# Qualitative Behaviors of Functional Differential Equations of Third Order with Multiple Deviating Arguments

Authors: Cemil Tunç
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper considers nonautonomous functional differential equations of the third order with multiple constant deviating arguments. Using the Lyapunov-Krasovskii functional approach, we find certain sufficient conditions for the solutions to be stable and bounded. We give an example to illustrate the theoretical analysis made in this work and to show the effectiveness of the method utilized here.


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## Body

## 1. Introduction

In this paper, we consider nonautonomous differential equation of the third order with constant multiple deviating arguments,τ, τi, (i=1,2,…,n) as follows:
(1.1)x′′′(t)+a(t)x′′(t)+∑i=1nbi(t)gi(x′(t-τi))+g(x′(t))+h(x(t-τ))=p(t,x(t),x(t-τ1),…,x(t-τn),x′,…,x′(t-τn)…,x′′(t),…,x′′(t-τn)).
Writing (1.1) as a system of first order equations, we have
(1.2)x′=y,y′=z,z′=-a(t)z-∑i=0nbi(t)gi(y)-h(x)-g(y)+∑i=1nbi(t)∫t-τitgi′(y(s))z(s)ds+∫t-τth′(x(s))y(s)ds+p(t,x,…,x(t-τn),y,…,y(t-τn),z,…,z(t-τn)),
where τ and τi,    (i=1,2,…,n), are positive constants, that is, fixed delays; the functions a,    bi, g,    gi,h,   and p are continuous for their all respective arguments and the primes in (1.1) denote differentiation with respect to t,  t∈ℜ+=[0,∞). It is also assumed that the derivatives a'(t)≡(d/dt)a(t),   bi′(t)≡(d/dt)bi(t),   gi′(y)≡(d/dy)gi(y), and h'(x)≡(d/dx)h(x) exist and are continuous; throughout the paper x(t),y(t),   andz(t) are abbreviated as x,y, and z, respectively. Finally, the existence and uniqueness of solutions of (1.1) are assumed and all solutions considered are supposed to be real valued.To the best of our knowledge from the literature, in the last five decades, there has been much attention paid to the discussion of stability and boundedness of solutions of nonlinear differential equations of the third order without a deviating argument. For a comprehensive treatment of the subject, we refer the readers to the book of Reissig et al. [1] as a survey and the papers of Ademola et al. [2], Afuwape et al. [3], Ezeilo [4–13], Ezeilo and Tejumola [14, 15],  Mehri and Shadman [16], Ogundare [17], Ogundare and Okecha [18], Omeike [19, 20], Ponzo [21, 22], Swick [23–25], Tejumola [26, 27],  Tunç [28–34], Tunç and Ateş [35], Tunç and Ayhan [36], and the references cited in these papers for some works on the topic.Besides, first, in 1973, Sinha [37] studied the stability of solutions of a third order nonlinear differential equation with a deviating argument. Later, some authors dealt with the stability and boundedness of solutions for various third order nonlinear differential equations with a deviating argument. For some related works, one can refer to the papers of Afuwape and Omeike [38], Omeike [39], Sadek [40, 41], Tejumola and Tchegnani [42], Tunç [43–59], Yao and Meng [60], Zhu [61], and the references thereof.It should be noted that throughout all the above mentioned papers, Lyapunov’s functions or the Lyapunov-Krasovskii functionals have been used as a basic tool to prove the results established there. It is also worth mentioning that the most effective method to study the stability and boundedness of solutions of nonlinear differential equations of higher orders without or with a deviating argument in the literature is still the Lyapunov’s direct method, despite its use for a past long period by now.The motivation for this paper comes from the above mentioned papers and the books. Our results improve and include the results existing in the literature. This work makes also a contribution to the existing studies made in the literature.


