# Stability and Uniform Boundedness in Multidelay Functional Differential Equations of Third Order

Authors: Cemil Tunç; Melek Gözen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiì functional approach, we give certain sufficient conditions to guarantee the asymptotic stability and uniform boundedness of the solutions.


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## Body

## 1. Introduction

Differential equations of third order are valuable tools in the modeling of many phenomena in various fields of science and engineering (Chlouverakis and Sprott [1], Cronin-Scanlon [2], Eichhorn et al. [3], Friedrichs [4], Linz [5], and Rauch [6]). In reality, the stability and boundedness of solutions of certain nonlinear differential equations of the third order have been received intensive attentions by authors (Ademola et al. [7], Afuwape and Castellanos [8], Chukwu [9], Ezeilo ([10, 11]), Hara [12], Mehri and  Shadman [13], Ogundare and Okecha [14], Omeike [15], Reissig et al. [16], Swick [17], Tejumola ([18, 19]),   Tunç [20–33], and Yoshizawa [34]).In 2009, Omeike [15] considered the nonlinear differential equation of the third order with the constant delay r(>0):
(1)x⃛+a(t)x¨+b(t)g(x˙)+c(t)h(x(t-r))=p(t),
and he discussed the stability and boundedness of solutions of this equation.In this paper, instead of the above equation, we consider the nonautonomous differential equation of the third order with multiple deviating arguments:(2)x⃛+a(t)x¨+nb(t)g(x˙)+c(t)∑i=1khi(x(t-ri))=p(t),
where ri are certain positive constants, and a(t),b(t),c(t),g(x˙),h(x) and p(t) are real valued and continuous functions in their respective arguments with g(0)=h(0)=0,k=n. The existence and uniqueness of the solutions of (2) are also assumed.The motivation for this paper is a result of the researches mentioned regarding ordinary differential equations of the third order. It follows that the equation discussed in [15] is a special case of (2). Our aim is to improve the results established in [15] from one deviating argument to the multiple deviating arguments for the asymptotic stability and uniform boundedness of solutions. This work contributes to and complements previously known results on the topic in the literature, and it may be useful for researchers working on the qualitative behaviors of solutions. It should be noted that in recent years scores of papers have been published on the qualitative behaviors of solutions (stability of solutions, boundedness of the solutions, existence of the periodic solutions, etc.) of the functional differential equations of the second order with multiple deviating arguments. However, very little attention was given to stability and boundedness of functional differential equations of the third order with multiple deviating arguments ([32]). Therefore, it is worth investigating the qualitative behaviors of solutions in multidelay functional differential equations of the third order. This case is the novelty of the present paper. It should also be noted that the results to be established here are different from those in Tunç [20–33] and the literature.


