# Existence of Periodic Solutions to Nonlinear Differential Equations of Third Order with Multiple Deviating Arguments

Authors: Cemil Tunç
Journal: International Journal of Differential Equations
 (2012)

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## Abstract

We establish certain new sufficient conditions which guarantee the existence of periodic solutions for a nonlinear differential equation of the third order with multiple deviating arguments. Using the Lyapunov functional approach, we prove a specific theorem and provide an example to illustrate the theoretical analysis in this work and the effectiveness of the method utilized here.


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## Body

## 1. Introduction

It is known that functional differential equations, in particular, that delay differential equations can be used as models to describe many physical, biological systems, and so forth. In reality, many actual systems have the property aftereffect, that is, the future states depend not only on the present, but also on the past history, and after effect is also known to occur in mechanics, control theory, physics, chemistry, biology, medicine, economics, atomic energy, information theory, and so forth (Burton [1], Kolmanovskii and  Myshkis [2]). Therefore, it is important to investigate the qualitative behaviors of functional differential equations.In 1978, using the known theorem of Yoshizawa [3, Theorem 37.2], Chukwu [4] found certain sufficient conditions that guarantee the existence of a periodic solution to nonlinear-linear differential of the third order with the constant deviating argument h (>0):
(1.1)x′′′+f(x,x′,x′′)x′′+g(x(t-h),x′(t-h))+i(x(t-h))=p(t,x,x′,x(t-h),x′(t-h),x′′).Later, in 1992, Zhu [5] considered the nonlinear differential equation of the third order with the constant deviating argument r (>0):
(1.2)x′′′+ax′′+ϕ(x′(t-r))+f(x)=p(t),
and he discussed the existence of periodic solutions for this equation when  p(t) is a periodic function of period T, T>0.In 2000, Tejumola and Tchegnani [6] considered the nonlinear differential equation of the third order with the constant deviating argument τ (>0):
(1.3)x′′′+f(t,x,x′,x′′)+g(t,x(t-τ),x′(t-τ))+h(x(t-τ))=P1(t,x,x′,x′′,x(t-τ),x′(t-τ)).
The authors established certain sufficient conditions on the existence of periodic of solutions of this equation.In 2010, Tunç [7] established certain sufficient conditions for the existence of a periodic solution for the nonlinear differential equation of the third order with the constant deviating argument r (>0):
(1.4)x′′′+ψ(x′)x′′+g(x′(t-r))+f(x)=p(t,x,x(t-r),x′,x′(t-r),x′′).However, a review to date of the literature indicates that the existence of periodic solutions to the nonlinear differential equation of the third order with multiple deviating arguments has not been investigated. The paper considers the nonlinear differential equation of the third order with multiple constant deviating argumentsτi,(i=1,2,…,n):
(1.5)x′′′+ψ(x′)x′′+∑i=1ngi(x′(t-τi))+f(x)=p(t,x,x(t-τ1),…,x′,…,x′(t-τn),x′′).The equation (1.5) is stated in system form as follows:
(1.6)x′=y,y′=z,z′=-ψ(y)z-∑i=1ngi(y)-f(x)+∑i=1n∫t-τitgi′(y(s))z(s)ds+p(t,x,x(t-τ1),…,y(t-τn),z),
where τi  are  positive constants, that is,  τi are constant deviating arguments, which are determined in Section 2. It is assumed that the functions  ψ,   gi,  f, and p(·) are continuous in their respective arguments on ℜ, ℜ, ℜ, and  ℜ+×ℜ2n+3,  (ℜ+=[0,∞), ℜ=(-∞,∞)), respectively; gi(0)=f(0)=0 and p(·) is periodic in t of period T,T≥τi, the derivatives gi′(y)≡(d/dy)gi(y) exist and are also continuous; throughout what follows  x(t), y(t), and z(t) are abbreviated as x,y, and z, respectively.The motivation for this paper is a result of the research mentioned regarding ordinary differential equations with a deviating argument. Our aim is to achieve the results established in [5, 7] to (1.5) with multiple deviating arguments. Our results generalize the results established on the existence of periodic solution in [5, 7]. This paper is the first known publication regarding the existence of periodic solution for differential equations of the third order with multiple deviating arguments.In order to reach our main result, this paper offers fundamental information regarding the general nonautonomous delay periodic differential system. Consider the delay periodic system:(1.7)x˙(t)=F(t,xt),xt=x(t+θ),-r≤θ≤0,t≥0,
where F:[0,∞)×CH→ℜn is a continuous mapping, F(t+T,φ)=F(t,φ) for all φ∈C and for some constant T>0. We assume that F takes closed bounded sets into bounded sets of ℜn. Here (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},CH  is the open H-ball in C,C=C([-r,0],ℜn).Theorem 1.1.
Suppose thatF(t,φ)∈C-0(φ) and F(t,φ) is periodic in t of period T,T≥r, and consequently for any α>0 there exists an L(α)>0 such that φ∈Cα implies  |F(t,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 with |φ(0)|≥H (H may be large), and that V(t,φ) satisfies the following conditions.(i)
Continuous increasing functionsa(s) and b(s) exist, satisfying a(s)>0,b(s)>0 for s≥H and a(s)→∞ as s→∞, such that
(1.8)a(|ϕ(0)|)≤V(t,ϕ)≤b(‖ϕ‖),when|φ(0)|≥H.(ii)
A continuous and positive functionw(s) exists such that
(1.9)V˙(t,ϕ)≤-w(|φ(0)|)fors≥H.(iii)
A constantH1>0,H1>H, exists such that
(1.10)rL(γ*)<H1-H,where γ*>0 is a constant which is determined in the following way.
Using the condition onV(t,φ), constants α>0,β>0, and γ>0 exist such that b(H1)≤a(α),b(α)≤a(β), and b(β)≤a(γ),   γ*  is defined by  b(γ)≤a(γ*).
Under these conditions, a periodic solution of (1.7) of period T  exists.  In particular, the relation rL(γ*)<H1-H is always satisfied if r is sufficiently small (see Yoshizawa [3]).


