# New Qualitative Results for Solutions of Functional Differential Equations of Second Order

Authors: Cemil Tunç; Sultan Erdur
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.


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## Body

## 1. Introduction

Differential equations of second order with and without delay(s) can find a wide range of applications in atomic energy, biology, chemistry, control theory, economy, engineering technique fields, information theory, medicine, physics, population dynamics, and so forth (see Burton [1], El’sgol’ts [2], Hale [3], Krasovskii [4], Smith [5], and Yoshizawa [6]). During investigations, we would naturally be inclined to compute the solutions of differential equations of second order with and without delay(s) explicitly or numerically. However, as we know from practice, there are very few such equations, for example, linear equations with constant coefficients, but without delay(s), for which this can be effectively done. Further, it should be noted that finding analytical or explicit solutions of differential equations of second order with delay(s) is more difficult, even if, to the best of our knowledge, there is no general method in the literature to find the explicit solutions of those equations. In addition, most of the times, it is impossible to find analytical solutions for those equations. The problem therefore is to find convenient techniques that will be useful in obtaining some qualitative information such as stability, instability, convergence, global existence, integrability, boundedness of solutions, existence of periodic solutions, and so forth about the elusive solutions of ordinary or delay differential equations.From the past till now, various methods have been constructed and are still discussed in order to investigate the various qualitative behaviors of solutions of ordinary or delay differential equations without solving those equations.However, here, we would only like to summarize some works that can be found in the literature and methods used during the investigations of the existence of the periodic solutions, stability, asymptotic stability, square integrability, and boundedness of solutions of ordinary and functional differential equations of second order.Yoshizawa [6] considered the following nonlinear differential equation of second order with constant delay:(1)x′′+ϕt,x′+fxt-τ=ptand he investigated the existence of ω-periodic solutions of this equation by using the second method of Lyapunov.Zhao at al. [7] obtained sufficient conditions for the existence of ω-periodic solutions of the below nonlinear differential equation of second order with constant delay(2)x′′+ax′+gxt-τ=ptby the second method of Lyapunov.Cong [8] considered a class of nonlinear differential equations of second order in the form(3)ddtptdxdt+ft,x=0.Cong [8] proved that there exists a unique 2π-periodic solution of those differential equations under Landesman–Lazer type conditions by applying the Leray–Schauder principle.Guo and Xu [9] studied the existence of periodic solutions of a differential equation of second order with a deviating argument by means of Mawhin’s continuation theorem. In [9], a new result on the existence of periodic solutions is obtained.Ji and Dong [10] discussed the existence and uniqueness of periodic solutions for a class of nonlinear differential equations of second order by using a comparison theorem and Leray-Schauder degree theory. The results obtained in [10] generalize and refine a recent work that can be found in the literature.Tian and Zeng [11] studied the existence of periodic solutions to the second-order functional differential equation(4)x′′t+ft,xt,xt-τtx′tn+atx2t+btxt=pt,n≥2,by applying Mawhin’s continuation theorem of coincidence degree theory. In [11], some new results on the existence of at least two periodic solutions to this equation are obtained.Li and Li [12] obtained existence results of positive ω-periodic solutions for the following functional differential equation of second order with multiple variable delays:(5)u′′t+atut=ft,ut,ut-τ1t,…,ut-τnt.In [12], the existence conditions concern the first eigenvalue of the associated linear periodic boundary problem and the discussion is based on the fixed-point index theory in cones.Li and Zhang [13] established several criteria for the existence, multiplicity, and nonexistence of positive periodic solutions of the following system(6)x′′+Atx=ft,xby combining some new properties of Green’s function together with Krasnoselskii’s fixed-point theorem on the compression and expression of cones.Zu [14] studied periodic solutions for the following nonlinear second-order ordinary differential equation:(7)x′′=ft,x,x′.By constructing upper and lower boundaries and using Leray-Schauder degree theory, the author presented a result about the existence and uniqueness of a periodic solution for the above second-order ordinary differential equation with some assumptions.Tunç and Yazgan [15] took into consideration the following nonlinear differential equation of second order with multiple fixed delays:(8)x′′+fx,x′+gx,x′x′x′+hx+∑i=1ngixt-τi=ptand they obtained the sufficient conditions for the existence of periodic solutions of this delay equation by the second method of Lyapunov.Ma and Lu [16] showed the existence of positive T-periodic solutions of the below second-order functional differential equation:(9)u′′t-p2ut+λgtfut-τt=0.The approach in [16] is based on global bifurcation theorem.Jia and Shao [17] established sufficient conditions for the existence and uniqueness of periodic solutions of an ordinary differential equation of second order by applying Mawhin’s continuation theorem of coincidence degree theory.Ardjouni and Djoudi ([18, 19]) discussed the existence of periodic