# Stability, Boundedness, and Existence of Periodic Solutions to Certain Third-Order Delay Differential Equations with Multiple Deviating Arguments

Authors: A. T. Ademola; B. S. Ogundare; M. O. Ogundiran; O. A. Adesina
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

The behaviour of solutions for certain third-order nonlinear differential equations with multiple deviating arguments is considered. By employing Lyapunov’s second method, a complete Lyapunov functional is constructed and used to establish sufficient conditions that guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results not only are new but also include many outstanding results in the literature. Finally, the correctness and effectiveness of the results are justified with examples.


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## Body

## 1. Introduction

Differential equations of second and third order with and without delay are essential tools in scientific modeling of problems from many fields of sciences and technologies, such as biology, chemistry, physics, mechanics, electronics, engineering, economy, control theory, medicine, atomic energy, and information theory. Many authors have proposed different methods, in the literature, to discuss qualitative bahaviour of solutions to nonlinear differential equations. Here, we will single out two methods. In this direction, we can mention Lyapunov’s second method which demands the construction of a suitable positive definite function (or functional) whose derivative is negative definite; that is, it involves finding the system of closed surfaces that contained the origin and are converging to it. The second method is the frequency domain technique which involves the study of position of the characteristics polynomial roots in the complex plane to obtain certain matrix inequalities which must be positive.The qualitative behaviors of solutions of differential equations of third order have been discussed extensively and are still receiving attention of authors because of their practical applications. In this regard, we can mention the works of Burton [1, 2], Driver [3], Hale [4], and Yoshizawa [5, 6] which contain general results on the subject matters and expository papers of Abou-El-Ela et al. [7], Ademola et al. [8–10], Adesina [11], Afuwape and Omeike [12], Chukwu [13], Gui [14], Omeike [15, 16], Sadek [17], Tejumola and Tchegnani [18], Tunç et al. [19–28], Yao and Wang [29], and Zhu [30] and the references cited therein.Recently, Tunç [27] employed Lyapunov’s second method to prove two results on stability and boundedness of nonautonomous differential equations with constant delay(1)x⃛+ft,x,xt-τ,x˙,x˙t-τ,x¨,x¨t-τ+gx˙t-τ+hxt-τ=pt,x,xt-τ,x˙,x˙t-τ,x¨,x¨t-τ.Furthermore, Ademola [9] discussed existence and uniqueness of a periodic solution to the third-order differential equation(2)x⃛+ft,x,x˙,x¨x¨+∑i=1ngit,xt-τit,x˙t-τit+∑i=1nhit,xt-τit=pt,x,xt-τit,x˙,x˙t-τit,x¨,x¨t-τi.Unfortunately, the problem of uniform asymptotic stability, uniform ultimate boundedness, and existence and uniqueness of periodic solutions of the third-order delay differential equation (3), where all the nonlinear terms (specifically, the forcing term pi and the function fi) are sum of multiple deviating arguments, is yet to be investigated. This is not unconnected with the difficulties associated with the construction of suitable complete Lyapunov functional. The aim of this paper is to fill this gap. We will consider(3)x⃛+∑i=1nfit,x,xt-τit,x˙,x˙t-τit,x¨,x¨t-τit+∑i=1ngix˙t-τit+∑i=1nhixt-τit=∑i=1npit,x,xt-τit,x˙,x˙t-τit,x¨,x¨t-τit,where fi, gi, hi, and pi are continuous functions in their respective arguments on R+×R3n+3, R, R+, and R+×R3n+3, respectively, with R+=[0,∞) and R=(-∞,∞). The dots indicate differentiation with respect to the independent variable t. Equation (3) is equivalent to the system of first-order delay differential equation(4)x˙=y,y˙=z,z˙=∑i=1npit,x,xt-τit,y,yt-τit,z,zt-τit-∑i=1nfit,x,xt-τit,y,yt-τit,z,zt-τit-∑i=1ngiyt-∑i=1nhixt+∑i=1n∫t-τittgiyszsds+∑i=1n∫t-τitthi′xsysds,where 0≤τi(t)≤γ, γ>0 is a constant to be determined latter, and the derivatives gi′ and hi′ for all i(i=1,2,3,…,n) exist and are continuous for all x and y with hi(0)=0. This work is motivated by the recent works in [9, 27]. Our results are new; in fact according to our observation from relevant literature, this is the first paper where both the functions fi and the forcing term pi contain sum of multiple deviating arguments. For the next section, for easy references, we recall the main mathematical tools that will be used in the sequel. Our main results are stated and proved in Section 3 while in the last section, examples are given.


