Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 414901, 7 pages

http://dx.doi.org/10.1155/2013/414901

Research Article

Periodic Solutions and Nontrivial Periodic Solutions for a Class of Rayleigh-Type Equation with Two Deviating Arguments

Meiqiang Feng

School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China

Received 27 May 2013; Accepted 27 August 2013

Academic Editor: Henryk Hudzik

Copyright © 2013 Meiqiang Feng. 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.

The Rayleigh equation with two deviating arguments is studied. By using Leray-Schauder index theorem and Leray-Schauder fixed point theorem, we obtain some new results on the existence of periodic solutions, especially for the existence of nontrivial periodic solutions to this equation. The results are illustrated with two examples, which cannot be handled using the existing results.

1. Introduction

Consider the Rayleigh equation with two deviating arguments in the form of

where

The dynamic behaviors of Rayleigh equation have been widely investigated due to their applications in many fields such as physics, mechanics, and the engineering technique fields. For example, an excess voltage of ferroresonance, known as some kind of nonlinear resonance having long duration that arises from the magnetic saturation of inductance in an oscillating circuit of a power system, and a boosted excess voltage can give rise to some problems in relay protection. To probe this mechanism, a mathematical model was proposed in [1–3], which is a special case of the Rayleigh equation with two delays. This implies that (1) can represent analog voltage transmission. In a mechanical problem, usually represents a damping or friction term, represents the restoring force, is an externally applied force, and is the time lag of the restoring force (see [4]). Some other examples in practical problems concerning physics and engineering technique fields can be found in [5–7].

At the same time, the periodic solutions for Rayleigh equations with two deviating arguments have been studied by authors [8–10] under the assumption of

It is not difficult to see that if , then the periodic solution obtained in [8–10] must be nontrivial. But if , then the periodic solution obtained in [8–10] may be trivial under the assumption of or . And if the periodic solution is unique, then it must be trivial. Thus, it is worth discussing the existence of the nontrivial periodic solutions of Rayleigh equations with two deviating arguments in this case.

The main purpose of this paper is to establish sufficient conditions for the existence of periodic solution, especially for the existence of nontrivial periodic solutions of (1) by using the Leray-Schauder index theory. We remark that our methods are different from those used in [8–10] to some degree. In particular, two examples are also given to illustrate the effectiveness of our results.

For ease of exposition, throughout this paper, we assume that .

On the other hand, the following assumptions are used in this paper.

, , , , , , , , and ., and there exists such that , for all (or , for all ). is differentiable with respect to , and there exist , , , such that

There exist such that , for all , , . There exists , such that for , we have

There exist integers such that , .

The main results of the present paper are as follows.

Theorem 1. If ()–() hold, and , then (1) has at least one periodic solution.

The proof of Theorem 1 will be given in Section 3.

Remark 2. If , then the periodic solution obtained by Theorem 1 must be nontrivial. If , we could not conclude whether or not the periodic solution is nontrivial.

So we give the following conditions:

, exist and are continuous, ..

Theorem 3. If ()–() hold, and , , then (1) has at least one nontrivial periodic solution.

The proof of Theorem 3 will be given in Section 4.

2. Preliminaries

In this section, to establish the periodic solutions of (1), we provide some background definitions and some well-known results, which are crucial in our arguments.

Let be a real Banach space, and let be a completely continuous operator.

Definition 4 (see [11, 12]). If is an isolated fixed point of , then the fixed point index at of is defined by

where is a neighborhood of , which satisfies that is the unique fixed point in of .

Definition 5 (see [11, 12]). If there exits with such that , , then is called an eigenvalue of operator and is called the eigenfunction of operator corresponding to .

Definition 6. Let be continuous. is said to be periodic on if

Lemma 7 (see [11, 12], index theorem of Leray-Schauder). Suppose that is a fixed point of , is Fréchet differentiable at , and is not the eigenvalue of . Then is an isolated fixed point of , and

where is equal to the sum of the algebraic multiplicities of all of the eigenvalues of .

Lemma 8 (see [11, 12], fixed point theorem of Leray-Schauder). Let be a real Banach space and be a completely continuous operator. If

is bounded, then has a fixed point , where

Lemma 9. Suppose and . Then .

Lemma 10 (see [13]). Let be constant, be periodic with period , and . Then for any which is periodic with period , we have

3. Proof of Theorem 1

In this section, we will use Lemma 8 to prove Theorem 1.

Proof. Let

Then and are real Banach spaces endowed with the norm

respectively.

Choosing with , then

has only trivial solution in .

In fact, it is easy to see that the general solution of is

By the periodic boundary conditions we obtain that is its unique solution in . Then for ,

has unique solution . Writing , then is a completely continuous operator.

Define an operator by

Then is continuous and bounded. Let . Then is also a completely continuous operator. By Lemma 8, if

is bounded in , then has a fixed point in . Thus, (1) has periodic solution.