## 2. Main Results

Letp(·)=0.Our first result is given by the following theorem.Theorem 2.1.
In addition to the basic assumptions imposed to the functionsa(t), b(t), g, gi, and h appearing in (1.1), we assume that there exist positive constants a, α, β, δ, b, bi, Bi, c, c1, and Li such that the following conditions hold:(i)
gi(0)=g(0)=h(0)=0,a(t)≥2α+a,
βi≤bi(t)≤Bi,
(2.1)0<c1≤h′(x)≤c,αb-c>δ,gi(y)y≥bi,g(y)y≥b,(y≠0),|gi′(y)|≤Li,(ii)
[∑i=1n{αbibi(t)}-c]y2≥2-1αa'(t)y2+∑i=1nbi′(t)∫0ygi(η)dη.
If(2.2)τ0<min{αbc+2αc+α∑i=1n(BiLi),2αc+∑i=1n[(2+α)BiLi]},
then the zero solution of (1.1) is stable.Proof.
Define the Lyapunov-Krasovskii functionalV(·)=V(t,xt,yt,zt) as
(2.3)2V(·)=z2+2αyz+2∑i=0nbi(t)∫0ygi(η)dη+2α∫0xh(ξ)dξ+2∫0yg(η)dη+αa(t)y2+2h(x)y+λ∫-τ0∫t+sty2(θ)dθds+∑i=1nλi∫-τi0∫t+stz2(θ)dθds,
where λ and λi are some positive constants to be chosen later.
Using the assumptionsgi(y)/y≥bi, g(y)/y≥b, (y≠0),h(0)=0, and 0<c1≤h'(x)≤c, we have
(2.4)2bi(t)∫0ygi(η)dη=2bi(t)∫0ygi(η)ηηdη≥(βibi)y2,2∫0yg(η)dη=2∫0yg(η)ηηdη≥by2,h2(x)=2∫0xh(ξ)h′(ξ)dξ≤2c∫0xh(ξ)dξ,
so that
(2.5)2V(·)≥(z+αy)2+b[y+b-1h(x)]2+2α∫0xh(ξ)dξ-1bh2(x)+∑i=1n(βibi)y2+α(α+a)y2+λ∫-τ0∫t+sty2(θ)dθds+∑i=1nλi∫-τi0∫t+stz2(θ)dθds≥(z+αy)2+b[y+b-1h(x)]2+2α∫0xh(ξ)dξ-2cb∫0xh(ξ)dξ+∑i=1n(βibi)y2+α(α+a)y2+λ∫-τ0∫t+sty2(θ)dθds+∑i=1nλi∫-τi0∫t+stz2(θ)dθds.
On the other hand, it is obvious that
(2.6)2α∫0xh(ξ)dξ-2cb∫0xh(ξ)dξ=2b-1(αb-c)∫0xh(ξ)dξ≥c1b-1(αb-c)x2,
so that
(2.7)2V(·)≥(z+αy)2+b[y+b-1h(x)]2+c1b-1(αb-c)x2+α(α+a)y2+∑i=1n(βibi)y2+λ∫-τ0∫t+sty2(θ)dθds+∑i=1nλi∫-τi0∫t+stz2(θ)dθds.
Hence, we can obtain some positive constants Dj, (j=1,2,3), such that
(2.8)V(·)≥D1x2+D2y2+D3z2≥D4(x2+y2+z2),
where D4=min{D1,D2,D3}, since the integrals λ∫-τ0∫t+sty2(θ)dθds and ∑i=1nλi∫-τi0∫t+stz2(θ)dθds are nonnegative.
Let(x,y,z) be a solution of (1.2). Differentiating the Lyapunov-Krasovskii functional V(·) along this solution, we find
(2.9)ddtV(·)=-[α∑i=1n{bi(t)gi(y)}y-1+αg(y)y-1-h′(x)-2-1αa′(t)]y2+∑i=1nbi′(t)∫0ygi(η)dη-(a(t)-α)z2+z∑i=1nbi(t)∫t-τitgi′(y(s))z(s)ds+z∫t-τth′(x(s))y(s)ds+αy∑i=1nbi(t)∫t-τitgi′(y(s))z(s)ds+αy∫t-τth′(x(s))y(s)ds+λτy2-λ∫t-τty2(s)ds+∑i=1n(λiτi)z2-∑i=1nλi∫t-τitz2(s)ds.