## 2. Preliminary Results

Consider the following general nonlinear nonautonomous delay differential equation(5)X˙=dXdt=Ft,Xt,Xt=Xt+θ,-r≤θ<0,t≥0,where F:R+×CH→Rn is a continuous mapping and F(t+ω,ϕ)=F(t,ϕ) for all ϕ∈C and for some positive constant ω. We assume that F takes closed bounded sets into bounded sets in Rn. (C,·) is the Banach space of continuous function φ:[-r,0]→Rn with supremum norm, r>0; for H>0, we define CH⊂C by CH={φ∈C:φ<H}, and  CH is the open H-ball in C, C=C([-r,0],Rn).Definition 1 (see [2]).
A continuous functionW:R+→R+ with W(0)=0, W(s)>0 if s≠0 and W strictly increasing is a wedge. (We denote wedges by W or Wi, where i is an integer.)Definition 2 (see [2]).
The zero solution of (5) is asymptotically stable if it is stable and if for each t0≥0 there is η>0 such that ϕ≤η implies that(6)Xt,t0,ϕ⟶0ast⟶∞.Definition 3 (see [1]).
An elementψ∈CH is in the ω-limit set of ϕ, say Ω(ϕ), if X(t,0,ϕ) is defined on R+ and there is a sequence {tn}, tn→∞ as n→+∞, with Xtn(ϕ)-ψ→0 as n→∞, where Xtn(ϕ)=X(tn+θ,0,ϕ) for -r≤θ<0.Definition 4 (see [30]).
A setQ⊂CH is an invariant set if, for any ϕ∈Q, the solution X(t,0,ϕ) of (5) is defined on R+ and Xt(ϕ)∈Q for t∈R+.Lemma 5 (see [6]).
Suppose thatF(t,ϕ)∈C¯0(ϕ) and F(t,ϕ) is periodic in t of period ω, ω≥r, and consequently for any α>0 there exists L(α)>0 such that ϕ∈Cα implies Ft,ϕ≤L(α). Suppose that a continuous Lyapunov functional V(t,ϕ) exists, defined on t∈R+, ϕ∈S∗, S∗ is the set of ϕ∈C such that ϕ0≥H (H may be large), and V(t,ϕ) satisfies the following conditions:(i)
a(ϕ0)≤V(t,ϕ)≤b(ϕ), where a(r) and b(r) are continuous, increasing, and positive for r≥H and a(r)→∞ as r→∞;(ii)
V˙5(t,ϕ)≤-c(ϕ0), where c(r) is continuous and positive for r≥H.Suppose that there exists H1>0, H1>H, such that(7)hLγ∗<H1-H,where γ∗>0 is a constant which is determined in the following way: By the condition on V(t,ϕ) there exist α>0, β>0, and γ>0 such that b(H1)≤a(α), b(α)≤a(β), and b(β)≤a(γ). γ∗ is defined by b(γ)≤a(γ∗). Under the above conditions, there exists a periodic solution of (5) of period ω. In particular, relation (7) can always be satisfied if h is sufficiently small.Lemma 6 (see [6]).
Suppose thatF(t,ϕ) is defined and continuous on 0≤t≤c, ϕ∈CH, and there exists a continuous Lyapunov functional V(t,ϕ,φ) defined on 0≤t≤c, ϕ, and φ∈CH which satisfy the following conditions:(i)
V(t,ϕ,φ)=0 if ϕ=φ;(ii)
V(t,ϕ,φ)>0 if ϕ≠φ;(iii)
for the associated system(8)x˙t=Ft,xt,y˙t=Ft,yt,we haveV(8)′(t,ϕ,φ)≤0, where, for ϕ=H or φ=H, we understand that the condition V(8)′(t,ϕ,φ)≤0 is satisfied in case V′ can be defined.Then, for given initial value ϕ∈CH1, H1<H, there exists a unique solution of (5).Lemma 7 (see [6]).
