Abstract and Applied Analysis

Volume 2013 (2013), Article ID 157140, 4 pages

http://dx.doi.org/10.1155/2013/157140

Research Article

Affine-Periodic Solutions for Dissipative Systems

Yu Zhang,¹ Xue Yang,1,2 and Yong Li¹

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.

Abstract.

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.

1. Introduction

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 [13]). 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.

Definition 3. The system (1) is said to be dissipative or ultimately bounded, if there is and for any , there are and such that for ,

where denotes the solution of (1) with the initial value .

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,

Definition 5. The system (15) is said to be -affine-dissipative; if there is and for any , there are and such that

whenever ; here, denotes the solution of (15) at initial value .

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

whenever .

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.

Proof. Let

Then,

By assumption, , there is such that

Thus,

By Theorem 1, (1) has an affine-periodic solution. This finishes the proof.

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.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grants nos. 11171132 and 11201173).

References

O. A. Arino, T. A. Burton, and J. R. Haddock, “Periodic solutions to functional differential equations,” Proceedings of the Royal Society of Edinburgh A, vol. 101, pp. 253–271, 1985.

T. A. Burton and S. Zhang, “Unified boundedness, periodicity, and stability in ordinary and functional differential equations,” Annali di Matematica Pura ed Applicata, vol. 145, no. 1, pp. 129–158, 1986.

J. Chu, J. Sun, and P. J. Y. Wong, “Existence for singular periodic problems: a survey of recent reaults,” Abstract and Applied Analysis, Article ID 420835, 17 pages, 2013.

H. Ergoren and C. Tunc, “Anti-periodic solutions for a class of fourth-order nonlinear differential equations with multiple deviating arguments,” Azerbaijan Journal of Mathematics, vol. 3, no. 1, pp. 111–126, 2013.

H. Gao, Y. Li, and B. Zhang, “A fixed point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations,” Fixed Point Theory, vol. 12, no. 1, pp. 91–112, 2011.

J. K. Hale and O. Lopes, “Fixed point theorems and dissipative processes,” Journal of Differential Equations, vol. 13, no. 2, pp. 391–402, 1973.

T. Küpper, Y. Li, and B. Zhang, “Periodic solutions for dissipative-repulsive systems,” Tohoku Mathematical Journal, vol. 52, no. 3, pp. 321–329, 2000.

Y. Li, Q. D. Zhou, and X. R. Lu, “Periodic solutions and equilibrium states for functional differential inclusions with nonconvex right-hand side,” Quarterly of Applied Mathematics, vol. 55, no. 1, pp. 57–68, 1997.

G. Makay, “Periodic solutions of dissipative functional differential equations,” Tohoku Mathematical Journal, vol. 46, pp. 417–426, 1994.

T. I. Seidman and O. Klein, “Periodic solutions of isotone hybrid systems, Discrete Contin,” Dynamic Systems Service B, vol. 18, no. 2, pp. 483–493, 2013.

T. Yoshizawa, Stability Theory by Liapunov's Second Method, The Mathematical Society of Japan, Tokyo, Japan, 1966.

Y. Yuan and Z. Guo, “On the existence and stability of periodic solutions for a nonlinear neutral functional differential equation,” Abstract and Applied Analysis, vol. 2013, Article ID 175479, 8 pages, 2013.

W. A. Horn, “Some fixed point theorems for compact maps and flows in Banach spaces,” Transactions of the American Mathematical Society, vol. 149, pp. 391–404, 1970.