Now suppose that , satisfying . Then is a solution of

Let , be the maximum point and minimum point of on , respectively. Then

Noticing , we have

and hence, there exists (or ) such that

which implies that

From we know that (see [10]). So we have

and then,

where is the norm of .

Multiplying (19) with and integrating from to , then we have

By we know that

By Hölder's inequality, from (26) and (27) we have

Since , are -periodic, differentiable, and

we have

Combining (28) and (30) with and we obtain

By Lemma 9 we have

By (32) and Hölder's inequality we have

From (33) we have

By and Lemma 10 we have

Thus, it follows from (31), (32), (34), and (35) that

Combining this with , we know that there exists such that . Then

Multiplying (19) with and integrating from to , then we have

where

Thus,

Selecting such that , then we have

Thus, from (35) and (41) we know that . It follows that

is bounded. Therefore, by Lemma 8, we obtain that has a fixed point , where , and hence, it follows that (1) has a periodic solution.

4. Proof of Theorem 3

In this section, we will use Lemma 7 to prove Theorem 3.

Proof. Let be defined and same as in Section 3. We prove that is Fréchet differentiable at and

In fact, by , we have . Let

We have

where , . Thus,

which implies that exists and .

Now, we prove that is not the eigenvalue of .

In fact, if there exists satisfying , then is a solution of

Integrating (47) from to , we obtain

By , there exists such that .

Multiplying (47) with and integrating it from to , we have

By we have

which implies that , and hence . It follows that is not the eigenvalue of . By Lemma 7 we have

Next we prove that . We can prove that

has only one eigenvalue in (0, 1) and the algebraic multiplicity of it is .

In fact, it is easy to see that is the eigenvalue of (52) if and only if , that is,

In this case, (52) degenerates to

It is not difficult to see that all the solutions of (54) are . Therefore the geometric multiplicity of is .

If , then , and hence, satisfies

which implies that

Integrating (56) from to , we obtain , which implies that . It follows that the algebraic multiplicity of is also , and therefore, . So . Letting small enough, then we have

On the other hand, by the proof of Theorem 1, we know that there exists such that if , satisfying , then . Suppose for . By the homotopy invariance of topological degree, we have

where . It follows from (57) and (58) that

which implies that has a fixed point in and is a nontrivial periodic solution of (1).

5. Two Examples

In this section, we give two examples to demonstrate the results obtained in previous sections.

Example 1. Consider the forced Rayleigh-type equation with period :

where

Conclusion. Equation (60) has at least one periodic solution with period .

Proof. By (61) and (62), it is not difficult to see that the condition holds, , and

On the other hand, let , , , , , and . If , then

Hence, ()–() are satisfied. Thus, by Theorem 1, (60) has at least one periodic solution with period .

Example 2. If we replace , , and in Example 1 by

, then we can obtain that , for all , and

Let , , , , , , , and . If ; then

Hence, ()–() are satisfied. Thus, by Theorem 3, (60) has at least one nontrivial periodic solution with period .

Acknowledgments

This work is sponsored by the project NSFC (11171032) and Beijing Excellent Training Grant (2010D005007000002).

References

G. J. Ji, Z. X. Wang, and D. W. Lai, “Existence of periodic solutions to overvoltage models in electric power systems,” Acta Mathematica Scientia A, vol. 16, no. 1, pp. 99–104, 1996 (Chinese).

Z. Wang and D. Lai, “A delay differential equation appeared in the study of overvoltage,” in Report to the Italian Symposium, vol. 12, 1984.

C. H. Feng, “Existence and uniqueness of almost periodic solutions for some delayed differential equations appearing in a power system,” Acta Mathematica Sinica, vol. 46, no. 5, pp. 931–936, 2003 (Chinese).

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, vol. 178, Academic Press, Orlando, Fla, USA, 1985.

J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977.

T. Yashizaw, “Asymptotic behavior of solutions of differential equations,” in Differential Equation: Qualitative Theory (Szeged, 1984), vol. 47 of Colloquia mathematica Societatis János Bolyai, pp. 1141–1172, North-Holland, Amsterdam, 1987.

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, NY, USA, 1993.

L. Peng, B. Liu, Q. Zhou, and L. Huang, “Periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Journal of the Franklin Institute, vol. 343, no. 7, pp. 676–687, 2006.

C. Huang, Y. He, L. Huang, and W. Tan, “New results on the periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Mathematical and Computer Modelling, vol. 46, no. 5-6, pp. 604–611, 2007.

B. Liu, “Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with two deviating arguments,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2108–2117, 2008.

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986.

D. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Ji Nan, China, 1985.

S. Lu and W. Ge, “Periodic solutions for a kind of second order differential equations with multiple with deviating arguments,” Applied Mathematics and Computation, vol. 146, pp. 195–209, 2003.