and positive periodic solutions for a class of nonlinear neutral differential equations of second order with variable delays by Burton-Krasnoselskii’s hybrid fixed-point theorem.Similarly, Lü et al. [20] and Tian [21] investigated the existence of multiple positive periodic solutions for certain ordinary differential equations of second order and a delay differential equation of second order, respectively. In addition, Zhang and Wang [22] studied the existence of periodic solutions for a class of second-order functional differential equations with deviating arguments by using the abstract continuation theorem of k-set contractive operator and some analysis techniques.Zhou [23] considered the existence of periodic solutions for a class of semilinear second-order differential equations of the form(10)-x′′+ft,x,x′x′+etgx=ht.By applying the viscosity solutions method and the classical upper-lower solutions method, as well as the Leray-Schauder fixed-point principle, the author established the existence of periodic solutions. The result of Zhou [23] improves and generalizes many results on the ropes mechanics equations in the existing literature.Wei [24] proved the existence and uniqueness of periodic solutions for second-order ordinary differential equation(11)x′′=ft,x,x′under some assumptions on the function f. The proofs in [24] are based on Schauder’s fixed-point theorem.Finally, more recently, Zhu and Li [25] discussed the existence of periodic solutions for the below differential equation of second order with multiple delays(12)-u′′t=ft,ut,ut-τ1,ut-τ2,…,ut-τnby using the monotone iterative method of upper and lower solutions.Besides, for some other related papers, one can look at the book of Yoshizawa [26], the paper of Tunç and Çinar [27], and the references that can be found in the sources mentioned above.In fact, through the papers or books presented above, it can be seen that the second method of Lyapunov has rarely been used to investigate the existence of periodic solutions of nonlinear differential equations of second order with and without delay(s) instead of the other mentioned methods. To the best of our knowledge, the basic reason for the lack of the papers by this method is to find suitable Lyapunov function(s) or functional(s), which give(s) meaningful results. In this paper, we study the existence of the periodic solutions by defining suitable new Lyapunov functionals. This is a contribution of this paper to the subject and literature.On the other hand, the problems of the stability, asymptotic stability, convergence, integrability, and boundedness of solutions of linear and nonlinear differential equations of second order with and without delay(s) can appear in various physical, engineering, and many other scientific models. These kinds of differential equations are significant in describing fluid mechanical, nonlinear elastic mechanical phenomena, investigation of stability and instability of geodesic on Riemannian manifolds, dynamics process in electromechanical systems of physics and engineering, and so on. Many important theoretical and applied results related to these properties of solutions of differential equations of second order with and without delay(s) can be found in the literature (see Ahmad and Rama Mohana Rao [28], Burton [18], Burton and Hering [29], El’sgol’ts [2], Hale [3], Heidel [30], Kato [31, 32], Korkmaz and Tunç [33], Krasovskii [4], Liu and Huang [34, 35], Luk [36], Malyseva [37], Muresan [38], Mustafa and Tunç [39], Napoles Valdes [40], Amano [41], Sugie et al. [42], Tunç [43–52], Tunç and Dinç [53], Tunç and Tunç [54–57], Ye et al. [58], Yoshizawa [6], Yu and Xiao [59], Yu and Zhao [60], Zhang [61], Zhou and Xiang [62], and Zhou and Jiang [63] and their references).In the sources mentioned, the second method of Lyapunov, perturbation theory, fixed-point method or theory, iterative techniques, the variation of constants formula, and some other tools are used to investigate the mentioned qualitative behaviors of solutions of linear and nonlinear differential equations of second order with and without delay(s). Here, for the sake of brevity, we would not like to give more details about these subjects. In addition, in view of the information given above, we would like to say that it is worthwhile to continue the investigation of the qualitative properties of the solutions of nonlinear differentials of second order with delay(s).In this paper, we consider the following functional nonlinear differential equation of second order:(13)atx′′+ϕt,x′+ht,x′t-τ+gx+fxt-τ=et,x,x′,where t∈R+,R+=[0,∞),τ∈R,τ>0, if fixed constant delay, x∈R,R=(-∞,∞),  a(t) is a continuous differentiable positive and ω-periodic function with ω>0,ω∈R,a(t+ω)=a(t);  ϕ,h, and e are continuous functions according to their related arguments and ω-periodic in t. That is, ϕ(t+ω,x′)=ϕ(t,x′),  h(t+ω,x′(t-τ))=h(t,x′(t-τ)), and e(t+ω,x,x′)=e(t,x,x′). Finally, f(x) and g(x) are continuous differentiable functions with f(0)=g(0)=0.The following system can be written from (13):(14)x˙=y,y˙=-a′taty-1atϕt,y-1atht,yt-τ-1atgx-1atfx+1at∫-τ0f′xt+θyt+θdθ+1atet,x,y.To the best of our knowledge from the literature, we did not find any paper on the existence of periodic solutions, stability, asymptotic stability, square integrability, and boundedness of solutions of a mathematical model like (13). The purpose of this paper is to give new sufficient hypotheses, five theorems, with an example by MATLAB-Simulink on existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of (13) by the second method of Lyapunov. By the results of this paper, we extend and improve some results that can be found in the references of this paper (see [1–63]). These are the contributions of this paper to the mentioned topics and relevant literature.


