# Lower Bounds of Periods of Periodic Solutions for a Class of  Differential Equations with Variable Delays

Authors: Xin-Ge Liu; Mei-Lan Tang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Based on generalized Wirtinger's inequality, periods of periodic solutions of the nonautonomous differential equations with variable delays are investigated. Based on Hölder inequality, lower bounds of periods of periodic solutions for a class of functional differential equations with variable delays are obtained by a simple method.


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## Body

## 1. Introduction

The existence and multiplicity of periodic solutions, bifurcations of periodic solutions, and stability of solutions of functional differential equations have attracted the attention of many mathematicians [1–5]. A lot of remarkable results have been achieved [6–10]. However, only a few works on periods of periodic solutions have been done (see, e.g., [11–13]). Supposefis Lipschitz continuous in a Banach space with constantLandx(t)is aT-periodic nonconstant solution ofx′(t)=f(x(t)). Lasota and Yorke [12] have showed thatTL≥4. Busenberg et al. [14] refined the earlier estimate ofTLin [12]; they [14] showed thatTL≥6. At the same time, they [14] also gave a simple proof of the better lower boundTL≥2πin spaces with the norm defined via an inner product. Mawhin and Walter [15] showed how some lower bounds on the period of the possible periodic solutions of autonomous ordinary differential equations due to Yorke [11] are easy consequences of the general principle. Zevin and Pinsky [16] investigated a class of Lipschitzian differential equations of even order; they obtained the minimal periods of periodic solutions. In 2012, Domoshnitsky et al. [17] investigated componentwise positivity of solutions to periodic boundary problem for linear functional differential system. Recently, Cheng and Zhang [18] proved a generalized Wirtingers inequality. Based on this inequality, they [18] studied estimates for lower bounds of periods of periodic solutions for the following autonomous delay differential equation:
(1)x˙(t)=-∑k=1n‍f(x(t-kr)),wherex∈Rp,f∈C(Rp,Rp), andr>0is a given constant. In their paper [18], delays are required to be constants with the form ofkr. In this paper, we will replace the constant delaykrwith the generalized delay functionrk(t)with|rk′(t)|<1,k=1,2,…,n. Furthermore, the method used in our paper is simpler than that in [18]. Lower bounds of periods of periodic solutions for a class of functional differential equations with variable delays are obtained.Consider the lower bounds of periods of periodic solutions for the following delay differential equations:(2)x˙(t)=-∑k=1n‍f(x(t-rk(t))),wherex∈Rp,f∈C(Rp,Rp),rk(t)=rk(t+T),rk(t)>0, and|rk′(t)|<1fort∈R.In order to estimate the lower bounds of periods of periodic solution of (2), we need the following definitions and lemmas.Definition 1.
For a positive constantL,f(x)∈C(Rp,Rp)is calledL-Lipschitz continuous if, for allx,y∈Rp,
(3)|f(x)-f(y)|≤L|x-y|,
where|·|denotes the Euclidean norm inRp.LetHT1(R,Rp)be the Hilbert space consisting of theT-periodic functionsxon R which together with weak derivatives belong toL2(0,T;Rp). For allx,y∈L2(0,T;Rp), let〈x,y〉=∫0T‍(x,y)dtand ∥x∥=〈x,x〉denote the inner product and the norm inL2(0,T;Rp), respectively, where(·,·)is the inner product inRp. LetS={x∣x(t)=(x1(t),x2(t),…,xp(t))}, wherexi(t)has the second derivative,i=1,2,…,p.Lemma 2 (see [8]).
Supposeτ∈Cω1andτ′(t)<1,forallt∈[0,ω]. Then the functiont-τ(t)has an inverseμ(t)satisfyingμ∈C(R,R) withμ(a+ω)=μ(a)+ω.Lemma 3 (see [18]).
Ifx∈HT1and∫0T‍x(t)dt=0, then
(4)∫0T‍|x(t)|2dt≤T24π2∫0T‍|x˙(t)|2dt.


