Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

1. Introduction

It is well known that the problem of the existence of periodic solutions of retarded functional differential equations of second order is not only very important in the background applications, but also of considerable significance in theory of differential equations. Besides, the scope of retarded functional differential equations is very general. For example, it contains ordinary differential equations, differential-difference equations, integrodifferential equations, and so on. The motivation of this paper is that in recent years the study of the existence of periodic solutions to various kinds of retarded functional differential equations of second order has become one of the most attractive topics in the literature. Especially, by using the famous continuation theorem of degree theory (see Gaines and Mawhin [1]), many authors have made a lot of interesting contributions to the topic for retarded functional differential equations of second order. Here, we would not like to give the details of these works.

On the other hand, amongst the achieved excellent results, one of them is the famous Yoshizawa’s theorem [2] for existence of periodic solutions of retarded functional differential equations, which has vital influence and has been widely used in the literature. This theorem has also been generally one of the best results in the literature from the past till now. It should be noted that in 1994, Zhao et al. [3] proved four sufficiency theorems on the existence of periodic solutions for a class of retarded functional differentials to have the existence of an ω-periodic solution. By this work, the authors proved that their theorems are better than Yoshizawa’s [2] periodic solutions theorem primarily by removing restrictions of the size of the constant delay h. An example of application was given at the end of the paper. Namely, in the same paper, the authors applied the following Theorem A to discuss the existence of an ω-periodic solution of the nonlinear delay differential equation of the second order:
(1)x′′(t)+ax′(t)+g(x(t-τ))=p(t),
where p(t) is an external force, g(x(t-τ)) is a delayed restoring force, the delay τ is a positive constant and the friction is proportional to the velocity, and a is a positive constant. It should be noted that a feedback system with friction proportional to velocity, an external force p(t), and a delayed restoring force g(x(t-τ)), (τ>0) may be written as (1) (see Burton [4]).

Consider the following general nonautonomous delay differential equation:
(2)x˙=F(t,xt),xt=x(t+θ),-h≤θ≤0,t≥0,
where F:ℜ×C→ℜn, C=C([-h,0],ℜn), h is a positive constant, and we suppose that F is continuous, ω-periodic, and takes closed bounded sets into bounded sets of ℜn; and such that solutions of initial value problems are unique, h can be either larger than ω, or equal to or smaller than ω. Here (C,∥·∥) is the Banach space of continuous function ϕ:[-h,0]→ℜn with supremum norm; h>0, CH is the open H-ball in C; CH∶={ϕ∈C([-h,0],ℜn):∥ϕ∥<H}. Standard existence theory, see Burton [4], shows that if ϕ∈CH and t≥0, then there is at least one continuous solution x(t,t0,ϕ) such that on [t0,t0+α) satisfying (2) for t>t0, xt(t,ϕ)=ϕ and α is a positive constant. If there is a closed subset B⊂CH such that the solution remains in B, then α=∞. Further, the symbol |·| will denote a convenient norm in ℜn with |x|= max1≤i≤n|xi|. Let us assume that C(t)={ϕ:[t-α,t]→ℜn∣ϕ is continuous} and ϕt denotes the ϕ in the particular C(t), and that ∥ϕt∥=maxt-α≤s≤t|ϕ(t)|. Finally, by the periodicity, we mean that that there is an ω>0 such that F(t,ϕ) is ω-periodic in the sense that if x(t) is a solution of (2) so is x(t+ω).

Definition 1.

Solutions of (2) are uniform bounded at t=0 if for each B1 there exists B2 such that [ϕ∈C,∥ϕ∥<B1,t≥0] imply that |x(t,0,ϕ)|<B2 (see Burton [4]).

Definition 2.

Solutions of (2) are uniform ultimate bounded for bound B at t=0 if for each B3>0 there exists a K>0 such that [ϕ∈C,∥ϕ∥<B3,t≥K] imply that |x(t,0,ϕ)|<B (see Burton [4]).

The first theorem given in Zhao et al. [3] is the following.

Theorem A.

If the solutions of (2) are ultimately bounded by the bound B, then

equation (2) has an ω-periodic solution and is bounded by B,

if (2) is autonomous, then (2) has an equilibrium solution and is bounded by B (see Zhao et al. [3]).

Regarding (1) Zhao et al. [3] proved the following theorem as example of application.

Theorem B.

Assume that the following conditions hold:

p(t) is an ω-periodic continuous function, g is a continuous differentiable function,

lim|x|→∞g(x)sgnx=∞, and there is a bounded set Ω containing the origin such that |g′(x)|≤c on Ωc; Ωc is the complement of the set Ω,

τc<a.

Then, (1) has an ω-periodic motion.

Moreover, when p(t)=K, K is a constant, under the above conditions (1) has a constant motion x=c0, and the constant c0 satisfies g(c0)=K. In fact, from the condition (2), there is a bounded set Ω1 containing the origin such that |g′(x)|≤c and ∫0xg(s)ds>0 on Ω1c; Ω1c is the complement of the set Ω1.