## 2. Existence of Periodic Solutions

We now establish our some basic assumptions.(A) Hypotheses. We suppose that the following hypotheses hold:(15)A1a0≥at≥1,a′t≥0,a0∈R,t∈R+.A2ϕt,yy>αat>0fory≥A,y≠0,ht,yt-τy≥b0>0fory≥A,y≠0,t∈R+,y∈R,g0=0,gxsgn⁡x→∞forx→∞,g′x≤L0,f0=0,fxsgn⁡x→∞forx→∞,f′x≤L,x∈R,where α, A, b0, L0, L, and b0∈R are some positive real constants with b0≥1 and τ<α/2L.(16)A3et,x,y≤αaty4,t∈R+,x,y∈R.Our first theorem for the existence of periodic solutions of system (14) can be given below.Theorem 1.
If hypotheses(A1)-(A3) hold, then system (14) has a ω-periodic solution.Proof.
LetV0=V0(t,xt,yt) be a Lyapunov functional defined by(17)V0=2at∫0xfsds+2at∫0xgsds+y2+α2τ∫-τ0∫s0y2θdθds+L∫-τ0∫s0yθdθds.
It is obvious that the Lyapunov functionalV0 is positive definite.
By calculating the time derivative of the Lyapunov functionalV0 with respect to t along system (14) and by usage of hypotheses (A1)-(A3) of Theorem 1, we have(18)ddtV0≤-2a′taty2-2atϕt,yyy2-2atht,yt-τyy2+2aty∫-τ0f′xt+θyt+θdθ+α2y2+Lτy-L∫-τ0yt+θdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ.
It is clear that(19)-2a′tat≤0by hypothesis (A1) of Theorem 1. Then, from (18), we can write(20)dV0dt≤-2αy2+2at∫-τ0f′xt+θytyt+θdθ+α2y2+Lτy-L∫-τ0yt+θdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ≤-2αy2+2∫-τ0Lytyt+θdθ+α2y2+Lτy-L∫-τ0yt+θdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθby hypotheses (A1)-(A3) of Theorem 1.
LetL<α/2τ. Then, we have(21)dV0dt≤-∫-τ0α2τy2t-ατytyt+θ+α2τy2t+θdθ-32αy2+α2y2+Lτy-L∫-τ0yt+θdθ+α2τ∫-τ0y2tdθ=-α2τ∫-τ0yt-yt+θ2dθ-αy2+Lτy-L∫-τ0yt+θdθby the last inequality. If τ<α/2L and y(t)≥c=max⁡{A,2Lτ/α+1}, then we can obtain(22)dV0dt≤-12αy2.
Case I. We assume that y(t)≤c,c∈R,c>0, and B0 is a positive constant. Then, it is clear that(23)2yϕt,yat+ϕt,yat+ht,yt-τat≤B0.Suppose that x(t)≥d,d∈R,d>0 and define the Lyapunov functional(24)Vt,xt,yt=V0t,xt,yt+yt.Then, the time derivative of functional V along system (14) is given by(25)ddtV=ddtV0+ddtyt=ddtV0-a′taty-1atϕt,y-1atht,yt-τ-1atgx-1atfx+1at∫-τ0f′xt+θyt+θdθ+et,x,yat.
