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:
(1)x′′+cx′-ax(t)+bxm(t-τ(t))=p(t),
where τ(t) and p(t) are almost periodic functions on R, m>1, and a, b, and c 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
(2)x′′+c(t)x′-a(t)x(t)+b(t)xm(t-τ(t))=p(t),
where c(t), a(t), b(t), and p(t) are continuous functions on R, m>1 is a constant, and τ(t)≥0 is continuous functions on R. There exists a constant τ such that τ=supt∈Rτ(t). 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 a1 be a constant. Define
(3)y(t)=dx(t)dt+a1x(t).
Then system (2) can be transformed into the following equivalent system:
(4)dx(t)dt=-a1x(t)+y(t),dy(t)dt=-(c(t)-a1)y(t)+[a(t)-a1(a1-c(t))]x(t)-b(t)xm(t-τ(t))+p(t).
Let BC((-∞,0],R) denote the space of bounded continuous functions φ:[-∞,0]→R with the supremum norm ∥·∥. According to Burton [38], Hale [39], and Yoshizawa [40], we know that for a(t), b(t), c(t), and p(t) continuous, given a continuous initial function φ∈BC((-∞,0],R) and a vector y0∈R, there exists a solution of (4) on an interval [0,T) satisfying the initial condition and satisfying (4) on [0,T). If the solution remains bounded, then T=+∞. We denote such a solution by (x(t),y(t))=(x(t,φ,y0),y(t,φ,y0)). Let y(s)=y(0) for all s∈[-τ,0]. It follows that (x(t),y(t)) can be defined on [-τ,+∞].

Definition 1.

Let u(t):R→R be continuous function in t. u(t) is said to be T-antiperiodic on R if
(5)u(t+T)=-u(t),
for all t∈R.

Definition 2.

Let Z*(t)=(x*(t),y*(t)) be an antiperiodic solution of (4) with initial value (φ*(t),y0*)∈BC((-∞,0],R)×R×R. If there exist constants λ>0 and M>1 such that for every solution Z(t)=(x(t),y(t)) of (4) with an initial value φ=(φ(t),y0)∈BC((-∞,0],R)×R,
(6)max{|x(t)-x*(t)|,|y(t)-y*(t)|}≤Mmax{∥φ(t)-φ*(t)∥,|y0-y0*|}e-λt
for all t>0, where
(7)∥φ(t)-φ*(t)∥=supt∈(-∞,0]|φ(t)-φ*(t)|.
Then Z*(t) is said to be globally exponentially stable.

Throughout this paper, we make the following assumptions.

There exists a constant T>0 such that
(8)a(t+T)=a(t),b(t+T)=b(t),c(t+T)=c(t),τ(t+T)=τ(t),p(t+T)=-p(t),
for all t,u∈R.

There exists a constant p+ such that, for all t>0,
(9)p+=supt∈R|p(t)|.

There exists a constant λ>0 such that λ-(a1-1)<0 and
(10)λ-inft∈R(c(t)-a1)+supt∈R|a(t)-a1(a1-c(t))|+b+[(m-1)δm-1+1]eλτ<0.
The organization of this paper is as follows. In Section 2, we give some preliminary results. In Section 3, we derive the existence of T-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 (x~(t),y~(t)) is a solution of (4) with initial conditions
(11)x~(s)=φ(s),y~(s)=y0,max{|x~(s)|,|y0|}<δ,s∈[-τ,0],
where δ satisfies
(12)inft∈R(c(t)-a1)δ-supt∈R[a(t)-a1(a1-c(t))]δ-b+δm-p+>0.
Then,
(13)max{|x~(t)|,|y~(t)|}<δ,
for all t≥0.

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 t1>0 such that
(14)max{|x~(t1)|,|y~(t1)|}=|x~(t1)|=δ,max{|x~(t)|,|y~(t)|}<δ,
where t∈[-τ,t1).

Case 2. There exists t2>0 such that
(15)max{|x~(t2)|,|y~(t2)|}=|y~(t2)|=δ,max{|x~(t)|,|y~(t)|}<δ,
where t∈[-τ,t2).

If Case 1 holds true, we can calculate the upper left derivative of |x~(t)| as follows:
(16)0≤D+(|x~(t1)|)≤-a1|x(t1)|+|y(t1)|≤-(a1-1)δ<0,
which is a contradiction. Then (13) holds.

If Case 2 holds true, we can calculate the upper left derivative of |y~(t)| as follows:
(17)0≤D+(|y~(t2)|)≤-(c(t2)-a1)|y(t2)|+|[a(t2)-a1(a1-c(t2))]x(t2)-b(t)xm(t2-τ(t2))+p(t2)[a(t2)-a1(a1-c(t2))]|≤-inft∈R(c(t)-a1)|y(t2)|+supt∈R[a(t)-a1(a1-c(t))]|x(t2)|+b+|xm(t2-τ(t2))|+p+≤-[inft∈R(c(t)-a1)δ-supt∈R[a(t)-a1(a1-c(t))]δ≤2--b+δm-p+inft∈R(c(t)-a1)δ-supt∈R[a(t)-a1(a1-c(t))]δ]<0,
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 (x~(t),y~(t)) can be defined on [0,+∞).

