Research Article | Open Access

# Antiperiodic Solutions for a Kind of Nonlinear Duffing Equations with a Deviating Argument and Time-Varying Delay

**Academic Editor:**Shao-Ming Fei

#### Abstract

This paper deals with a kind of nonlinear Duffing equation with a deviating argument and time-varying delay. By using differential inequality techniques, some very verifiable criteria on the existence and exponential stability of antiperiodic solutions for the equation are obtained. Our results are new and complementary to previously known results. An example is given to illustrate the feasibility and effectiveness of our main results.

#### 1. Introduction

In recent years, Duffing equations have attracted much attention due to its wide range of applications in many practical problems such as in physics, mechanics, and the engineering fields. Many results on various Duffing equations are available (see [1–12]). However, to the best of our knowledge, there are few results on the antiperiodic solutions of Duffing equations. Many authors argue that in many applied science fields the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations [13–36]. This motivates us to focus on the existence and stability of antiperiodic solutions for Duffing equations. In 2010, Peng and Wang [37] consider the existence of positive almost periodic solutions for the following nonlinear Duffing equation with a deviating argument: where and are almost periodic functions on , , and , , and are constants. By applying some analysis technique, Peng and Wang [37] obtained the results on the existence of positive almost periodic solutions for system (1).

In this paper, we will consider the antiperiodic solutions of the following more general Duffing equation with a deviating argument and time-varying delay which takes the form where , , , and are continuous functions on , is a constant, and is continuous functions on . There exists a constant such that . By using differential inequality techniques, a series of new sufficient conditions for the existence, uniqueness, and exponential stability of antiperiodic solutions of system (2) are established. In addition, an example is presented to illustrate the effectiveness of our main results.

Let be a constant. Define Then system (2) can be transformed into the following equivalent system: Let denote the space of bounded continuous functions with the supremum norm . According to Burton [38], Hale [39], and Yoshizawa [40], we know that for , , , and continuous, given a continuous initial function and a vector , there exists a solution of (4) on an interval satisfying the initial condition and satisfying (4) on . If the solution remains bounded, then . We denote such a solution by . Let for all . It follows that can be defined on .

*Definition 1. *Let be continuous function in . is said to be -antiperiodic on if
for all .

*Definition 2. *Let be an antiperiodic solution of (4) with initial value . If there exist constants and such that for every solution of (4) with an initial value ,
for all , where
Then is said to be globally exponentially stable.

Throughout this paper, we make the following assumptions.(H1)There exists a constant such that for all .(H2)There exists a constant such that, for all , (H3)There exists a constant such that and The organization of this paper is as follows. In Section 2, we give some preliminary results. In Section 3, we derive the existence of -antiperiodic solution, which is globally exponentially stable. An example is provided to illustrate the effectiveness of our main results in Section 4.

#### 2. Preliminary Results

In this section, we will first present two important lemmas which are used in what follows.

Lemma 3. *Let (H1)-(H2) hold. Suppose that is a solution of (4) with initial conditions
**
where satisfies
**
Then,
**
for all .*

*Proof. *By way of contradiction, we assume that (13) do not hold. Then one of the following two cases must occur. *Case 1.* There exists such that
where .*Case 2.* There exists such that
where .

If Case 1 holds true, we can calculate the upper left derivative of as follows:
which is a contradiction. Then (13) holds.

If Case 2 holds true, we can calculate the upper left derivative of as follows:
which is a contradiction. Then (13) holds.

*Remark 4. *It follows from the boundedness of this solution and the theory of functional differential equations in [36] that can be defined on .

Lemma 5. *Suppose that (H1)–(H3) hold. Let be the solution of (4) with initial values , and let be the solution of (4) with initial value . Then there exists a constant such that
**
for all .*

*Proof. *Let , . Then,
In the sequel, we define the Lyapunov functional as follows:
Calculating the upper left derivative of along the solution of system (20) with the initial value , we have
Let be an arbitrary real number and set
Then by (21), we have
Thus we can claim that
Otherwise, one of the following cases must occur.*Case (a)*. There exists such that
*Case (b)*. There exists such that
If Case (a) holds, then it follows from (21) and (26) that
Then,
which contradicts (H3). Then (25) holds.

If Case (b) holds, then it follows from (22) and (27) that
Then,
which contradicts (H3). Then (25) holds. It follows that
for all . This completes the proof of Lemma 5.

*Remark 6. *If is a -antiperiodic solution of (4), it follows from Lemma 5 and Definition 2 that is globally exponentially stable.

#### 3. Main Results

In this section, we present our main result that there exists the exponentially stable antiperiodic solution of (1).