In view of hypotheses(A1)-(A3) of Theorem 1, we have(26)dVdt≤-2a′taty2-2atϕt,yy-2atht,yt-τy+2aty∫-τ0f′xt+θyt+θdθ+2atet,x,yy+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+Lτy-L∫-τ0yt+θdθ-a′taty-1atϕt,y-1atht,yt-τ-1atgx-1atfx+1at∫-τ0f′xt+θyt+θdθ+et,x,yat.By hypotheses (A1)-(A3) of Theorem 1, inequality (23), and the last estimate, we obtain(27)dVdt≤2yϕt,yat+1atϕt,y+1atht,yt-τ+α2c2-α2τ∫-τ0y2tdθ-2L∫-τ0ytyt+θdθ+α2τ∫-τ0y2t+θdθ+αc2+Lτy-L∫-τ0yt+θdθ-a′taty-1atgx-1atfx+L∫-τ0yt+θdθ+et,x,yat≤B0+αc22-α2τ∫-τ0yt-yt+θ2dθ+αc2+Lτc+a0c+αc4-1atgx-1atfx.Since a0≥a(t)≥1, it follows that(28)dVdt≤B0+αc22+αc2+a0c+3αc4-gx+fx.In view of the hypotheses fxsgn⁡x→∞ for x→∞ and g(x)sgn⁡x→∞ for x→∞, it is obvious that(29)dVdt≤-1when x→∞.
Case II.We assume that(30)yt≤c,xt≤-d,c,d∈R,c>0,d>0.It is known that the function a(t) is bounded and y≤c. Then, inequality (23) holds.
We define a Lyapunov functional by(31)Vt,xt,yt=V0t,xt,yt-yt.The time derivative of the Lyapunov functional V along system (14) implies that(32)ddtV=ddtV0-ddtyt=dV0dt+a′taty+1atϕt,y+1atht,yt-τ+1atgx+1atfx-1at∫-τ0f′xt+θyt+θdθ-et,x,yat.Hence, using inequality (18), we obtain(33)ddtV≤-2a′taty2-2atϕt,yy-2atht,yt-τy+2aty∫-τ0f′xt+θyt+θdθ+2atet,x,yy+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+Lτy-L∫-τ0yt+θdθ+a′taty+1atϕt,y+1atht,yt-τ+1atgx+1atfx-1at∫-τ0f′xt+θyt+θdθ-et,x,yat.After that, in view of hypotheses of Theorem 1, it follows that(34)ddtV≤-2atϕt,yy+2L∫-τ0ytyt+θdθ+α2c2+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2tdθ+α2τ∫-τ0y2t+θdθ+α2τ∫-τ0y2tdθ+Lτy-L∫-τ0yt+θdθ+a′taty+1atϕt,y+1atht,yt-τ+1atgx+1atfx+1at∫-τ0f′xt+θyt+θdθ+αc4.
By taking into consideration the hypotheses of Theorem1, it can be obtained that(35)ddtV≤B0+αc22-α2τ∫-τ0yt-yt+θ2dθ+αc2+Lτc+βc+αc4+1atgx+1atfx≤B0+αc22+αc2+βc+3αc4+gx+fx.Since fxsgn⁡x→∞ for x→∞ and g(x)sgn⁡x→∞ for x→∞, it can be easily concluded that(36)dVdt≤-1.
Case III. Let(37)xt≤d,yt≥c,c,d∈R,c>0,d>0.