Suppose that a continuous Lyapunov functionalV(t,ϕ) exists, defined on t∈R+, ϕ<H, and 0<H1<H which satisfies the following conditions:(i)
a(ϕ)≤V(t,ϕ)≤b(ϕ), where a(r) and b(r) are continuous, increasing, and positive;(ii)
V˙(5)(t,ϕ)≤-c(ϕ), where c(r) is continuous and positive for r≥0;then the zero solution of (5) is uniformly asymptotically stable.Lemma 8 (see [1]).
LetV:R+×C→R be continuous and locally Lipschitz in ϕ. If(i)
W0(Xt)≤V(t,Xt)≤W1(Xt)+W2(∫t-r(t)tW3(Xt(s))ds);(ii)
V˙5(t,Xt)≤-W4(Xt)+N, for some N>0, where Wi(i=0,1,2,3,4) are wedges,then Xt of (5) is uniformly bounded and uniformly ultimately bounded for bound B.


## 3. Main Results

We will give the following notations before we state our main results. Let(9)∑i=1nfit,x,xt-τit,y,yt-τit,z,zt-τit=∑i=1nfi·,∑i=1npit,x,xt-τit,y,yt-τit,z,zt-τit=∑i=1npi·.For the first case of consideration set ∑i=1npi(·)≡0, system (4) reduces to(10)x˙=y,y˙=z,z˙=-∑i=1nfi·-∑i=1ngiyt-∑i=1nhixt+∑i=1n∫t-τittgiyszsds+∑i=1n∫t-τitthi′xsysds,where fi, gi, hi, and τi are functions defined in Section 1. Let (xt,yt,zt) be any solution of (10); the continuously differentiable functional employed in the proof of our results is V=V(t,xt,yt,zt) defined as(11)2V=2∑i=1nα+ai∫0xhiξdξ+4∑i=1n∫0ygiηdη+4y∑i=1nhix+2∑i=1nα+aiyz+2z2+∑i=1nα2+β+ai2y2+∑i=1nβbix2+2∑i=1naiβxy+2βxz+∫-τit0∫t+stλ1x2θ+λ2y2θ+λ3z2θdθds,where α and β are fixed positive constants satisfying(12)bi-1ci<α<ai∀ii=1,2,3,…,n;0<β<min⁡bi,∑i=1nαbi-ciA1-1,∑i=1nai-αA2-1,with(13)A1≔21+ai+δi-1y-1giy-bi2,A2≔41+δi-1z-1fi·-ai2.λ1, λ2, and λ3 are nonnegative constants which will be determined later.Remark 9.
The Lyapunov functional defined in (11) is an improvement on the one used in [9].At last, we now state our main results and give their proofs.Theorem 10.
Further to the assumptions on the functionsfi, gi, hi, and τi, suppose that, for all i (i=1,2,3,…,n),  ai, δi, ci, Bi, ρ and γ are positive constants and for all t≥0:(i)
fi·/z≥ai for all z≠0;(ii)
bi≤gi(y)/y≤Bi for all y≠0;(iii)
hi(0)=0,hi(x)/x≥δi for all x≠0;(iv)
hi′(x)≤ci for all x and aibi-ci>0;(v)
τi(t)≤γ, τi′(t)≤ρ, ρ∈(0,1); and if(14)γ<min⁡∑i=1nδiBi+ci-1,∑i=1naibi-ciA3-1,∑i=1nai-αA4-1,where(15)A3≔Bi+ciα+ai+ci1-ρ-1α+β+ai+2,A4≔22Bi+ci+Bi1-ρ-1α+β+ai+2.then the trivial solution of (10) is uniformly asymptotically stable.Remark 11.
(i) Ifi=1, f1(·)=φ(x˙), g1(x˙(t-τ1(t)))=g(x˙(t-τ(t))), and h1(x(t-τ1(t)))=h(x(t-τ(t))), (10) reduces to the system considered in [29] and some of our hypotheses agree with the hypotheses obtained therein.
(ii) Wheni=1, the functions f1(·)=g(x,x˙)x¨, g1(x˙(t-τ1(t)))=f(x(t-τ), x˙(t-τ)), and h1(x(t-τ1(t)))=h(x(t-τ)) the above result includes that discussed in [24].
(iii) Wheneveri=1, f1(·)=h(x˙)x¨, g1(x˙(t-τ1(t)))=g(x˙(t-τ(t))), and h1(x(t-τ1(t)))=h(x(t-τ(t))), (10) specializes to that studied in [12]. Thus, the result of Theorem 10 coincides with results in [12] if i=1.