Using the assumptions of Theorem2.1 and the estimate 2|mn|≤m2+n2, we get
(2.10)[α∑i=1n(bi(t)gi(y))y-1+αg(y)y-1-h′(x)-2-1αa′(t)]y2-∑i=1nbi′(t)∫0ygi(η)dη≥[α∑i=1n(bibi(t))+αb-c-2-1αa′(t)]y2-∑i=1nbi′(t)∫0ygi(η)dη=[∑i=1n{αbibi(t)}-c]y2-2-1αa'(t)y2-∑i=1nbi′(t)∫0ygi(η)dη+(αb)y2≥(αb)y2,[a(t)-α]z2≥(α+a)z2,z∑i=1nbi(t)∫t-τitgi′(y(s))z(s)ds≤12∑i=1n(BiLiτi)z2+12∑i=1n(BiLi)∫t-τitz2(s)ds,αy∑i=1nbi(t)∫t-τitgi′(y(s))z(s)ds≤12∑i=1n(αBiLiτi)y2+12∑i=1n(αBiLi)∫t-τitz2(s)ds,z∫t-τth′(x(s))y(s)ds≤c2τz2+c2∫t-τty2(s)ds,αy∫t-τth′(x(s))y(s)ds≤αc2τy2+αc2∫t-τty2(s)ds,
so that
(2.11)ddtV(·)≤-12αby2-az2-12[αb-{α∑i=1n(BiLiτi)+(2λ+αc)τ}]y2-12[2α-{∑i=1n(BiLiτi)+2∑i=1n(λiτi)+cτ}]z2+[2-1(1+α)c-λ]∫t-τty2(s)ds+[2-1(1+α)∑i=1n(BiLi)-∑i=1nλi]∫t-τitz2(s)ds.
Let λ=(1/2)(1+α)c and ∑i=1nλi=(1/2)(1+α)∑i=1n(BiLi). Hence,
(2.12)ddtV(·)≤-12αby2-az2-12[αb-{α∑i=1n(BiLiτi)+(2λ+αc)τ}]y2-12[2α-{∑i=1n(BiLiτi)+2∑i=1n(λiτi)+cτ}]z2.
Let τ0=max{τ,τ1,τ2,…,τn}. Then
(2.13)ddtV(·)≤-12αby2-az2-12[αb-{α∑i=1n(BiLi)+(2λ+αc)}τ0]y2-12[2α-{∑i=1n(BiLi)+2∑i=1nλi+c}τ0]z2.
Thus, if
(2.14)τ0<min{αbc+2αc+α∑i=1n(BiLi),2αc+∑i=1n[(2+α)BiLi]},
then
(2.15)ddtV(·)≤-12αby2-az2.
This completes the proof of Theorem 2.1 (Burton [62], Hale [63], and Krasovskiĭ [64]).Letp(·)=0.Our second main result is given by the following theorem.Theorem 2.2.
In addition to all the assumptions of Theorem2.1, we assume that the condition
(2.16)|p(·)|≤|q(t)|
holds, where |q|∈L1(0,∞). If
(2.17)τ0<min{αbc+2αc+α∑i=1n(BiLi),2αc+∑i=1n[(2+α)BiLi]},
then, there exists a finite positive constant K such that the solution x(t) of (1.1) defined by the initial function
(2.18)x(t)=ϕ(t),x′(t)=ϕ′(t),x′′(t)=ϕ′′(t)
satisfies
(2.19)|x(t)|≤K,|x′(t)|≤K,|x′′(t)|≤K
for all t≥t0, where ϕ∈C2([t0-r,t0],ℜ).Proof.
Under the assumptions of Theorem2.2, the time derivative of the Lyapunov-Krasovskii functional V(·) satisfies
(2.20)ddtV(·)≤-12αby2-az2+(αy+z)p(·).