Lemma 5.

Suppose that (H1)–(H3) hold. Let Z*(t)=(x*(t),y*(t)) be the solution of (4) with initial values (φ*(s),y0*)∈C([-τ,0],R)×R, and let Z(t)=(x(t),y(t)) be the solution of (4) with initial value (φ(s),y0)∈C([-τ,0],R)×R. Then there exists a constant M>1 such that
(18)max{|x(t)-x*(t)|,|y(t)-y*(t)|}≤Mmax{∥φ(t)-φ*(t)∥,|y0-y0*|}e-λt
for all t>0.

Proof.

Let u¯(t)={x(t)-x*(t)}, v¯(t)={y(t)-y*(t)}. Then,
(19)du¯(t)dt=-a1u¯(t)+v¯(t),dv¯(t)dt=-(c(t)-a1)v¯(t)+[a(t)-a1(a1-c(t))]u¯(t)-b(t)[xm(t-τ(t))-x*m(t-τ(t))].
In the sequel, we define the Lyapunov functional as follows:
(20)V1(t)=|u¯(t)|eλt,V2(t)=|v¯(t)|eλt.
Calculating the upper left derivative of Vi(t)(i=1,2) along the solution (u¯(t),v¯(t)) of system (20) with the initial value (φ(t)-φ*(t),y0-y0*), we have
(21)D+(V1(t))≤λ|u¯(t)|eλt+D+(|u¯(t)|)eλt≤λ|u¯(t)|eλt+eλtsign(u¯(t))[-a1u¯(t)+v¯(t)]≤eλt[(λ-a1)|u¯(t)|+|v¯(t)|],(22)D+(V2(t))≤λ|v¯(t)|eλt+D+(|v¯(t)|)eλt≤λ|v¯(t)|eλt+eλtsign(v¯(t))×{-(c(t)-a1)v¯(t)[x*m((t))]+[a(t)-a1(a1-c(t))]u¯(t)-b(t)×[xm(t-τ(t))-x*m(t-τ(t))]}≤eλt{[λ-(c(t)-a1)]|v¯(t)|+b+[mδm-1]|(t-τ(t))|≤1eλt+supt∈R|a(t)-a1(a1-c(t))||u¯(t)|≤1eλt+b+[mδm-1+1]|u¯(t-τ(t))|}.
Let M>1 be an arbitrary real number and set
(23)ϱ=max{∥φ-φ*∥,|y0-y0*|}>0.
Then by (21), we have
(24)V1(t)=|u¯(t)|eλt<Mϱ,V2(t)=|v¯(t)|eλt<Mϱ,∀t∈[-τ,0].
Thus we can claim that
(25)V1(t)=|u¯(t)|eλt<Mϱ,V2(t)=|v¯(t)|eλt<Mϱ,∀t>0.
Otherwise, one of the following cases must occur.

Case (a). There exists T1>0 such that
(26)V1(T1)=Mϱ,Vi(t)<Mϱ,∀t∈(-τ,T1),i=1,2.

Case (b). There exists T2>0 such that
(27)V2(T2)=Mϱ,Vi(t)<Mϱ,∀t∈(-τ,T2),i=1,2.
If Case (a) holds, then it follows from (21) and (26) that
(28)0≤D+(V1(T1))≤(λ-a1)|u¯(T1)|eλT1+|v¯(T1)|eλT1≤[λ-(a1-1)]Mϱ.
Then,
(29)0≤λ-(a1-1),
which contradicts (H3). Then (25) holds.

If Case (b) holds, then it follows from (22) and (27) that
(30)0≤D+(V2(T2))≤[λ-inft∈R(c(t)-a1)]|v¯(T2)|eλT2+supt∈R|a(t)-a1(a1-c(t))||u¯(T2)|eλT2+b+[mδm-1+1]|u¯(T2-τ(T2))|eλ(T2-τ(T2))eλτ(T2)≤[λ-inft∈R(c(t)-a1)]Mϱ+supt∈R|a(t)-a1(a1-c(t))|Mϱ+b+[mδm-1+1]Mϱeλτ(T2)≤{λ-inft∈R(c(t)-a1)+supt∈R|a(t)-a1(a1-c(t))|+b+[mδm-1+1]eλτλ-inft∈R+supt∈R|a(t)-(a1)|}Mϱ.
Then,
(31)0≤λ-inft∈R(c(t)-a1)+supt∈R|a(t)-a1(a1-c(t))|+b+[mδm-1+1]eλτ,
which contradicts (H3). Then (25) holds. It follows that
(32)max{|x(t)-x*(t)|,|y(t)-y*(t)|}≤Mmax{∥φ(t)-φ*(t)∥,|y0-y0*|}e-λt
for all t>0. This completes the proof of Lemma 5.

Remark 6.

If Z*(t)=(x*(t),y*(t)) is a T-antiperiodic solution of (4), it follows from Lemma 5 and Definition 2 that Z*(t) 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 T-antiperiodic solution Z*(t)=(x*(t),y*(t)). Moreover, this solution is globally exponentially stable.

Proof.