We now define a Lyapunov functional by(38)Vt,xt,yt=V0t,xt,yt+cdxt.Then, calculating the time derivative of the functional V with respect to t, using inequality (18) and hypotheses (A1)-(A3) of Theorem 1, we have(39)ddtV=ddtV0+cdyt≤-2a′taty2-2atϕt,yyy2-2atht,yt-τyy2+2aty∫-τ0f′xt+θyt+θdθ+2atet,x,yy+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+Lτy-L∫-τ0yt+θdθ+cdy≤-2αy2+2L∫-τ0ytyt+θdθ+α2y2+α2τ∫-τ0y2tdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+α2y+cdy≤-2αy2+α2y2-α2τ∫-τ0yt-yt+θ2dθ+αy2+α2y+cdy≤-2αy2+α2y2+αy2+α2y+cdy≤-12αy2+cdy≤-14αy2.
Case IV. Let(40)xt≤d,yt≤-c,c,d∈R,c>0,d>0.We define a Lyapunov functional by(41)Vt,xt,yt=V0t,xt,yt-cdxt.Like before, we can obtain(42)ddtV=ddtV0-cdyt≤-2a′taty2-2atϕt,yyy2-2atht,yt-τyy2+2aty∫-τ0f′xt+θyt+θdθ+2atet,x,yy+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+Lτy-L∫-τ0yt+θdθ-cdy≤-2αy2+2L∫-τ0ytyt+θdθ+α2y2+α2τ∫-τ0y2tdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ-cdy≤-2αy2+α2y2-α2τ∫-τ0yt-yt+θ2dθ+αy2+α2y-cdy≤-2αy2+α2y2+αy2-cdy≤-12αy2-cdy≤-14αy2by (18) and hypotheses (A1)-(A3) of Theorem 1.
Case V. We suppose that x(t)≥d for c,d∈R,c>0,d>0,  y(t)≥c or x(t)≤-d,  y(t)≤-c.
We define a Lyapunov functionalV by(43)Vt,xt,yt=V0t,xt,yt+c.
Case VI. Further, in the case of x(t)≥d,  y(t)≤-c, or x(t)≤-d,  y(t)≥c, we define Lyapunov functional by(44)Vt,xt,yt=V0t,xt,yt-c.
In view of the above two cases, that is, Cases V and VI, and the hypotheses of Theorem1, since c is a positive constant, the time derivative of functional V along system (14) leads to the following:(45)ddtV=ddtV0≤-2a′taty2-2atϕt,yyy2-2atht,yt-τyy2+2aty∫-τ0f′xt+θyt+θdθ+2atet,x,yy+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ+Lτy-L∫-τ0yt+θdθso that(46)ddtV≤-2αy2+2L∫-τ0ytyt+θdθ+α2y2+α2τ∫-τ0y2tdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ≤-2αy2+α2y2-α2τ∫-τ0yt-yt+θ2dθ+αy2+α2y≤-2αy2+α2y2+αy2=-12αy2≤-14αy2.
Hence, it is obvious that the time derivative of functionalV, that is, d/dtV, is negative semidefined for the above two cases. In addition, we can see that(47)2at∫0xfsds+2at∫0xgsds+y2-c≤2∫0xfsds+2∫0xgsds+y2-c≤Vt,xt,yt≤2∫0xfsds+2∫0xgsds+c+a2τ∫-τ0∫τ0y2θdθds+L∫-τ0∫τ0yθdθds.Hence, “wedges” can be easily found (see Burton [1]), bounding the functional V0 from bottom to top. Then, we can conclude that system (14) has a periodic solution of period ω (see Yoshizawa ([6, Theorem 37.2])).