In this paper, we consider the following nonlinear differential equation of second order with multiple constant delays, τi(>0):
(3)x′′(t)+{f(x(t),x′(t))+g(x(t),x′(t))x′(t)}x′(t)+h(x(t))+∑i=1ngi(x(t-τi))=p(t),
where τi are fixed constants delay with t-τi≥0; the primes in (3) denote differentiation with respect to t∈ℜ, f, g, h, gi, and p are continuous functions in their respective arguments on ℜ2, ℜ2, ℜ, ℜ, and ℜ, respectively, and also depend only on the arguments displayed explicitly. The continuity of these functions is a sufficient condition for existence of the solution of (3). It is also assumed as basic that the functions f, g, h, and gi satisfy a Lipschitz condition in x,x′,x(t-τ1),x(t-τ2),…,x(t-τn). By this assumption, the uniqueness of solutions of (3) is guaranteed. The derivatives dgi/dx≡gi′(x) exist and are continuous. It should be noted that throughout the paper, sometimes, x(t) and y(t) are abbreviated as x and y, respectively.

We write (3) in system form as follows:
(4)x′=y,y′=-{f(x,y)+g(x,y)y}y-h(x)-∑i=1ngi(x)+∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+p(t),
where gi′(x(t+s))=dgi/dx.

It is clear that (3) is a particular case of (2). It should be noted that the reason or the motivation for taking into consideration (3) comes from the following modified Liénard type equation of the form:
(5)x′′(t)+{f(x(t))+g(x(t))x′(t)}x′(t)+h(x(t))=e(t).
These types of equations have great applications in theory and applications of the differential equations. Therefore, till now, the qualitative behaviors, the stability, boundedness, global existence, existence of periodic solutions, and so forth, of these type differential equations have been studied by many researchers, and the researches on these topics are still being done in the literature. For example, we refer the readers to the books of Ahmad and Rao [5], Burton [4], Gaines and Mawhin [1], and the papers of Constantin [6], Graef [7], Huang and Yu [8], Jin [9], Liu and Huang [10], Nápoles Valdés [11], Qian [12], Tunç [13–21], C. Tunç and E. Tunç [22], Zhao et al. [3], Zhou [23], and the references cited in these works.

We here give certain sufficient conditions to guarantee the existence of an ω-periodic solution of (3). This paper is inspired by the mentioned papers and that in the literature. Our aim is to generalize and improve the application given in [3] for (3). This paper has also a contribution to the investigation of the qualitative behaviors of retarded functional differential equations of second order, and it may be useful for researchers who work on the above mentioned topics. Finally, without using the famous continuation theorem of degree theory, which belongs to Gaines and Mawhin [1], we prove the following main result. This case makes the topic of this paper interesting.

2. Main Result

Our main result is the following.

Theorem 3.

We assume that there are positive constants a, a-, b, τ, and ci such that the following conditions hold:

a-≥f(x,y)+g(x,y)y≥a,

lim|x|→∞h(x)sgnx=∞, and there is a bounded set Ω containing the origin such that |h′(x)|≤b on Ωc; Ωc is the complement of the set Ω,

lim|x|→∞gi(x)sgnx=∞, and there is a bounded set Ω containing the origin such that |gi′(x)|≤ci on Ωc; Ωc is the complement of the set Ω,

p(t) is an ω-periodic continuous function.

If τ∑i=1nci<a, then (3) has an ω-periodic solution.

Moreover, p(t)=K, K-constant, under the above conditions (3) has a constant motion x=c0, and the constant c0 satisfies h(c0)+gi(c0)=K.

Proof.

Define the Lyapunov-Krasovskii functional V=V(xt,yt):
(6)V=12y2+∫0xh(s)ds+∑i=1n∫0xgi(s)ds+∑i=1nλi∫-τi0∫t+sty2(θ)dθds,
where λi are some positive constants to be determined later in the proof.

Evaluating the time derivative of V along system (4), we get
(7)V˙=-{f(x,y)+g(x,y)y}y2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1nλi∫-τi0(y2(t)-y2(t+s))ds=-{f(x,y)+g(x,y)y}y2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1n(λiτi)y2-∑i=1nλi∫-τi0y2(t+s)ds.
By noting the assumption |gi′(x)|≤ci of the theorem and the estimate 2|αγ|≤α2+γ2, one can obtain the following estimates:
(8)y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds≤∑i=1n∫-τi0|gi′(x(t+s))||y(t)||y(t+s)|ds≤12∑i=1n∫-τi0ci(y2(t)+y2(t+s))ds≤12∑i=1n(ciτi)y2+12∑i=1nci∫-τi0y2(t+s)ds.
Then, it follows that
(9)V˙≤-{f(x,y)+g(x,y)y}y2+yp(t)+12∑i=1n(ci+2λi)τiy2+12∑i=1n(ci-2λi)∫-τi0y2(t+s)ds.
Let λi=ci/2 and τ=max{τ1,…,τn}. In fact, these choices imply that
(10)V˙≤-{f(x,y)+g(x,y)y-τ∑i=1nci-|p(t)||y|-1}y2≤-(a-τ∑i=1nci-|p(t)||y|-1)y2.