Let v(t)=(v1(t),v2(t))=(x(t),y(t)) be a solution of (4) with initial conditions (11). Thus according to Lemma 3, the solution v(t) is bounded and (13) holds. From (4), for any natural number p, we derive
(33)((-1)p+1x(t+(p+1)T))′=(-1)p+1x′(t+(p+1)T)=(-1)p+1[-a1x(t+(p+1)T)+y(t+(p+1)T)]=-a1(-1)p+1x(t+(p+1)T)+(-1)p+1y(t+(p+1)T),((-1)p+1y(t+(p+1)T))′=(-1)p+1y′(t+(p+1)T)=(-1)p+1{-(c(t+(p+1)T)-a1)y(t+(p+1)T)xm[((p+1)T)]=(-11)p+1+[a(t+(p+1)T)-a1(a1-c(t+(p+1)T))]=(-11)p+1×x(t+(p+1)T)=(-11)p+1-b(t+(p+1)T)xm=(-11)p+1×(t+(p+1)T-τ(t+(p+1)T))=(-11)p+1+p(t+(p+1)T)xm[a(t+(p+1)T)-a1(a1-c(t+(p+1)T))]}=-(c(t)-a1)(-1)p+1y(t+(p+1)T)+[a(t)-a1(a1-c(t))](-1)p+1x(t+(p+1)T)-b(t)(-1)p+1xm(t+(p+1)T-τ(t))+p(t).
Thus (-1)p+1v(t+(p+1)T) are the solutions of (4) on R for any natural number p. Then, from Lemma 5, there exists a constant M>1 such that
(34)|(-1)p+1vi(t+(p+1)T)-(-1)kvi(t+pT)|≤Me-λ(t+pT)sup-τ≤s≤0max1≤i≤2|vi(s+T)+vi(s)|≤2e-λ(t+pT)Mδ,∀t+pT>0,i=1,2.
Thus, for any natural number q, we have
(35)(-1)q+1vi(t+(q+1)T)=vi(t)+∑k=0q[(-1)k+1vi(t+(k+1)T)=vi(t)+∑k=0q-(-1)kvi(t+kT)].
Hence,
(36)|(-1)q+1vi(t+(q+1)T)|≤|vi(t)|+∑k=0q|(-1)k+1vi(t+(k+1)T)≤|vi(t)|+∑k=0q-(-1)kvi(t+kT)|,
where i=1,2. By (35), we can choose a sufficiently large constant N>0 and a positive constant γ such that
(37)|(-1)p+1vi(t+(p+1)T)-(-1)kvi(t+pT)|≤γ(e-λT)k,∀k>N,i=1,2,
on any compact set of R. It follows from (36) and (37) that {(-1)qv(t+qT)} uniformly converges to a continuous function Z*(t)=(x*(t),y*(t)) on any compact set of R.

Now we show that Z*(t) is T-antiperiodic solution of (4). Firstly, Z*(t) is T-antiperiodic, since
(38)Z*(t+T)=limq→∞(-1)qv(t+T+qT)=-lim(q+1)→∞(-1)q+1v(t+(q+1)T)=-Z*(t).
In the sequel, we prove that Z*(t) is a solution of (4). Noting that the right-hand side of (4) is continuous, (33) shows that {((-1)q+1v(t+(q+1)T))′} uniformly converges to a continuous function on any compact subset of R. Thus, letting q→∞ on both sides of (33), we can easily obtain
(39)dx*(t)dt=-a1x*(t)+y*(t),dy*(t)dt=-(c(t)-a1)y*(t)+[a(t)-a1(a1-c(t))]x*(t)-b(t)x*m(t-τ(t))+p(t).
Therefore, Z*(t) is a solution of (4). Applying Lemma 5, we can easily check that Z*(t) 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,
(40)x′′+(10+|sint|)x′-(0.5+|cost|)x(t)+0.2sintx2(t-0.01|sint|)=0.2sint,
has exactly one π-antiperiodic solution.

Proof.

Let
(41)y(t)=dx(t)dt+0.5x(t).
Then system (40) can be transformed into the following equivalent system:
(42)dx(t)dt=-0.5x(t)+y(t),dy(t)dt=-(10+|sint|-0.5)y(t)+[0.5+|cost|-0.5(0.5-(10+|sint|))]x(t)-0.2sintx2(t-0.01|sint|)+0.2sint.
Corresponding to system (3) and (4), we have
(43)a1=0.5,a(t)=0.5+|cost|,b(t)=0.2sint,c(t)=10+|sint|,m=2,p(t)=0.2sint,τ(t)=0.01|sint|.
Let δ=0.5, λ=0.02. Then τ=0.01, p+=0.2. It is easy to check that all the conditions in Theorem 7 are fulfilled. Hence we can conclude that system (42) has exactly one π-antiperiodic solution. Moreover, this π-periodic solution is globally exponentially stable. Thus system (40) has exactly one π-antiperiodic solution, and all solutions of system (40) exponentially converge to this π-antiperiodic solution. This result is illustrated in Figure 1.

Time response of state variables x and y.

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 (J[2012]2100), Governor Foundation of Guizhou Province ([2012]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 ([2013]14).

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