## 2. Main Results

Sincerk′(t)<1, by Lemma 2, the inverse oft-rk(t)exists. Letμk(s)be the inverse oft-rk(t).Theorem 4.
Letx be a nonconstant T-periodic solution of the nonautonomous delay differential equation (2) andx∈S. Suppose that the functionf:Rp→RpisL-Lipschitz continuous andrk(t)=rk(t+T),rk(t)>0,|rk′(t)|<1. ThenT>2π/nL.Proof.
Sincexis a nonconstant T-periodic solution of the nonautonomous delay differential equation (2), for allt,u∈R, we have
(5)|x˙(t+u)-x˙(t)|=|∑k=1n‍f(x(t+u-rk(t+u)))-∑k=1n‍f(x(t-rk(t)))|≤∑k=1n‍|f(x(t+u-rk(t+u)))-f(x(t-rk(t)))|≤L∑k=1n‍|x(t+u-rk(t+u))-x(t-rk(t))|.
We claim that ifu≠0, then, fort∈R, there exists at least oneksuch that
(6)u-rk(t+u)+rk(t)≠0.
Otherwise, ifu-rk(t+u)+rk(t)=0fork=1,2,…,n, then
(7)x(t+u-rk(t+u))-x(t-rk(t))=0.
From (5), one has
(8)|x˙(t+u)-x˙(t)|=0.
Noting thatx∈S, we obtain
(9)|x¨(t)|=0.
Noting thatx(t)=x(t+T), then∫0T‍x˙(t)dt=0. From Lemma 3, we have
(10)2π∥x˙∥≤T∥x¨∥.
Then|x˙|=0.xis a constantT-periodic solution. This contradicts the assumption thatxis a nonconstantT-periodic solution.
For simplicity of proof, we suppose thatu-rk(t+u)+rk(t)≠0fork=1,2,…,n, (5) can be rewritten as
(11)|x˙(t+u)-x˙(t)u|≤L∑k=1n‍|x(t+u-rk(t+u))-x(t-rk(t))u-rk(t+u)+rk(t)|×|u-rk(t+u)+rk(t)u|.
Letu→0; one has
(12)|x¨(t)|≤L∑k=1n‍|x˙(t-rk(t))|[1-rk′(t)]=L∑k=1n‍|x˙(t-rk(t))|[1-rk′(t)]1/2[1-rk′(t)]1/2.
Applying Hölder inequality gives
(13)|x¨(t)|≤L∑k=1n‍|x˙(t-rk(t))|[1-rk′(t)]1/2[1-rk′(t)]1/2≤L{∑k=1n‍|x˙(t-rk(t))|2[1-rk′(t)]}1/2×{∑k=1n‍[1-rk′(t)]}1/2<(2n)1/2L{∑k=1n‍|x˙(t-rk(t))|2[1-rk′(t)]}1/2.
Raising both sides of inequality (13) to power2and integrating both sides from0toT, we have
(14)∫0T‍|x¨(t)|2dt<2nL2∫0T‍∑k=1n‍|x˙(t-rk(t))|2[1-rk′(t)]dt.
Sinceμk(s)is the inverse of t-rk(t), by using Lemma 2, we have
(15)∫0T‍|x¨(t)|2dt<2nL2∑k=1n‍∫0T‍|x˙(t-rk(t))|2[1-rk′(t)]dt=2nL2∑k=1n‍∫-rk(0)T-rk(T)‍|x˙(s)|2×[1-rk′(μk(s))]11-rk′(μk(s))ds=2nL2∑k=1n‍∫-rk(0)T-rk(T)‍|x˙(s)|2ds=2nL2∑k=1n‍∫0T‍|x˙(s)|2ds=2n2L2∫0T‍|x˙(s)|2ds.
That is,
(16)∥x¨∥<2nL∥x˙∥.
Sincex(t)=x(t+T), obviously,∫0T‍x˙(t)dt=0. By Lemma 3, we have2π∥x˙∥≤T∥x¨∥. So
(17)T>2πnL.Remark 5.
When delayrk(t)=kr,k=1,2,…,n, from the second inequality of (13), we can easily obtain Theorem  1 in [18].We can easily obtain the following result.Corollary 6.
Letxbe a nonconstantT-periodic solution of the nonautonomous delay differential equation (2) andx∈S. Suppose that the functionf:Rp→RpisL-Lipschitz continuous andrk(t)=rk(t+T),rk(t)>0,-1/n≤rk′(t)<1. ThenT≥2π/n(n+1)L.



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