Theorem 7. *Assume that (H1)–(H3) are fulfilled. Then (4) with the initial condition (11) has exactly one -antiperiodic solution . Moreover, this solution is globally exponentially stable.*

*Proof. *Let be a solution of (4) with initial conditions (11). Thus according to Lemma 3, the solution is bounded and (13) holds. From (4), for any natural number , we derive
Thus are the solutions of (4) on for any natural number . Then, from Lemma 5, there exists a constant such that
Thus, for any natural number , we have
Hence,
where . By (35), we can choose a sufficiently large constant and a positive constant such that
on any compact set of . It follows from (36) and (37) that uniformly converges to a continuous function on any compact set of .

Now we show that is -antiperiodic solution of (4). Firstly, is -antiperiodic, since
In the sequel, we prove that is a solution of (4). Noting that the right-hand side of (4) is continuous, (33) shows that uniformly converges to a continuous function on any compact subset of . Thus, letting on both sides of (33), we can easily obtain
Therefore, is a solution of (4). Applying Lemma 5, we can easily check that is globally exponentially stable. The proof of Theorem 7 is completed.

#### 4. An Example

In this section, we give an example to illustrate our main results obtained in previous sections.

*Example 8. *The following two-order Duffing equation with two deviating arguments,
has exactly one -antiperiodic solution.

*Proof. *Let
Then system (40) can be transformed into the following equivalent system:
Corresponding to system (3) and (4), we have
Let , . Then , . It is easy to check that all the conditions in Theorem 7 are fulfilled. Hence we can conclude that system (42) has exactly one -antiperiodic solution. Moreover, this -periodic solution is globally exponentially stable. Thus system (40) has exactly one -antiperiodic solution, and all solutions of system (40) exponentially converge to this -antiperiodic solution. This result is illustrated in Figure 1.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010, no. 11201138), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J2100), Governor Foundation of Guizhou Province (53), Natural Science and Technology Foundation of Guizhou Province (2014), Scientific Research Fund of Hunan Provincial Education Department (no. 12B034), and Natural Science Innovation Team Project of Guizhou Province (14).