## 2. Main Results

Letp(t)≡0.Theorem 1.
One assumes that there exist positive constantsa,b,c,ρi,α,μi,δ1,δ2,δ5,δ6, and L such that the following conditions hold:(i)
h(0)=g(0)=0,hi(x)/x≥μi,x≠0,hi′(x)≤ρi,g(y)/y≥b>0,y≠0(i=1,2,…,k),(ii)
0<δ1≤c(t)≤b(t),-L≤b′(t)≤c′(t)≤0,0<a<a(t).If(3)bρi>α>1a,12a′(t)≤δ2<δ1(nb-α∑i=1kρi),∑i=1kri<min{2δ5Lc(α+2),δ6Lcα},then every solutionx≡x(t) of (2) is uniform bounded and satisfies
(4)x(t)⟶0,x˙(t)⟶0,x¨(t)⟶0ast⟶∞.Remark 2.
It should be noted that it follows from (ii) thatb(t) and  c(t) are nonincreasing functions on [0,∞). Therefore, since these functions are continuous on this interval and bounded below by δ1>0, they are bounded on [0,∞) and the limit of each exists as t→∞.  Since L in (ii) is an arbitrary selected bound, we can also assume the following estimates:
(5)0<δ1≤c(t)≤b(t)≤L,limt→∞c(t)=c0,limt→∞b(t)=b0,δ1≤c0≤b0≤L.Proof.
We write (2) in the system form as follows:
(6)x˙=y,y˙=z,z˙=-a(t)z-nb(t)g(y)-c(t)∑i=1khi(x)+c(t)∑i=1k∫t-rithi′(x(s))y(s)ds.
Define a Lyapunov-Krasovskiì functional ([35]) V(·)=V(t,xt,yt,zt) by
(7)V(·)=c(t)H(x)+nαb(t)G(y)+αc(t)y∑i=1khi(x)+12a(t)y2+12αz2+yz+∑i=1kλi∫-ri0∫t+sty2(θ)dθds,
where H(x)≡∫0x∑i=1khi(s)ds,G(y)≡∫0yg(ξ)dξ, and λi are certain positive constants, which will be determined later in the proof.
This functional can be arranged as follows:(8)V(·)=V1+12V2+∑i=1kλi∫-ri0∫t+sty2(θ)dθds,
where
(9)V1=c(t)[H(x)+nαb(t)c(t)G(y)+αy∑i=1khi(x)],V2=a(t)y2+αz2+2yz.
Using the assumptions of Theorem1, it follows that αa(t)≥αa>1 since α>a-1 and αa(t)-1>0.
Thus, there exist constantsδ1>0 and δ3>0 such that
(10)V2=a(t)y2+αz2+2yz=a(t)[y+z2a(t)]2+14a(t)[4αa(t)-1]z2≥12δ3y2+12δ3z2,V1≥δ1[H(x)+12nαby2+αy∑i=1khi(x)]
since b(t)/c(t)≥1,c(t)≥δ1>0,andg(y)/y≥b>0implythatG(y)≥(1/2)by2. Further, using the assumptions of Theorem 1 and k=n,  it follows that
(11)H(x)+12nαby2+αy∑i=1khi(x)=12[2H(x)+nαby2+2αy∑i=1khi(x)]=12{∑i=1khi2αb[(by+h1(x))2+(by+h2(x))2+⋯+(by+hn(x))2]+2H(x)-αb∑i=1khi2(x)}=12αb{(by+h1(x))2+(by+h2(x))2+⋯+(by+hn(x))2}+∑i=1k∫0x(1-αhi′(s)b)hi(s)ds≥∑i=1k∫0x(1-αρib)hi(s)ssds≥∑i=1k∫0x(1-αρib)μisds≥12∑i=1k(1-αρib)μix2=δ42x2
so that
(12)V1≥δ1δ42x2,
where
(13)δ4=∑i=1k(1-αρib)μi.