(iv) When∑i=1nfi(·)=ax¨, ∑i=1ngi(x˙(t-τi(t)))=bx˙, ∑i=1nhi(x(t-τi(t)))=cx, and ∑i=1npi(·)=0, (3) reduces to linear constant coefficients differential equations and conditions (i) to (v) of Theorem 10 specialize to the corresponding Routh-Hurwitz criteria a>0, ab>c, and c>0.
(v) Wheni=1, f1(·)=a1x¨, g1(x˙(t-τ1(t)))=f2(x˙(t-τ1(t))), and hi(x(t-τ1(t)))=a3x(t), (10) specializes to that discussed in [28]. Theorem 10 coincides with the stability result in [28].
(vi) When∑i=1nfi(·)=f(x¨), ∑i=1ngi(x˙(t-τi(t)))=g(x˙), ∑i=1nhi(x(t-τi(t)))=h(x), and ∑i=1npi(·)=p(t,x,x˙,x¨), then (3) reduces to the ordinary differential equation studied in [31].
(vii) Ifi=1 and τi(t)=τ then (3) coincides with (2) discussed in [27]; hence our hypotheses coincide with that of Tunç in [27].
(viii) Wheneveri=1, f1(·)=f(x,x˙)x¨, g1(x˙(t-τ1(t)))=g(x(t-τ(t)), x˙(t-τ(t))), and h1(x(t-τ1(t)))=h(x(t-τ(t))), (10) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for sup⁡{h′(x)}=c>0 which is replaced by a more general condition.
(ix) Finally, the functionsfi(·) and pi(·) used in this paper extend the works in [7–10, 12, 24, 27–29, 31].In what follows, we will state and prove a result that would be useful in the proof of Theorem10 and subsequent ones.Lemma 12.
Under the hypotheses of Theorem10 there exist positive constants D0=D0(ai,bi,ci,δi,α,β), D1=D1(ai,bi,ci,Bi,α,β), and D2=D2(λ1,λ2,λ3) such that for all (xt,yt,zt)∈R3(16)D0x2t+y2t+z2t≤Vt,xt,yt,zt≤D1x2t+y2t+z2t+D2∫-τit0∫t+stx2θ+y2θ+z2θdθds.Furthermore, there exists a constant D3=D3(ai,bi,ci,Bi,δi,α,β,γ,ρ,λ1,λ2,λ3)>0 such that(17)V˙10=ddtVt,xt,yt,zt10≤-D3x2t+y2t+z2t.Proof.
Let(xt,yt,zt) be any solution of (10); since hi(0)=0, (11) can be recast in the form(18)V=∑i=1nbi-1∫0xα+aibi-2hi′ξhiξdξ+2∑i=1n∫0yη-1giη-biηdη+∑i=1nbi-1hix+biy2+12βy2+12∑i=1nβx+aiy+z2+12αy+z2+12∑i=1nβbi-βx2+12∫-τit0∫t+stλ1x2θ+λ2y2θ+λ3z2θdθds.From hypotheses (ii), (iii), and (iv) and the fact that the double integrals appearing in inequality (18) are nonnegative, it follows that there exists a constant K0>0 such that(19)V≥K0x2+y2+z2for all x, y, and z, where(20)K0=min∑i=1nbi-1α+aibi-2ciδi+∑i=1nbi-1min⁡∑i=1nδi,∑i=1nbi+12min⁡β,∑i=1nai,1+12∑i=1nβbi-β,∑i=1nbi-1min⁡∑i=1nδi,∑i=1nbi+12min⁡β,∑i=1nai,1+12min⁡α,1+12β,12min⁡β,∑i=1nai,1+12min⁡α,1.Estimate (19) establishes the lower inequality in (16) with K0=D0, respectively. Moreover, from inequality (19) we find that V(t,xt,yt,zt)=0 if and only if (x2+y2+z2)=0 and V(t,xt,yt,zt)>0 if and only if (x2+y2+z2)≠0, and it follows that for all x, y, z(21)Vt,xt,yt,zt⟶+∞asx2+y2+z2⟶∞.Furthermore, since hi(0)=0, hi(x)≤cix for all x≠0, gi(y)≤Biy for all y≠0, and inequality 2x1x2≤x12+x22, there exists positive constants K1, K2 such that(22)V≤K1x2+y2+z2+K2∫-τit0∫t+stx2θ+y2θ+z2θdθds,where(23)K1=12max⁡∑i=1nα+ai+2ci+βai+bi+1,∑i=1n2Bi+ci+aiα+β+ai+α2+β,∑i=1nα+β+ai+2,K2=max⁡λ1,λ2,λ3.Estimate (22) establishes the upper inequality in (16) with K1=D1 and K2=D2, respectively. Hence, from inequalities (19) and (22) estimate (16) of Lemma 12 is established.