## 2. Main Result

The main result is the following theorem:letτ=max1≤i≤nτi.Theorem 2.1.
Suppose that positive constantsa,bi,  c,  m,  δ,  Li, and  τexist such that the following conditions hold:
(2.1)ab-c>0,f(0)=0,f(x)sgn⁡x>0∀x≠0,sup⁡{f′(x)}=c,f(x)sgn⁡x⟶∞as|x|⟶∞,gi(0)=0,gi(y)y≥bi,(y≠0),|gi′(y)|≤Li,0≤ψ(y)-a≤δ∀y,|p(⋅)|≤m.
If
(2.2)τ<min⁡{ab-c(2+μ)bN1,ab-c4M1},
then (1.5) has a periodic solution of period T, where μ=(ab+c)/2b, b=∑i=1nbi, M1=∑i=1n(μLi/2) and N1=∑i=1nLi.Proof.
Define a Lyapunov functionalV=V(xt,yt,zt)by:
(2.3)V(xt,yt,zt)=V1(xt,yt,zt)+V2(x,y,z)+1+∑i=1nLi∫-τi0∫t+st|z(θ)|dθds,
where
(2.4)V1(xt,yt,zt)=μ∫0xf(ξ)dξ+f(x)y+μ∫0yψ(η)ηdη+∑i=1n∫0ygi(η)dη+μyz+12z2+∑i=1nγi∫-τi0∫t+stz2(θ)dθds,V2(x,y,z)={zMsgn⁡x,|x|≥1,|z|≤Msgn⁡zsgn⁡x,|x|≥1,|z|≥MxzM,|x|≤1,|z|≤Mxsgn⁡z,|x|≤1,|z|≥M,M,(M>1),   and γi are certain positive constants; the constants γi will be determined later in the proof.
It follows thatV1(0,0,0)=0. In view of the assumptions ψ(y)≥a,gi(y)/y≥bi,(y≠0),f(0)=0, f(x)sgnx>0,(x≠0), and  sup{f′(x)}=c, obtain
(2.5)V1(xt,yt,zt)≥μ∫0xf(ξ)dξ+f(x)y+12μay2+∑i=1n∫0ygi(η)ηηdη+μyz+12z2+∑i=1nγi∫-τi0∫t+stz2(θ)dθds≥12b[by+f(x)]2+12by2[4∫0xf(ξ){∫0y(μb-f′(ξ))ηdη}dξ]+12(μy+z)2+12μ(a-μ)y2+∑i=1nγi∫-τi0∫t+stz2(θ)dθds.
Using the assumptions of Theorem 2.1, have
(2.6)a-μ=a-ab+c2b>0,μb-f′(x)≥ab-c2>0
so that
(2.7)V1(xt,yt,zt)≥D1x2+D2y2+D3z2≥D4(x2+y2+z2),
where D4=min{D1,D2,D3}.
It is also clear that the functionV2 is continuous and satisfies
(2.8)|V2|≤1.
In view of (2.3), (2.7), (2.8), and the assumptions of Theorem 2.1, it can be shown that V satisfies the condition (i) of Theorem 1.1.
Using a basic calculation, the time derivative ofV1  along solutions of (1.6) results in
(2.9)V˙1(xt,yt,zt)=f′(x)y2+μz2-μy∑i=1ngi(y)-ψ(y)z2+(μy+z)∑i=1n∫t-τitgi′(y(s))z(s)ds+∑i=1n(γiτi)z2-∑i=1nγi∫t-τitz2(s)ds+(μy+z)p(⋅).
The assumption|g′(y)|≤Li  and the estimate 2|m||n|≤m2+n2  imply