In view of the previous discussion, we can get(14)V(·)≥(δ1δ42)x2+δ34y2+δ34z2+∑i=1kλi∫-ri0∫t+sty2(θ)dθds.
Using a basic calculation, the time derivative ofV(·)  along solutions of (6) results in
(15)ddtV(·)=c′(t)H(x)+c(t)H′(x)+nαb′(t)G(y)+nαb(t)G′(y)+αc′(t)y∑i=1khi(x)+αc(t)y′∑i=1khi+αc(t)yx′∑i=1khi′(x)+12a′(t)y2+a(t)yy′+αzz′+y′z+yz′+∑i=1kλi∫-ri0[y2(t)-y2(t+s)]ds=c′(t)H(x)+c(t)y∑i=1khi(x)+nαb′(t)G(y)+nαb(t)zg(y)+αc′(t)y∑i=1khi(x)+αc(t)z∑i=1khi(x)+αc(t)y2∑i=1khi′(x)+12a′(t)y2+a(t)yz+αz×[∑i=1k∫t-rith′i(x(s))y(s)ds-a(t)z-nb(t)g(y)-c(t)∑i=1khi(x(t))+c(t)∑i=1k∫t-rithi′(x(s))y(s)ds]+z2+y[∫t-rith′i(x(s))y(s)ds-a(t)z-nb(t)g(y)-c(t)∑i=1khi(x(t))+c(t)∑i=1k∫t-rithi′(x(s))y(s)ds]+∑i=1kλiriy2-∑i=1kλi∫t-rity2(s)ds=c′(t)H(x)+αc′(t)y∑i=1khi(x)+nαb′(t)G(y)+12a′(t)y2-[nyb(t)g(y)-αc(t)y2∑i=1khi′(x)-∑i=1kλiriy2]-[αa(t)-1]z2+αzc(t)∑i=1k∫t-rithi′(x(s))y(s)ds+yc(t)∑i=1k∫t-rithi′(x(s))y(s)ds-∑i=1kλi∫t-rity2(s)ds.
Usinghi′(x)≤ρi,c(t)≤L, and the estimate 2|ef|≤e2+f2, we have
(16)αzc(t)∑i=1k∫t-rithi′(x(s))y(s)ds≤αc(t)|z|∑i=1k∫t-ritc|y(s)|ds≤12αc(t)z2∑i=1kriρi+12αc(t)∑i=1k∫t-ritρiy2(s)ds≤12Lαcz2∑i=1kri+12Lαc∑i=1k∫t-rity2(s)ds,yc(t)∑i=1k∫t-rithi′(x(s))y(s)ds≤c(t)|y|∑i=1k∫t-ritρi|y(s)|ds≤12c(t)y2∑i=1kriρi+12c(t)∑i=1k∫t-ritρiy2(s)ds≤12Lcy2∑i=1kri+12Lc∑i=1k∫t-rity2(s)ds,
where
(17)c=maxρi.
Noting the previous discussion, it follows that(18)ddtV(·)≤c′(t)H(x)+nαb′(t)G(y)+αc′(t)y∑i=1khi(x)+12a′(t)y2-[nyb(t)g(y)-αc(t)y2∑i=1khi′(x)-∑i=1kλiriy2]-12[2(αa(t)-1)-Lαc∑i=1kri]z2+12Lcy2∑i=1kri+∑i=1k[12Lc(α+1)-λi]∫t-rity2(s)ds.
Ify=0, then
(19)nyb(t)g(y)-αc(t)y2∑i=1khi′(x)-∑i=1kλiriy2=0.
Ify≠0, then it follows that
(20)nyb(t)g(y)-αc(t)y2∑i=1khi′(x)-∑i=1kλiriy2=[nb(t)g(y)y-αc(t)∑i=1khi′(x)-∑i=1kλiri]y2≥[nb(t)b-αc(t)∑i=1kρi-∑i=1kλiri]y2=c(t)[b(t)c(t)nb-α∑i=1kρi]y2-∑i=1kλiriy2≥δ1[nb-α∑i=1kρi-δ1-1∑i=1kλiri]y2.
Since0≤δ1≤c(t)≤b(t),1≤b(t)/c(t), and αa(t)≥αa>1, then
(21)12a′(t)y2-[nyb(t)g(y)-αc(t)y2∑i=1khi′(x)-∑i=1kλiriy2]≤[δ2-δ1(nb-α∑i=1kρi)+∑i=1kλiri]y2=-(δ5-∑i=1kλiri)y2,[2(αa(t)-1)-Lαc∑i=1kri]z2≥(δ6-Lαc∑i=1kri)z2,
where δ5=δ1(nb-α∑i=1kρi)-δ2>0 and δ6≡2(αa-1)>0.
Thus, we get(22)ddtV(·)≤c′(t)H(x)+nαb′(t)G(y)+αc′(t)y∑i=1khi(x)-{δ5-∑i=1kλiri}y2-12(δ6-Lαc∑i=1kri)z2+12Lcy2∑i=1kri+∑i=1k[12Lc(α+1)-λi]∫t-rity2(s)ds.