Next, the time derivative of the functional defined in inequality (11) with respect to the independent variable t, along a solution of system (10), is simplified to give(24)V˙10=U1-U2-β∑i=1ngiyy-bixy-β∑i=1nfi·z-aixz+2βyz-∑i=1nα+aifi·z-α2+ai2yz+∑i=1naiβy2+12λ1x2+λ2y2+λ3z2τit-121-τi′t∫t+stλ1x2s+λ2y2s+λ3z2sds,where(25)U1≔βx+∑i=1nα+aiy+2z∑i=1n∫t-τittgi′yszs+hi′xsysds,U2≔∑i=1nβhixxx2+∑i=1nα+aigiyy-2hi′xy2+∑i=1n2fi·z-α+aiz2.Now from the assumptions of Theorem 10 we find that(26)U1≤12∑i=1nβx2+α+aiy2+2z2Bi+ciτit+12∑i=1nα+β+ai+2∫t-τittciy2s+Biz2sdsfor all x, y, z and t≥0 and(27)U2≥∑i=1nβδix2+∑i=1nα+aibi-2ciy2-12∑i=1nai-αz2for all x, y, z. Using estimates U1 and U2 in (24), we find that(28)V˙10≤-12∑i=1nβδix2-12∑i=1nα+aibi-2ciy2-12∑i=1nai-αz2-∑j=34Uj-121-τi′tλ1∫t-τittx2sds+12∑i=1nβBi+ci+λ1τitx2-121-τi′tλ2-∑i=1nα+β+ai+2ci∫t-τitty2sds+βz2-121-τi′tλ3-∑i=1nα+β+ai+2Bi∫t-τittz2sds+∑i=1nβai+1y2+12∑i=1nBi+ciα+ai+λ2τity2+122∑i=1nBi+ci+λ3τitz2for all x, y, z, where(29)U3≔14∑i=1nβδix2+β∑i=1ny-1giy-bixy+14∑i=1nβδix2+β∑i=1nz-1fi·-aixz,U4≔12∑i=1nα+aibi-2ciy2+2∑i=1nα+aiz-1fi·-α2+ai2yz+∑i=1nai-αz2.Since β>0, δi>0, and(30)∑i=1nx+2δi-1y-1giy-biy2≥0,∑i=1nx+2δi-1z-1fi·-aiz2≥0for all x, y, z, we find(31)U3≥-β∑i=1nδi-1y-1giy-bi2y2-β∑i=1nδi-1z-1fi·-ai2z2for all y and z. Moreover, using the estimate(32)∑i=1nα+aiz-1fi·-α2+ai22<∑i=1nα+aibi-2ciai-αin U4 we obtain(33)U4≥12∑i=1nα+aibi-2ciy-∑i=1nai-αz2≥0for all y and z. Inserting estimates U3 and U4 in inequality (28) with hypothesis (v) of Theorem 10, choosing λ1=0, λ2=∑i=1n1-ρ-1(α+β+ai+2)ci>0, and λ3=∑i=1n1-ρ-1α+β+ai+2Bi>0, we have(34)V˙10≤-12∑i=1naibi-ci-Bi+ciα+ai+ci1-ρ-1α+β+ai+2γy2-14∑i=1nai-α-22Bi+ci+Bi1-ρ-1α+β+ai+2γz2-12∑i=1nαbi-ci-β∑i=1n1+ai+δi-1y-1giy-bi2y2-14∑i=1nai-α-β∑i=1n1+δi-1z-1fi·-ai2z2-12∑i=1nβδi-Bi+ciγx2for all x, y, z.
Now in view of the inequalities in (12) there exists a positive constant K3 such that(35)V˙10≤-K3x2+y2+z2∀x,y,z∈R3,where(36)K3=min12∑i=1nβδi-Bi+ciγ,12∑i=1naibi-ci-Bi+ciα+ai+ci1-ρ-1α+β+ai+2γ,14∑i=1nai-α-22Bi+ci+Bi1-ρ-1α+β+ai+2γ.Inequality (35) establishes (17) with K3=D3, respectively. This completes the proof of Lemma 12.Proof of Theorem10.