(2.10)μy∑i=1n∫t-τitgi′(y(s))z(s)ds≤μ|y|∑i=1n∫t-τit|gi′(y(s))||z(s)|ds≤μ|y|∑i=1n∫t-τitLi|z(s)|ds≤μ2∑i=1n(Liτi)y2+μ2∑i=1nLi∫t-τitz2(s)ds,z∑i=1n∫t-τitgi′(y(s))z(s)ds≤|z|∑i=1n∫t-τit|gi′(y(s))||z(s)|ds≤|z|∑i=1n∫t-τitLi|z(s)|ds≤12∑i=1n(Liτi)z2+12∑i=1nLi∫t-τitz2(s)ds
so that
(2.11)V˙1(xt,yt,zt)≤f′(x)y2+μz2-μy∑i=1ngi(y)-ψ(y)z2+∑i=1n(γiτi)z2+μ2∑i=1n(Liτi)y2+12∑i=1n(Liτi)z2+∑i=1n(12Li+12μLi-γi)∫t-τitz2(s)ds+(μy+z)p(⋅).
Using the assumptions sup{f′(x)}=c>0,ψ(y)≥a and ab-c>0, and the estimation μ=(ab+c)/2b, it follows that
(2.12)V˙1(xt,yt,zt)≤-[μ∑i=1ngi(y)y-c-μ2∑i=1n(Liτi)]y2-[a-μ-12∑i=1n(Liτi)-∑i=1n(γiτi)]z2+∑i=1n(12Li+12μLi-γi)∫t-τitz2(s)ds|(μy+z)||p(⋅)|≤-[μ∑i=1ngi(y)y-c-μ2∑i=1n(Liτi)]y2-[ab-c2b-12∑i=1n(Li+2γi)τi]z2+12∑i=1n[(1+μ)Li-2γi]∫t-τitz2(s)ds+|(μy+z)||p(⋅)|.
If we choose γi=(1/2)∑i=1n(1+μ)Li and use the assumption |p(·)|≤m, then
(2.13)V˙1(xt,yt,zt)≤-[μ∑i=1ngi(y)y-c-μ2∑i=1n(Liτi)]y2-[ab-c2b-12∑i=1n(2+μ)Liτi]z2+μm|y|+m|z|.
An easy calculation fromV2(x,y,z) and (1.6) leads to
(2.14)V˙2(⋅)={1M{-ψ(y)z-∑i=1ngi(y)-f(x)+∑i=1n∫t-τitgi′(y(s))z(s)ds+p(⋅)}sgn⁡x,|x|≥1,|z|≤M0,|x|≥1,|z|≥MyzM+xM{-ψ(y)z-∑i=1ngi(y)-f(x)+∑i=1n∫t-τitgi′(y(s))z(s)ds+p(⋅)},|x|≤1,|z|≤Mysgn⁡z,|x|≤1,|z|≥M
so that
(2.15)V˙2(⋅)≤{-1Mf(x)sgn⁡x+{a+δ+m+∑i=1n|gi(y)|+∑i=1nLi∫t-τit|z(s)|ds},|x|≥1,|z|≤M0,|x|≥1,|z|≥M|y|+a+δ+m+∑i=1n|gi(y)|+∑i=1nLi∫t-τit|z(s)|ds,|x|≤1,|z|≤M|y|,|x|≤1,|z|≥M,
using the assumptions of Theorem 2.1.
First, we considerV in the domainmax{|y|-K,|z|-M}≥0, where the constants K and M are large enough, which will be determined later. We have to discuss the following two cases.
Case
1
0 (|y|≥K≥1, and x,z are arbitrary). In this case, it follows that:
(2.16)V˙2(⋅)≤(a+δ+m)+|y|+∑i=1n|gi(y)|+∑i=1nLi∫t-τit|z(s)|ds.
By the estimates (2.3), (2.13), (2.16), and  τ=max1≤i≤nτi, we get
(2.17)V˙(xt,yt,zt)≤-[μ∑i=1ngi(y)y-c-μ2∑i=1n(Liτi)]y2+∑i=1n|gi(y)|-[ab-c2b-(2+μ)2∑i=1n(Liτi)]z2+(μm+1)|y|+m|z|+(a+δ+m)+∑i=1nLi∫t-τit|z(s)|ds+∑i=1n(Liτi)|z|-∑i=1nLi∫t-τit|z(s)|ds≤-[μ∑i=1ngi(y)y-c-μ2∑i=1n(Liτi)]y2+∑i=1n|gi(y)|-[ab-c2b-(2+μ)2∑i=1n(Liτi)]z2+(μm+1)|y|+m|z|+(a+δ+m)|y|+∑i=1n(Liτi)|z|.
