizawa’s theorem, it is proved that a dissipative affine-periodic system admits affine-periodic solutions. This result reveals some oscillation mechanism in nonlinear systems.
Consider the system where is continuous and ensures the uniqueness of solutions with respect to initial values. Fix . The system (1) is said to be -periodic if for all . For this -periodic system, a major problem is to seek the existence of -periodic solutions. Actually, some physical systems also admit the certain affine-periodic invariance. For example, let , and This affine-periodic invariance exhibits two characters: periodicity in time and symmetry in space. Obviously, when , the invariance is just the usual periodicity; when , the invariance implies the usual antisymmetry in space. When , the invariance shows the rotating symmetry in space. Hence, the invariance also reflects some properties of solutions in geometry. Now, (2) is said to possess the affine-periodic structure. For this affine-periodic system, we are concerned with the existence of affine-periodic solutions with
In the qualitative theory, it is a basic result that the dissipative periodic systems admit the existence of periodic solutions. The related topics had ever captured the main field in periodic solutions theory from the 1960s to the 1990s. For some litratures, see, for example, [1–12].
In the present paper, we will see whether (1) admits affine-periodic solutions or not if (1) is affine-dissipative. Here, (1) is said to be affine-dissipative if are ultimately bounded. Our main result is the following.
Theorem 1. Let . If the system (1) is -affine-periodic, that is, and affine-dissipative, then it admits a -affine-periodic solution ; that is,
The paper is organized as follows. In Section 2, we use the asymptotic fixed-point theorem, for example, Horn’s fixed-point theorem to prove Theorem 1. Section 3 deals with the case of functional differential equations, where an anagolous version is given and the proof is sketched. Finally, in Section 4, we illustrate some applications.
2. Proof of Theorem 1
In order to prove Theorem 1, we first recall some preliminaries.
Lemma 2 (Horn’s fixed-point theorem ). Let be a Banach space, and let be convex sets, where is compact, relatively open with respect to , and closed. Assume that is continuous and satisfies Then, has a fixed point in .
The following is a usual definition.
For the affine-periodic system (1), we have the following.
Definition 4. The system (1) is said to be -affine-dissipative, if there is and for any , there are and such that whenever .
Proof of Theorem 1. Define the map by and set where By uniqueness and the affine periodicity of , is still the solution of (1) for each . Therefore, It follows from (8) that Thus, Horn’s fixed-point theorem implies that has a fixed point in ; that is, . Also, uniqueness yields This completes the proof of Theorem 1.
3. A Version to Functional Differential Equations
Consider the functional differential equation (FDE) where is continuous, takes any bounded set in to a bounded set in , and ensures the uniqueness of solutions with respect to initial values, where , , and . Moreover, is -affine-periodic; that is,
We are in position to state another main result.
Theorem 6. If the system (15) is -affine-periodic-dissipative, then it admits a -affine-periodic solution ; that is,
Proof. Define the map by and set where where . Then, (17) and the constructions imply that Hence, has a fixed point via Horn’s theorem. The uniqueness implies that is the desired affine-periodic solution of (15). The proof is complete.
4. Some Applications
First, we observe a simple example to show the meanings of affine-periodic solutions.
Example 7. Consider the equation Put . The general solution of (24) is Obviously, for given , and any solution satisfies which implies that (24) is -periodic-dissipative. By Theorem 1, (24) has an -affine-periodic solution. This solution is just and different from the usual periodic solutions!
As usual, Lyapunov’s method is flexible in studying the existence of affine-periodic solutions. The following results illustrate applications in this aspect.
Theorem 8. Assume that there exists a Lyapunov’s function such that(i) is of ;(ii), , where is continuous in , and , ;(iii)Uniformly in , Then, the system (1) has a -affine-periodic solution.
Proof. Let denote the solution of (1) with the initial value . Put
By assumption (iii), is bounded and closed. In the following, we will prove that for each , there are and such that
In fact, given that , implies on the maximal interval that ; we have This shows that there is such that Note that which together with the construction of yields If , and there is a such that then we also have Of course, in case of , we have Taking these cases into account, we choose
Now, the existence of affine-periodic solutions is an immediate consequence. The proof is complete.
Theorem 9. Assume that where satisfies Then, (1) has an affine-periodic solution.
Example 10. Consider the system where ; , , . Let Then In the following, we only consider the case . Otherwise, set for . Take . Notice that for , , Hence, by Theorem 8, has a -affine -periodic solution. Now, if leting be a reduced fraction and , , then the -affine -periodic solutions are just -subharmonic ones; if (the set of rational vectors), then there is a such that these affine -periodic solutions are -periodic ones; if , then these solutions are quasiperiodic ones with frequency .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grants nos. 11171132 and 11201173).
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