Let(xt,yt,zt) be any solution of system (10), in view of the inequalities in (19), (22), and (35); all the assumptions of Lemma 7 hold. Thus by Lemma 7 the trivial solution of system (10) or (3) for pi(·)=0 is uniformly asymptotically stable. This completes the proof of Theorem 10.Next, we will consider the case ofpi(·)≠0, and we have the following result.Theorem 13.
If hypotheses (i)–(v) and the inequality in (14) of Theorem 10 hold and(37)pi·≤ϕit+φitx+y+z,for all i  (i=1,2,…,n), (x,y,z)∈R3, and t≥0, where ϕi(t) and φi(t) are continuous functions satisfying(38)ϕit≤di,0<di<∞and there exists an ϵi>0 such that(39)0≤φit≤ϵi,then(i)
the solutions of system (4) are uniformly bounded and uniformly ultimately bounded;(ii)
equation (4) has a unique periodic solution of period ω.Remark 14.
(i) Wheneveri=1, f1(·)=f(x,x˙)x¨, g1(x˙(t-τ1(t)))=g(x(t-τ(t)),x˙(t-τ(t))), h1(x(t-τ1(t)))=h(x(t-τ(t))), and p1(·)=p(t,x,x˙,x¨,x(t-τ(t)),x˙(t-τ(t))), system (4) is a particular case of that studied in [7]. Our hypotheses coincide with that in [7] except for sup⁡{h′(x)}=c>0 which is replaced by a more general condition in ours.
(ii) Ifi=1, f1(·)=φ(x,x˙)x¨, g1(x˙(t-τ1(t)))=g(x˙(t-τ(t))), h1(x(t-τ1(t)))=f(x(t-τ(t))), and p1(·)=p(t,x,x˙,x¨), system (4) reduces to that considered in [16].
(iii) Wheni=1, the functions f1(·)=g(x,x˙)x¨, g1(x˙(t-τ1(t)))=f(x(t-τ),x˙(t-τ)), h1(x(t-τ1(t)))=h(x(t-τ)), and p1(·)=p(t,x,x˙,x(t-τ),x˙(t-τ),x¨), the above result includes that discussed in [24].
(iv) Wheneveri=1, f1(·)=h(x˙)x¨, g1(x˙(t-τ1(t)))=g(x˙(t-τ(t))), h1(x(t-τ1(t)))=h(x(t-τ(t))), and p1(·)=p(t,x,x˙,x(t-τ),x˙(t-τ),x¨), (4) specializes to that studied in [12].
(v) Wheni=1, f1(·)=a1x¨, g1(x˙(t-τ1(t)))=f2(x˙(t-τ1(t))), hi(x(t-τ1(t)))=a3x(t), p1(·)=p(t,x,x˙,x(t-τ1(t)),x˙(t-τ1(t)),x¨), and φi(t)=0, system (4) reduces to that considered in [28]. Theorem 13 coincides with the boundedness result in [28].
(vi) Ifi=1, and φi(t)≡0 in inequality (37), our result specializes to that studied in [9, 27].
(vii) Wheneveri=1, in inequality (37) the result in Theorem 13 reduces to that discussed in [8].
Hence, Theorem13 includes and improves the results in [7–9, 12, 16, 24, 27, 28].Proof of Theorem13.
(i) Let(xt,yt,zt) be any solution of system (4); the time derivative of the functional V≡V(t,xt,yt,zt) defined in system (11) along a solution of system (4) is(40)V˙4=V˙10+∑i=1nβx+α+aiy+2zpi·.Using inequality (35), the above relation becomes(41)V˙4≤-K3x2+y2+z2+K4x+y+z∑i=1npi·,where K4=max⁡{β,∑i=1n(α+ai),2}. Applying inequality (37), we find that(42)V˙4≤-K3x2+y2+z2+K4x+y+z∑i=1nϕit+K4x+y+z2∑i=1nφit.From estimates (38) and (39) and on choosing K3>K4∑i=1n(di+ϵi), there exist constants K5>0 and K6>0 such that(43)V˙4≤-K5x2+y2+z2+K6,∀x,y,z∈R3,where K5≔K3-K4∑i=1n(di+ϵi) and K6≔3K4∑i=1ndi.