We now consider the term(2.18)μ∑i=1ngi(y)y-∑i=1n|gi(y)|-cy2-μ2∑i=1n(Liτi)y2
and define
(2.19)h=ab+3c2(ab+c)<1.
Then, there exists a constant K1, (K1>1),satisfying (1-1/μ|y|)≥h  for |y|≥K1, so that, when |y|≥K1,
(2.20)μ∑i=1ngi(y)y-∑i=1n|gi(y)|-cy2-μ2∑i=1n(Liτi)y2=μ∑i=1ngi(y)y(1-1μ|y|)y2-(c-∑i=1nμLiτi2)y2≥(ab+c)b2bab+3c2(ab+c)y2-(c-∑i=1nμLiτi2)y2=-(ab-c4-∑i=1nμLiτi2)y2≤-(ab-c4-M1τ)y2.
Hence
(2.21)V˙(xt,yt,zt)≤-(ab-c4-M1τ)y2-(ab-c2b-(2+μ)N12τ)z2+(μm+1+a+δ+m)|y|+(m+N1τ)|z|,
where
(2.22)M1=∑i=1nμLi2,N1=∑i=1nLi.
If
(2.23)τ<min⁡{ab-c(2+μ)bN1,ab-c4M1},
then the above estimate  implies
(2.24)V˙(xt,yt,zt)≤-δ1(y2+z2)+(μm+a+δ+m+1)|y|+(m+N1τ)|z|
for a positive constant δ1.
Let(2.25)ρ1=max⁡{μm+δ+a+m+1,m+N1τ}
so that
(2.26)V˙(xt,yt,zt)≤-δ1(y2+z2)+ρ1(|y|+|z|)=-δ12(y2+z2)-δ12[(|y|-ρ1δ1)2+(|z|-ρ1δ1)2-2ρ12δ12]≤-δ12(y2+z2)
provided that   |y|≥(2+1)ρ1δ1-1.
LetK=max{(2+1)ρ1δ1-1,K1}. If  |y|≥K, then
(2.27)V˙(xt,yt,zt)≤-δ12(y2+z2).Case20 (|z|≥M, and x,y are arbitrary). Then
(2.28)V˙2(x,y,z)≤|y|.
By following a similar method shown in the first case, select γi=(1/2)∑i=1n(1+μ)Li and take
(2.29)τ<min⁡{ab-c(2+μ)bN1,ab-c4M1},
one can easily obtain
(2.30)V˙(xt,yt,zt)≤-δ2(y2+z2)+(μm+1)|y|+(m+N1τ)|z|≤-δ2(y2+z2)+ρ2(|y|+|z|)≤-δ22(y2+z2)
for some positive constants δ2 and ρ2 provided  that |z|≥M=K.
At the end, we considerV in max{|y|-K,|z|-M}≤0.
Let|x|≥H>1, where the constant H will be determined later. Hence
(2.31)V˙2(⋅)≤-1Mf(x)sgn⁡x+max⁡|y|≤K∑i=1n|gi(y)|+{a+δ+m+∑i=1nLi∫t-τit|z(s)|ds}.
It follows from (2.3), (2.13), and (2.31) that
(2.32)V˙(xt,yt,zt)≤-(ab-c4-M1τ)y2-(ab-c2b-(2+μ)N12τ)z2+μm|y|+(m+N1τ)|z|-1Mf(x)sgn⁡x+max⁡|y|≤K∑i=1n|gi(y)|+a+δ+m+∑i=1nLi∫t-τit|z(s)|ds-∑i=1nLi∫t-τit|z(s)|ds≤-(ab-c4-M1τ)y2-(ab-c2b-(2+μ)N12τ)z2-1Mf(x)sgn⁡x+max⁡|y|≤K∑i=1n|gi(y)|+(a+δ+m+N1Mτ+μmK+mM).
Since f(x)sgnx→∞  as |x|→∞ and  |x|≥H>1, then we have