Letλi=(1/2)Lc(α+1). Hence,
(23)ddtV(·)≤c′(t)H(x)+nαb′(t)G(y)+αc′(t)y∑i=1khi(x)-{δ5-12Lc(α+2)∑i=1kri}y2-12(δ6-Lαc∑i=1kri)z2.
Ifc′(t)=0, then c′(t)H(x)+nαb′(t)G(y)+αc′(t)y∑i=1khi(x)=nαb′(t)G(y)≤0 since b′(t)≤0 and G(y)≥0. For those t's such that c′(t)<0, we have
(24)c′(t)H(x)+nαb′(t)G(y)+αc′(t)y∑i=1khi(x)=c′(t)[H(x)+nαb′(t)c′(t)G(y)+αy∑i=1khi(x)]≤c′(t)[H(x)+nαG(y)+αy∑i=1khi(x)]≤c′(t)δ4H(x)≤0.
Thus,(25)ddtV(·)≤-{δ5-12Lc(α+2)∑i=1kri}y2-12(δ6-Lαc∑i=1kri)z2.
Therefore, if(26)∑i=1kri<min{2δ5Lc(α+2),δ6Lαc},
then we have
(27)ddtV(·)≤-β(y2+z2)forsomeβ>0.
The proof for Theorem1 is complete.Letp(t)≠0.Theorem 3.
One assumes that all the assumptions of Theorem1 and the assumption
(28)∫0t|p(s)|ds<∞
hold. If
(29)bρi>α>1a,12a′(t)≤δ2<δ1(nb-α∑i=1kρi),∑i=1kri<min{2δ5Lc(α+2),δ6Lcα},
then all solutions of (2) are bounded.Proof.
Equation (2) is equivalent to the system
(30)x˙=y,y˙=z,z˙=-a(t)z-nb(t)g(y)-c(t)∑i=1khi(x(t))+c(t)∑i=1k∫t-rithi′(x(s))y(s)ds+p(t).
Along any solution(x(t),y(t),z(t)) of (6), we have
(31)V˙(3)(·)=V˙(2)(·)+(y+αz)p(t).
SinceV˙(2)(·)≤0, then it follows that
(32)V˙(3)(·)≤(y+αz)p(t)≤(|y|+α|z|)|p(t)|≤δ8(|y|+|z|)|p(t)|,
where δ8≡max{1,α}. Noting that |m|<1+m2, we get
(33)V˙(3)(·)≤δ8(2+y2+z2)|p(t)|≤2δ8|p(t)|+δ8∥X∥2|p(t)|≤2δ8|p(t)|+(δ8δ7)V(·)|p(t)|
recalling that δ7∥X∥2≤V(·).
Letη=max(2δ8,δ8/δ7), then
(34)V˙(3)(·)≤η|p(t)|+ηV(·)|p(t)|
or
(35)V˙(3)(·)-ηV(·)|p(t)|≤η|p(t)|.
Multiplying each side of this estimate by the integrating factorexp(-η∫0t|p(s)|ds), we get
(36)V˙(3)(·)exp(-η∫0t|p(s)|ds)-ηV(·)|p(t)|exp(-η∫0t|p(s)|ds)≤η|p(t)|exp(-η∫0t|p(s)|ds).
Integrating each side of this estimate from 0 tot, we obtain
(37)V(·)exp(-η∫0t|p(s)|ds)-V(0)≤1-exp(-η∫0t|p(s)|ds)
or
(38)V(·)≤V(0)exp(η∫0t|p(s)|ds)+exp(η∫0t|p(s)|ds)-1,
where (0,x(0),y(0),z(0))=0.
Since∫0t|p(s)|ds≤A for all t, this implies
(39)V(·)≤V(0)eηA+(eηA-1)fort≥0.
Since the right-hand side of the last estimate is a constant andV(·)→∞ when x2+y2+z2→∞,  it follows that there exists a positive constant D such that
(40)|x(t)|≤D,|y(t)|≤D,|z(t)|≤D∀t≥0.
From the system (30) this implies that
(41)|x(t)|≤D,|x˙(t)|≤D,|x¨(t)|≤D∀t≥0.
The proof for Theorem3 is complete.



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