Using the estimates |m|<1+m2 and D4(x2+y2+z2)≤V(·),it follows that
(2.21)ddtV(·)≤(α|y|+|z|)|p(·)|≤D5(2+y2+z2)|q(t)|≤2D5|q(t)|+D5D4-1V(·)|q(t)|,
where D5=max{1,α}.
Integrating the above estimate from0 to t, using the assumption |q|∈L1(0,∞) and the Gronwall-Bellman inequality (see Gronwall [65] and Mitrinović [66]), we can conclude that all solutions of (1.1) are bounded.Example 2.3.
Consider the nonlinear differential equation of the third order with two deviating arguments as follows:(2.22)x′′′(t)+(11+11+t2)x′′(t)+2(1+e-t)x′(t-τ1)+2(2+e-t)x′(t-τ2)+4x′(t)+x(t-τ)+arctgx(t-τ)=41+t2+x2(t)+x2(t-τ1)+x'2(t-τ2).
Writing (2.22) as a system of first order equations, we obtain
(2.23)x′=y,y′=z,z′=-(11+11+t2)z-2(5+2e-t)y-x-arctgx+2(1+e-t)∫t-τ1tz(s)ds+2(2+e-t)∫t-τ2tz(s)ds+∫t-τty(s)ds+∫t-τty(s)1+x2(s)ds+41+t2+⋯+y2(t-τ2).
It follows that (2.22) is special case of (1.1), and when we compare (2.22) with (1.1) we obtain the following estimates:
(2.24)a(t)=11+11+t2≥11=2×5+1,α=5,a=1,b1(t)=1+1et,1≤1+1et≤2,β1=1,B1=2,g1(y)=2y,g1(0)=0,g1(y)y=2=b1,(y≠0),g1′(y)=2=L1,∫0yg1(η)dη=∫0y2ηdη=y2,b2(t)=2+1et,2≤2+1et≤3,β2=2,B2=3,g2(y)=2y,g2(0)=0,g2(y)y=2=b2,(y≠0),g2′(y)=2=L2,∫0yg2(η)dη=∫0y2ηdη=y2,g(y)=4y,g(0)=0,g(y)y=4=b,(y≠0),h(x)=x+arctgx,h(0)=0,h′(x)=1+11+x2,0<2-1<h′(x)≤2,c1=2-1,c=2,a′(t)=-2t(1+t2)2,b1′(t)=-1et,b2′(t)=-1et,(t≥0),p(t,x,…,z)=41+t2+⋯+y2(t-τ2)≤41+t2=q(t).
In view of the above discussion, it follows that
(2.25)αb-c=4>0,[∑i=12{αbibi(t)}-c]y2≥2-1αa′(t)y2+∑i=12bi′(t)∫0ygi(η)dη=[αb1b1(t)+αb2b2(t)-c]y2=[5(1+e-t)+10(2+e-t)-4]y2=(21+15e-t)y2,(t≥0),12αa′(t)y2+b1′(t)∫0yg1(η)dη+b1′(t)∫0yg1(η)dη=-[5t(1+t2)2]y2-2e-ty2,(t≥0),(21+15e-t)y2≥-[5t(1+t2)2]-2e-ty2,∫0∞|q(s)|ds=∫0∞41+s2ds=2π<∞,
that is, |q|∈L1(0,∞), and
(2.26)τ0<min{αbc+2αc+α∑i=12(BiLi),2αc+∑i=12[(2+α)BiLi]}=min{518,536}=536.
Thus, all the assumptions of Theorems 2.1 and 2.2 hold. This shows that the zero solution of  (2.22) is stable and all solutions of the same equation are bounded, when p(·)=0 and ≠0, respectively.



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