The inequalities in (19), (22), and (43) establish the hypotheses of Lemma 8. Hence by Lemma 8, the solution (xt,yt,zt) of system (4) is uniformly bounded and uniformly ultimately bounded.
(ii) From estimate (42), using the inequalities in (38) and (39), we have(44)V˙4≤-K3-3K4∑i=1nϵix2+y2+z2+31/2K4∑i=1ndix2+y2+z21/2.Choosing ϵi(i=1,2,…,n) sufficiently small such that K7≔K3-3K4∑i=1nϵi>0 and K8≔31/2K4∑i=1ndi we have(45)V˙4≤-K9x2+y2+z2∀x,y,z∈R3,provided that x2+y2+z21/2≥K10≔2K8K7-1>0 where K9≔(1/2)K7>0. In view of (19), (21), (22), and (45) all assumptions of Lemmas 5 and 6 are met. Hence by Lemmas 5 and 6, system (4) has a unique periodic solution of period ω. This completes the proof of Theorem 13.Next, ifpi(·) in system (4) is replaced by pi(t)≠0, we have(46)x˙=y,y˙=z,z˙=∑i=1npit-∑i=1nfi·-∑i=1ngiy-∑i=1nhix+∑i=1n∫t-τittgi′yszsds+∑i=1n∫t-τitthi′xsysds,where fi, gi, and hi are the functions defined in Section 1, and pi:R+→R, we have the following result.Theorem 15.
If hypotheses (i)–(v) and estimate (14) of Theorem 10 hold, and(47)∫0tpisds≤qi,0≤qi<∞,i=1,2,…,n,then for any given finite constants x0, y0, z0 there exists a constant D4=D4(x0,y0,z0,α,β,ai,bi,ci,δi) such that any solution (xt,yt,zt) of system (46) determined by x0=x0, y0=y0, z0=z0, for t=0, satisfies(48)xt≤D4,yt≤D4,zt≤D4for all t>0.Remark 16.
ifi=1, f1(·)=f1(t,x)x¨, g1(x˙(t-τ1(t)))=f2(t,x)x˙+g0(t,x), hi(x(t-τ1(t)))=gi(x(t-τ(t))), and p1(·)=p(t), (46) reduces to that considered in [19]. Our results are quite different from this because of the non-Liapunov approach used in [19].Proof of Theorem15.
Let(xt,yt,zt) be any solution of system (46). In view of the hypotheses (i)–(v) and estimate (14), inequality (19) holds. The derivative of the functional V defined in system (11) with respect to the independent variable t along a solution of system (46) is(49)V˙48=V˙10+∑i=1nβx+α+aiy+2zpit.By inequality (35), V˙(10)≤0 for all (x,y,z)∈R3, and from the fact that x1≤1+x12, it follows that(50)V˙48≤3K4∑i=1npit+K4x2+y2+z2∑i=1npit,for all x, y, z and t≥0. Also from inequality (19), the above inequality becomes(51)V˙48-K0-1K4V∑i=1npit≤3K4∑i=1npit.Solving this first-order differential inequality by multiplying each side by (52)exp⁡-K0-1K4∑i=1n∫0tpisds,integrating from 0 to t, and employing inequality (47), we find that(53)V=Vt,xt,yt,zt≤V0+1eK0-1K4∑i=1nqi-1,where V(0)=V(0,x0,y0,z0).
Engaging inequality (19), we have(54)xt≤K11,yt≤K11,zt≤K11for all t>0, where(55)K11≔V0+1eK0-1K4∑i=1nqi-1K0-11/2.Equating K11=D4, the inequalities in (48) are satisfied. This completes the proof of Theorem 15.


## 4. Examples

Example 1.