## 3. Stability of Solutions

Lete(t,x,y)≡0.(B) Hypothesis.It is assumed that the following hypothesis holds:(48)A4ϕt,0=0,ϕt,yy>αat>0fory∈R,y≠0,ht,0=0,ht,yt-τy≥b0>0fort∈R+,y∈R,y≠0,g0=0,gxx≥g0,x∈R,x≠0,f0=0,fxx≥f0,x≠0,f′x≤M,x∈R,where α,b0, f0, g0, and M∈R are some positive real constants with b0≥1 and τ<α/2M.Theorem 2.
If hypotheses(A1) and (A4) hold, then the zero solution of system (14) is stable.Proof.
Consider the Lyapunov functionalV1=V1(t,xt,yt) defined by(49)V1=2at∫0xfsds+2at∫0xgsds+y2+α2τ∫-τ0∫s0y2θdθds.It is clear that(50)V1t,0,0=0and(51)V1=2at∫0xfsssds+2at∫0xgsssds+y2+α2τ∫-τ0∫s0y2θdθds≥f0+g0a0x2+y2=σx2+y2,where σ=min⁡{(f0+g0)a0-1,1}, by hypotheses (A1) and (A4).
Differentiating the Lyapunov functionalV1 along system (14) and considering hypotheses (A1) and (A4), we obtain(52)ddtV1≤-2atϕt,yyy2-2atht,yt-τyy2+2aty∫-τ0f′xt+θyt+θdθ+α2τ∫-τ0y2tdθ-α2τ∫-τ0y2t+θdθ.
Hence,(53)dV1dt≤-32αy2-2b0a0y2+2at∫-τ0f′xt+θytyt+θdθ-α2τ∫-τ0y2t+θdθ.≤-32αy2-2b0a0y2+∫-τ0My2t+y2t+θdθ-α2τ∫-τ0y2t+θdθ≤-32α+2b0a0-Mτy2-α2τ-M∫-τ0y2t+θdθby hypotheses (A1) and (A4) of Theorem 2.
Letτ<α/2M. Then, it follows that(54)dV1dt≤-α+2a0b0-1y2.This result completes that the zero solution of system (14) is stable.Corollary 3.
If hypotheses(A1) and (A4) hold, then the zero solution of system (14) is uniformly stable.Theorem 4.
If hypotheses(A1) and (A4) hold, then the zero solution of system (14) is asymptotically stable.Proof.
In the proof of this theorem, we benefit from the Lyapunov functionalV1 given in the proof of Theorem 2.
In the light of the hypotheses of Theorem4, we can conclude that(55)dV1dt≤-α+2a0b0-1y2.Consider now the set defined by(56)S≡t,x,y:V˙1t,xt,yt=0.When we apply LaSalle’s invariance principle, we observe that (t,x,y)∈S implies that y=0, and hence x=ρ(ρ≠0 is a constant). The last estimate and system (14), together, necessarily imply that(57)1atgx+1atfx=0.When x=ρ, this equality can hold if and only if(58)gρ+fρ=0.Hence,(59)gρ+fρ=0⇔ρ=0,which implies that g(x)+f(x)=0⇔x=0. Therefore, we have x=y=0. In fact, this result shows that the largest invariant set contained in S is (t,0,0)∈S. Hence, we can conclude that the zero solution of system (14) is asymptotically stable. This completes the proof of Theorem 4.Theorem 5.
If hypotheses(A1) and (A4) hold, then the first derivatives of all solutions of (13) are square-integrable; that is, y(t)∈L2[0,∞), where L2[0,∞) is the space of all Lebesgue square-integrable functions on [0,∞).Proof.
Here, we also use the functionalV1 used in both theorems given just above. Notice the hypotheses of Theorem 5; we have(60)ddtV1t,xt,yt≤-ky2≤0,k=α+2a0b0-1>0.Integrating the last inequality from t0 to t, we have(61)V1t,xt,yt-V1t0,xt0,yt0≤-k∫t0tys2ds.From the above discussion, it can be seen that V1(t,xt,yt) is positive definite and a decreasing functional. Therefore, we can say that(62)V1t0,xt0,yt0=l,l>0,l∈R,and hence, it is clear that(63)k∫t0tys2ds≤V1t0,xt0,yt0-V1t,xt,yt≤V1t0,xt0,yt0.As the result of the above inequalities, we can conclude that(64)∫t0∞ys2ds≤k-1l.This result is the end of the proof of Theorem 5.


