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).