International Journal of Differential Equations

Volume 2012 (2012), Article ID 406835, 13 pages

doi:10.1155/2012/406835

Research Article

Existence of Periodic Solutions to Nonlinear Differential Equations of Third Order with Multiple Deviating Arguments

Cemil Tunç

Department of Mathematics, Faculty of Sciences, Yüzüncü Yıl University, 65080 Van, Turkey

Received 12 May 2012; Accepted 19 July 2012

Academic Editor: D. D. Ganji

Copyright © 2012 Cemil Tunç. 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.

We establish certain new sufficient conditions which guarantee the existence of periodic solutions for a nonlinear differential equation of the third order with multiple deviating arguments. Using the Lyapunov functional approach, we prove a specific theorem and provide an example to illustrate the theoretical analysis in this work and the effectiveness of the method utilized here.

1. Introduction

It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth. In reality, many actual systems have the property aftereffect, that is, the future states depend not only on the present, but also on the past history, and after effect is also known to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, and so forth (Burton [1], Kolmanovskii and Myshkis [2]). Therefore, it is important to investigate the qualitative behaviors of functional differential equations.

In 1978, using the known theorem of Yoshizawa [3, Theorem 37.2], Chukwu [4] found certain sufficient conditions that guarantee the existence of a periodic solution to nonlinear-linear differential of the third order with the constant deviating argument (>0):

Later, in 1992, Zhu [5] considered the nonlinear differential equation of the third order with the constant deviating argument (>0):

and he discussed the existence of periodic solutions for this equation when is a periodic function of period , .

In 2000, Tejumola and Tchegnani [6] considered the nonlinear differential equation of the third order with the constant deviating argument (>0:

The authors established certain sufficient conditions on the existence of periodic of solutions of this equation.

In 2010, Tunç [7] established certain sufficient conditions for the existence of a periodic solution for the nonlinear differential equation of the third order with the constant deviating argument (>0:

However, a review to date of the literature indicates that the existence of periodic solutions to the nonlinear differential equation of the third order with multiple deviating arguments has not been investigated. The paper considers the nonlinear differential equation of the third order with multiple constant deviating arguments :

The equation (1.5) is stated in system form as follows:

where are positive constants, that is, are constant deviating arguments, which are determined in Section 2. It is assumed that the functions ,   ,  , and are continuous in their respective arguments on , , , and ,  , , respectively; and is periodic in of period ,, the derivatives exist and are also continuous; throughout what follows , , and are abbreviated as ,, and , respectively.

The motivation for this paper is a result of the research mentioned regarding ordinary differential equations with a deviating argument. Our aim is to achieve the results established in [5, 7] to (1.5) with multiple deviating arguments. Our results generalize the results established on the existence of periodic solution in [5, 7]. This paper is the first known publication regarding the existence of periodic solution for differential equations of the third order with multiple deviating arguments.

In order to reach our main result, this paper offers fundamental information regarding the general nonautonomous delay periodic differential system. Consider the delay periodic system:

where is a continuous mapping, for all and for some constant . We assume that takes closed bounded sets into bounded sets of . Here is the Banach space of continuous function with supremum norm, ; for , we define by is the open -ball in ,.

Theorem 1.1. Suppose that and is periodic in of period , and consequently for any there exists an such that implies . Suppose that a continuous Lyapunov functional exists, defined on ,, is the set of such that with ( may be large), and that satisfies the following conditions.

(i)Continuous increasing functions and exist, satisfying , for and as , such that

(ii)A continuous and positive function exists such that

(iii)A constant ,, exists such that

where is a constant which is determined in the following way.

Using the condition on , constants ,, and exist such that ,, and ,    is defined by .

Under these conditions, a periodic solution of (1.7) of period exists. In particular, the relation is always satisfied if is sufficiently small (see Yoshizawa [3]).

2. Main Result

The main result is the following theorem:

let .

Theorem 2.1. Suppose that positive constants ,  ,  ,  ,  , and  exist such that the following conditions hold:

If

then (1.5) has a periodic solution of period , where , , and

Proof. Define a Lyapunov functional by:

where

,, and are certain positive constants; the constants will be determined later in the proof.

It follows that . In view of the assumptions ,,,, ,, and , obtain

Using the assumptions of Theorem 2.1, have

so that

where .

It is also clear that the function is continuous and satisfies

In view of (2.3), (2.7), (2.8), and the assumptions of Theorem 2.1, it can be shown that satisfies the condition (i) of Theorem 1.1.

Using a basic calculation, the time derivative of along solutions of (1.6) results in

The assumption and the estimate imply

so that

Using the assumptions , and , and the estimation , it follows that

If we choose and use the assumption , then

An easy calculation from and (1.6) leads to

so that

using the assumptions of Theorem 2.1.

First, we consider in the domain, where the constants and are large enough, which will be determined later. We have to discuss the following two cases.

Case (, and , are arbitrary). In this case, it follows that:

By the estimates (2.3), (2.13), (2.16), and , we get

We now consider the term

and define

Then, there exists a constant , satisfying for , so that, when ,

Hence

where

If

then the above estimate implies

for a positive constant .

Let

so that

provided that .

Let . If , then

Case (, and , are arbitrary). Then

By following a similar method shown in the first case, select and take

one can easily obtain

for some positive constants and provided that .

At the end, we consider in .

Let , where the constant will be determined later. Hence

It follows from (2.3), (2.13), and (2.31) that

Since as and , then we have

so that

In view of the above discussion, it follows that:

Subject to the evidence thus far, we can conclude that there exists a positive constant , which is large enough, such that

where . Thus, the Lyapunov functional satisfies all the assumptions of Theorem 1.1. The proof for Theorem 2.1 is complete.

Example 2.2. Consider the following nonlinear differential equation of the third order with two constant deviating arguments, :

which is a special case of (1.5).

This equation can be written in the system form as follows:

When we compare the system described to this point with (1.6), it follows the existence of the following estimates:

In view of the above estimates, it is seen that all the assumptions of Theorem 2.1 hold. This discussion verifies that (2.37) has a periodic solution of period ,.

Acknowledgment

The author would like to expresses his sincere thanks and best regards to the anonymous referees for their many helpful comments, corrections, and suggestions on the paper.

References

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

V. Kolmanovskii and A. Myshkis, Introduction To the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.

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

E. N. Chukwu, “On the boundedness and the existence of a periodic solution of some nonlinear third order delay differential equation,” Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, vol. 64, no. 5, pp. 440–447, 1978.

Y. F. Zhu, “On stability, boundedness and existence of periodic solution of a kind of third order nonlinear delay differential system,” Annals of Differential Equations, vol. 8, no. 2, pp. 249–259, 1992.

H. O. Tejumola and B. Tchegnani, “Stability, boundedness and existence of periodic solutions of some third and fourth order nonlinear delay differential equations,” Journal of the Nigerian Mathematics Society, vol. 19, pp. 9–19, 2000.

C. Tunç, “On the existence of periodic solutions to nonlinear third order ordinary differential equations with delay,” Journal of Computational Analysis and Applications, vol. 12, no. 1, pp. 191–201, 2010.