## 4. Boundedness of Solutions

Lete(t,x,y)≠0.(C) Hypothesis.It is assumed that the following hypothesis holds:(65)A5et,x,y≤θty,t∈R+,x,y∈R,where θ(t) is a nonnegative and continuous function for all t∈R+ such that θ∈L1[0,∞),L1[0,∞) is the space of all Lebesgue integrable functions on [0,∞).Theorem 6.
If hypotheses(A1), (A4), and (A5) hold, then all solutions of system (13) are bounded as t→∞.Proof.
Here, once again, we use the functionalV1 just given above. Notice the hypotheses of Theorem 6; we can have(66)ddtV1t,xt,yt≤2yet,x,y.In view of hypothesis (A5), we can get(67)ddtV1t,xt,yt≤2θty2≤2σ-1θtV1t,xt,yt.Integrating the former inequality from t0 to t, we have(68)V1t,xt,yt-V1t0,xt0,yt0≤2σ-1∫t0tθsV1s,xs,ysdsso that(69)V1t,xt,yt≤l+2σ-1∫t0tθsV1s,xs,ysds.By the Gronwall inequality, it follows that(70)V1t,xt,yt≤lexp⁡2σ-1∫t0∞θsds<∞.Let(71)lexp⁡2σ-1∫t0∞θsds=k1.In addition, we also have(72)x2+y2≤σ-1V1t,xt,yt.Hence, we can conclude that(73)x2+y2≤σ-1k1.This is the end of the proof of Theorem 6.Example 7.
We consider the following second-order nonlinear differential equation with constant delay:(74)x′′+3+sin⁡tx′+3+sin⁡t+x′2t-1,+2x+xt-1=18x′sin⁡t.When we compare this equation with (13), it follows that(75)ϕt,y=3+sin⁡ty,at=1,ht,yt-τ=3+sin⁡t+y2t-1,fx=x,gx=2xet,x,y=18ysin⁡t.It is obvious that the hypotheses of Theorem 1 are satisfied:(76)ϕt,yy=3+sin⁡t>α=1,ht,yt-τ=3+sin2⁡t+y2t-1≥3=b0>0,fxsgn⁡x=xsgn⁡x→∞forx→∞f′x=x=1≤2=L,et,x,y=18ysin⁡t≤αy4=y4,α=1,gxsgn⁡x=2xsgn⁡x→∞forx→∞,g′x=2=2≤L,at=1,a′t=0.Therefore, the given differential equation satisfies all hypotheses of Theorem1. Then, there exists a 2π-periodic solution of the above delay differential equation.The orbits of the solutions of the considered delay differential equation are shown by Figures1 and 2.Figure 1
Trajectory ofx(t) for Example 7.Figure 2
Trajectory ofy(t) for Example 7.Remark 8.
The differential equation with constant delay just given above by Example7 can be modified and the related graphs of the orbits can be drawn by MATLAB-Simulink to verify the stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions. Here, we omit the details.


## 5. Conclusion

With the help of the second method of Lyapunov, the qualitative properties such as the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of a class of nonlinear differential equations of second order with constant delay are investigated. On the mentioned topics, five new theorems are proven. The obtained results include and improve some results in the literature. We give an example to verify the applicability of the results by MATLAB-Simulink, which shows the behaviors of the orbits (see Figures1 and 2).



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