In view of the continuity and periodicity of the function p and the assumption a-τ∑i=1nci>0, it follows that there is a bounded set Ω2⊇Ω1 with Ω2 containing the origin and a positive constant μ such that
(11)μ≤a-cτ-|p(t)||y|forℜ×Ω2c.
Therefore, we can write
(12)V˙≤-μy2for(t,x,y)∈ℜ×Ω2c×Ω2c.

From the last estimate, we can arrive that the y-coordinate of the solutions of system (4) is ultimately bounded for a positive constant β. On the other hand, since p is a continuous periodic function, if |y|≤β and V1=V+y, then, subject to the assumptions of the theorem, it can be easily seen that there is a constant K1>0 on ℜ×Ω2c such that
(13)V˙1=V˙+y′=-{f(x,y)+g(x,y)y}y2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1n(λiτi)y2-∑i=1nλi∫-τi0y2(t+s)ds-{f(x,y)+g(x,y)y}y-h(x)-∑i=1ngi(x)+∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+p(t)≤-ay2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1n(λiτi)y2-∑i=1nλi∫-τi0y2(t+s)ds+a-|y|-h(x)-∑i=1ngi(x)+∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+p(t)≤-h(x)-∑i=1ngi(x)+K1.
Since h(x)→∞ and gi(x)→∞ as x→∞, then it can be chosen a positive constant B1 such that
(14)V˙1≤-0.5forx≥B1.
Therefore, we can conclude that there is a positive constant α1 such that the x-coordinate of the solutions of system (4) satisfies x≤α1 for |y|≤β.

In a similar manner, if |y|≤β and V2=V-y, then, subject to the assumptions of the theorem, it can be easily followed by the time derivative of the functional V2 that there is a constant K2>0 on ℜ×Ω2c such that
(15)V˙2=V˙-y′=-{f(x,y)+g(x,y)y}y2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1n(λiτi)y2-∑i=1nλi∫-τi0y2(t+s)ds+{f(x,y)+g(x,y)y}y+h(x)+∑i=1ngi(x)-∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds-p(t)≤-ay2+y∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds+yp(t)+∑i=1n(λiτi)y2-∑i=1nλi∫-τi0y2(t+s)ds+a-|y|+h(x)+∑i=1ngi(x)-∑i=1n∫-τi0gi′(x(t+s))y(t+s)ds-p(t)≤h(x)+∑i=1ngi(x)+K2.

Since h(x)→-∞ and gi(x)→-∞ as x→-∞, then it can be chosen a positive constant B2 such that
(16)V˙2≤-0.5forx≤-B2.

Then, we can conclude that there is a positive constant α2 such that the x-coordinate of the solutions of system (4) satisfies x≥-α2 for |y|≤β. On gathering the above whole discussion, one can see that the solutions of system (4) are ultimately bounded. Therefore, (3) has an ω-periodic motion (solution). When p(t)=K, K-constant, (3) has a constant motion x=c0. From (3), it can be seen that the constant x=c0 is given by h(c0)+gi(c0)=K.

Conflict of Interests

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

GainesR. E.MawhinJ. L.YoshizawaT.Stability theory by Liapunov's second methodZhaoJ. M.HuangK. L.LuQ. S.The existence of periodic solutions for a class of functional-differential equations and their applicationBurtonT. A.AhmadS.RaoM. R.ConstantinA.A note on a second-order nonlinear differential systemGraefJ. R.On the generalized Liénard equation with negative dampingHuangL. H.YuJ. S.On boundedness of solutions of generalized Liénard's system and its applicationJinZ.Boundedness and convergence of solutions of a second-order nonlinear differential systemLiuB.HuangL.Boundedness of solutions for a class of retarded Liénard equationNápoles ValdésJ. E.Boundedness and global asymptotic stability of the forced Liénard equationQianC. X.Boundedness and asymptotic behaviour of solutions of a second-order nonlinear systemTunçC.A note on the bounded solutionsTunçC.A note on boundedness of solutions to a class of non-autonomous differential equations of second orderTunçC.Boundedness results for solutions of certain nonlinear differential equations of second orderTunçC.Uniformly stability and boundedness of solutions of second order nonlinear delay differential equationsTunçC.On the boundedness of solutions of a non-autonomous differential equation of second orderTunçC.Stability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviating argumentTunçC.On the stability and boundedness of solutions of a class of nonautonomous differential equations of second order with multiple deviating argumentsTunçC.Stability to vector Lienard equation with constant deviating argumentTunçC.New results on the existence of periodic solutions for Rayleigh equation with state-dependent delayTunçC.TunçE.On the asymptotic behavior of solutions of certain second-order differential equationsZhouJ.Necessary and sufficient conditions for boundedness and convergence of a second-order nonlinear differential system