Consider the homogeneous third-order scalar delay differential equation(56)x⃛+∑i=1n3x¨+x¨5+sin⁡t+xt-τitx˙t-τit+x˙t-τitx¨t-τit+∑i=1n3x˙t-τit+x˙t-τit4+sin⁡2t+xt-τit+x˙t-τit+∑i=1n3xt-τit+xt-τit3+sin⁡2t=0.Reducing (56) to system of first-order delay differential equations by setting x˙=y and x¨=z we obtain(57)x˙=y,y˙=z,z˙=∑i=1n∫t-τitt3+4+sin⁡μ+x4+sin⁡μ+x+y2dμ+∑i=1n∫t-τitt3+13+sin⁡2μ2dμ-3x+x3+sin⁡2t-3y+y4+sin⁡t+x+y-∑i=1n3z+z5+sin⁡t+xt-τityt-τit+yt-τitzt-τit.Comparing system (10) with system (57), we have the following relations:(1)
The function∑i=1nfi(·)=3z+z/(5+sin⁡t+xt-τityt-τit+yt-τitzt-τit). It is clear from the above equation that(58)0<3=∑i=1nai≤∑i=1nfi·z,z≠0.(2)
The function∑i=1ngi(y)=3y+y/(4+sin⁡t+x+y). It is not difficult to show that(59)3=∑i=1nbi≤∑i=1ngiyy≤∑i=1nBi=4,y≠0.(3)
The function∑i=1nhi(x)=3x+x/(3+sin⁡2t), from where we obtain the following estimates:
(a)
3=∑i=1nδi≤∑i=1nhi(x)/x,  x≠0;(b)
∑i=1nhi′(x)≤∑i=1nci, ∑i=1n(aibi-ci)=5>0.(4)
The calculation of the following constants also follows:
(a)
4/3<α<3; we choose α=2;(b)
0<β<min⁡{4,1/4,1/4}=1/4 or 0<β<1/4, and we choose β=1/8;(c)
0<ρ<1, and we choose ρ=1/2;(d)
γ<min⁡{3/8,1/21,1/146}=1/146.All the assumptions of Theorem 10 are satisfied. Hence by Theorem 10 trivial solution of system (57) is uniformly asymptotically stable.Example 2.
Consider the nonhomogeneous third-order delay differential equation:(60)x⃛+∑i=1n3x¨+x¨5+sin⁡t+xt-τitx˙t-τit+x˙t-τitx¨t-τit+∑i=1n3x˙t-τit+x˙t-τit4+sin⁡2t+xt-τit+x˙t-τit+∑i=1n3xt-τit+xt-τit3+sin⁡2t=∑i=1n1+cos⁡2t+MtNi∗1+sin⁡2t+cos⁡2t+cos⁡2tsin⁡2tNi∗,where(61)Mt≔2+2sin⁡2t+cos⁡2t+cos⁡2tsin⁡2t,Ni∗≔1+x+x˙+x¨+xt-τit+x˙t-τit+x¨t-τit.As usual, system (60) is equivalent to(62)x˙=y,y˙=z,z˙=∑i=1n∫t-τitt3+4+sin⁡μ+x4+sin⁡μ+x+y2dμ+∑i=1n∫t-τitt3+13+sin⁡2μ2dμ-3x+x3+sin⁡2t-3y+y4+sin⁡t+x+y-∑i=1n3z+z5+sin⁡t+xt-τityt-τit+yt-τitzt-τit+∑i=1n1+cos⁡2t+MtQi∗1+sin⁡2t+cos⁡2t+cos⁡2tsin⁡2tQi∗,where(63)Qi∗≔1+x+y+z+xt-τit+yt-τit+zt-τit.Comparing systems (4) and (62) we observe that the function (64)∑i=1npi·=∑i=1n1+cos⁡2t+MtQi∗1+sin⁡2t+cos⁡2t+cos⁡2tsin⁡2tQi∗.Substituting appropriately for Qi(∗) and M(t), the above equation can be recast in the form(65)∑i=1npi·=∑i=1n1+11+cos⁡2t+11+sin⁡2t1+x+y+z+xt-τit+yt-τit+zt-τitor(66)∑i=1npi·≤∑i=1n1+11+cos⁡2t+11+sin⁡2t1+x+y+z.From inequalities (37) and (66) we obtain the following periodic functions (see Figures 1 and 2):(67)∑i=1nϕit=1+11+cos⁡2t,∑i=1nφit=11+sin⁡2t.It is not difficult to show that(68)0≤∑i=1nϕit≤∑i=1ndi=2,0≤∑i=1nφit≤∑i=1nϵi=1.The estimates in (68) and that of Example 1 satisfy the hypotheses of Theorem 13. Hence by Theorem 13,(i)
solutions of system (62) are uniformly bounded and uniformly ultimately bounded;(ii)
system (62) has a unique periodic solution of period ω=π/2.Figure 1
Periodic functionϕ(t)=ϕi(t).Figure 2
Periodic functionφ(t)=φi(t).



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