#### References

- Z. B. Cheng and J. L. Ren, “Harmonic and subharmonic solutions for superlinear damped Duffing equation,”
*Nonlinear Analysis: Real World Applications*, vol. 14, no. 2, pp. 1155–1170, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. U. Akhmet and M. O. Fen, “Chaotic period-doubling and OGY control for the forced Duffing equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 4, pp. 1929–1946, 2012. View at: Publisher Site | Google Scholar | MathSciNet - V. Marinca and N. Heri{\cS}anu, “Explicit and exact solutions to cubic DUFfing and double-well DUFfing equations,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 604–609, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Wang, “Novel existence and uniqueness criteria for periodic solutions of a Duffing type
*p*-Laplacian equation,”*Applied Mathematics Letters*, vol. 23, no. 4, pp. 436–439, 2010. View at: Publisher Site | Google Scholar | MathSciNet - B. Du, C. Bai, and X. Zhao, “Problems of periodic solutions for a type of Duffing equation with state-dependent delay,”
*Journal of Computational and Applied Mathematics*, vol. 233, no. 11, pp. 2807–2813, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Llibre and A. Rodrigues, “A note on the periodic orbits of a kind of Duffing equations,”
*Applied Mathematics and Computation*, vol. 219, no. 15, pp. 8358–8365, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y.-K. Li and L. Yang, “Anti-periodic solutions for Cohen-Grossberg neural networks with bounded and unbounded delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 7, pp. 3134–3140, 2009. View at: Publisher Site | Google Scholar | MathSciNet - B. Ahmad and B. S. Alghamdi, “Approximation of solutions of the nonlinear Duffing equation involving both integral and non-integral forcing terms with separated boundary conditions,”
*Computer Physics Communications*, vol. 179, no. 6, pp. 409–416, 2008. View at: Publisher Site | Google Scholar | MathSciNet - Q. Y. Zhou and B. W. Liu, “New results on almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 6–11, 2009. View at: Publisher Site | Google Scholar | MathSciNet - L.-Q. Peng, “Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments,”
*Mathematical and Computer Modelling*, vol. 45, no. 3-4, pp. 378–386, 2007. View at: Publisher Site | Google Scholar | MathSciNet - Q. Zhou and B. Liu, “New results on almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 6–11, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Y. Xu, “Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument,”
*Electronic Journal of Qualitative Theory of Differential Equations*, no. 80, pp. 1–9, 2012. View at: Google Scholar - J. Y. Shao, “Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 372, no. 30, pp. 5011–5016, 2008. View at: Publisher Site | Google Scholar - Q. Fan, W. Wang, and X. Yi, “Anti-periodic solutions for a class of nonlinear $n$th-order differential equations with delays,”
*Journal of Computational and Applied Mathematics*, vol. 230, no. 2, pp. 762–769, 2009. View at: Publisher Site | Google Scholar | MathSciNet - T. Zhang, Y. Li, and E. Xu, “Existence and stability of antiperiodic solution for a class of generalized neural networks with impulses and arbitrary delays on time scales,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 132790, 19 pages, 2010. View at: Publisher Site | Google Scholar - A. R. Aftabizadeh, S. Aizicovici, and N. H. Pavel, “On a class of second-order anti-periodic boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 171, no. 2, pp. 301–320, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Aizicovici, M. McKibben, and S. Reich, “Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 43, no. 2, pp. 233–251, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Gong, “Anti-periodic solutions for a class of Cohen-Grossberg neural networks,”
*Computers and Mathematics with Applications*, vol. 58, no. 2, pp. 341–347, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Liu, “An anti-periodic LaSalle oscillation theorem for a class of functional differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 223, no. 2, pp. 1081–1086, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Ou, “Anti-periodic solutions for high-order Hopfield neural networks,”
*Computers and Mathematics with Applications*, vol. 56, no. 7, pp. 1838–1844, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Peng and L. Huang, “Anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2434–2440, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z. Huang, L. Peng, and M. Xu, “Anti-periodic solutions for high-order cellular neural networks with time-varying delays,”
*Electronic Journal of Differential Equations*, vol. 2010, no. 5, pp. 1–9, 2010. View at: Google Scholar - A. P. Zhang, “Existence and exponential stability of anti-periodic solutions for HCNNs with timevarying leakage delays,”
*Advances in Difference Equations*, vol. 2013, article 162, 2013. View at: Publisher Site | Google Scholar - Y. Li, L. Yang, and W. Wu, “Anti-periodic solutions for a class of Cohen-Grossberg neutral networks with time-varying delays on time scales,”
*International Journal of Systems Science*, vol. 42, no. 7, pp. 1127–1132, 2011. View at: Publisher Site | Google Scholar | MathSciNet - L. Pan and J. Cao, “Anti-periodic solution for delayed cellular neural networks with impulsive effects,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 6, pp. 3014–3027, 2011. View at: Publisher Site | Google Scholar | MathSciNet - Y. Li and J. Shu, “Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 8, pp. 3326–3336, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Q. Fan, W. Wang, X. Yi, and L. Huang, “Anti-periodic solutions for a class of third-order nonlinear differential equations with a deviating argument,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 8, no. 12, pp. 1–12, 2011. View at: Google Scholar - W. Wang and J. Shen, “Existence of solutions for anti-periodic boundary value problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 2, pp. 598–605, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Y. Chen, J. J. Nieto, and D. O'Regan, “Anti-periodic solutions for fully nonlinear first-order differential equations,”
*Mathematical and Computer Modelling*, vol. 46, no. 9-10, pp. 1183–1190, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Peng and W. Wang, “Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays in leakage terms,”
*Neurocomputing*, vol. 111, pp. 27–33, 2013. View at: Publisher Site | Google Scholar - J. Y. Park and T. G. Ha, “Existence of anti-periodic solutions for quasilinear parabolic hemivariational inequalities,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 3203–3217, 2009. View at: Publisher Site | Google Scholar - Y. Yu, J. Shao, and G. Yue, “Existence and uniqueness of anti-periodic solutions for a kind of Rayleigh equation with two deviating arguments,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 71, no. 10, pp. 4689–4695, 2009. View at: Publisher Site | Google Scholar - X. Lv, P. Yan, and D. Liu, “Anti-periodic solutions for a class of nonlinear second-order Rayleigh equations with delays,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 11, pp. 3593–3598, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Li and L. Huang, “Anti-periodic solutions for a class of Liénard-type systems with continuously distributed delays,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 4, pp. 2127–2132, 2009. View at: Publisher Site | Google Scholar | MathSciNet - B. W. Liu, “Anti-periodic solutions for forced Rayleigh-type equations,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 5, pp. 2850–2856, 2009. View at: Publisher Site | Google Scholar | MathSciNet - P. Shi and L. Dong, “Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses,”
*Applied Mathematics and Computation*, vol. 216, no. 2, pp. 623–630, 2010. View at: Publisher Site | Google Scholar | MathSciNet - L. Peng and W. Wang, “Positive almost periodic solutions for a class of nonlinear Duffing equations with a deviating argument,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 6, pp. 1–12, 2010. View at: Google Scholar | MathSciNet - T. A. Burton,
*Stability and Periodic sSolutions of Ordinary and Functional Differential Equations*, Academic Press, Orlando, Fla, USA, 1985. View at: MathSciNet - J. K. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 1977. View at: MathSciNet - T. Yoshizawa, “Asymptotic behaviors of solutions of differential equations,” in
*Differential Equations: Qualitative Theory (Szeged, 1984)*, vol. 47 of*Colloquia Mathematica Societatis János Bolyai*, pp. 1141–1164, North-Holland, Amsterdam, The Netherlands, 1987. View at: Google Scholar

#### Copyright

Copyright © 2014 Changjin Xu and Maoxin Liao. 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.