(2.33)f(x)sgn⁡x≥2M{max⁡|y|≤K∑i=1n|gi(y)|+a+δ+m+N1Mτ+μmK+mM},
so that
(2.34)-f(x)sgn⁡x2M+{max⁡|y|≤K∑i=1n|gi(y)|+a+δ+m+N1Mτ+μmK+mM}≤0.
In view of the above discussion, it follows that:
(2.35)V˙(xt,yt,zt)≤-(ab-c4-M1τ)y2-(ab-c2b-(2+μ)N12τ)z2-12Mf(x)sgn⁡x.
Subject to the evidence thus far, we can conclude that there exists a positive constant R, which is large enough, such that
(2.36)V˙(xt,yt,zt)≤-w(u)foru2≥R2,
where u=(x2+y2+z2)1/2.  Thus, the Lyapunov functional  V(xt,yt,zt) satisfies all the assumptions of Theorem 1.1. The proof for Theorem 2.1 is complete.Example 2.2.
Consider the following nonlinear differential equation of the third order with two constant deviating arguments,τ1>0,τ2>0:
(2.37)x′′′+(4+11+(x′)2)x′′+4x′(t-τ1)+sin⁡x′(t-τ1)+4x′(t-τ2)+sin⁡x′(t-τ2)+11x=sin⁡t+cos⁡t3+sin⁡t+x2+⋅⋅⋅+x′2(t-τ2)+x′′2,
which is a special case of (1.5).
This equation can be written in the system form as follows:(2.38)x′=y,y′=z,z′=-(4+11+y2)z-(8y+2sin⁡y)-11x+∫t-τ1t(4+cos⁡y(s))z(s)ds+∫t-τ2t(4+cos⁡y(s))z(s)ds+sin⁡t+cos⁡t3+sin⁡t+x2+⋅⋅⋅+y2(t-τ2)+z2.
When we compare the system described to this point with (1.6), it follows the existence of the following estimates:
(2.39)ψ(y)=4+11+y2,0≤ψ(y)-4≤1,a=4,δ=1,g1(y)=4y+sin⁡y,g1(0)=0,(2.40)g1(y)y=4+sin⁡yy≥3,b1=3,g1′(y)=4+cos⁡y,|g1′(y)|=|4+cos⁡y|≤5=L1,g2(y)=4y+sin⁡y,g2(0)=0,g2(y)y=4+sin⁡yy≥3,b2=3,g2′(y)=4+cos⁡y,|g2′(y)|=|4+cos⁡y|≤5=L2,f(x)=11x,f(0)=0,f(x)sgn⁡x=11xsgn⁡x>0,(x≠0),f(x)sgn⁡x=11xsgn⁡x⟶∞as|x|⟶∞,f′(x)=11,c=11,ab-c=13>0,h=ab+3c2(ab+c)=5770<1,p(t,x,x(t-τ1),…,y(t-τ2),z)=sin⁡t+cos⁡t3+sin⁡t+x2+⋅⋅⋅+y2(t-τ2)+z2≤1=m,p(t+2π,x,x(t-τ1),…,y(t-τ2),z)=sin⁡(t+2π)+cos⁡(t+2π)3+sin⁡(t+2π)+⋅⋅⋅+z2=sin⁡t+cos⁡t3+sin⁡t+⋅⋅⋅+z2==p(t,x,x(t-τ1),…,y(t-τ2),z),T=2π.
In view of the above estimates, it is seen that all the assumptions of Theorem 2.1 hold. This discussion verifies that (2.37) has a periodic solution of period T,T=2π.



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*Source: 406835-2012-08-26.xml*