# Oscillations of Numerical Solutions for Nonlinear Delay Differential Equations in the Control of Erythropoiesis

Authors: Qi Wang; Jiechang Wen
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We consider the oscillations of numerical solutions for the nonlinear delay differential equations in the control of erythropoiesis. The exponentialθ-method is constructed and some conditions under which the numerical solutions oscillate are presented. Moreover, it is proven that every nonoscillatory numerical solution tends to the equilibrium point of the continuous system. Numerical examples are given to illustrate the main results.


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## Body

## 1. Introduction

The oscillatory and asymptotic behavior of solutions of delay differential equations has been the subject of intensive investigations during the past decades. A large number of articles has appeared in the literature, we refer to [1–4] and the references therein. The strong interest in this study is motivated by the fact that it has many useful applications in some mathematical models, such as ecology, biology, spread of some infectious diseases in humans, and so on. For more information on this investigation, the reader can see [5, 6] and the references therein.By contrast with the research on the oscillations of the analytic solutions, much studies have been focused on the oscillations of the numerical solutions for delay differential equations. In [7, 8], oscillations of numerical solutions in θ-methods and Runge-Kutta methods for a linear differential equation with piecewise constant arguments (EPCA, a special type of Delay Differential Equations) x′(t)+ax(t)+a1x([t-1])=0 were considered, respectively. More recently, Wang et al. [9] studied numerical oscillations of alternately advanced and retarded linear EPCA, the conditions of oscillations for the θ-methods are obtained. To the best of our knowledge, until now less attention had been paid for the oscillations of the numerical solutions for nonlinear delay differential equations except for [10]. Differently from [10], in our paper, we will investigate another nonlinear delay differential equation in the control of erythropoiesis and obtain some new results.Consider the following nonlinear delay differential equation:(1)x′(t)=β0ωμωμ+xμ(t-τ)-γx(t),
with conditions
(2)μ,ω,β0,γ>0,τ≥0.
Mackey and Glass [11] have proposed (1) as model of hematopoiesis (blood cell production). In (1), x(t) denotes the density of mature cells in blood circulation, τ is the time delay between the production of immature cells in the bone marrow and their maturation for release in the circulating blood stream, and the production is a monotonic decreasing function of x(t-τ). Equation (1) has been recently studied by many authors. Gopalsamy et al. [12] obtained sufficient and also necessary and sufficient conditions for all positive solutions to oscillate about their positive steady states. They also obtained sufficient conditions for the positive equilibrium to be a global attractor. Using the linearization method, Zaghrout et al. [13] considered (1) and gave a sufficient condition for oscillations of all solutions about the positive steady state x* and proved that every nonoscillatory positive solution of (1) tends to x* as t→∞. For more details of (1), we refer to Mackey and Milton [14] and Mackey [15]. Up to now, few results on the properties of numerical solutions for (1) were obtained. In the present paper, our main goal is to investigate some sufficient conditions under which the numerical solutions are oscillatory. We also consider the asymptotic behavior of nonoscillatory numerical solutions.The contents of this paper are as follows. In Section2, some necessary definitions and results for oscillations of the analytic solutions are given. In Section 3, we obtain a recurrence relation by applying the θ-methods to the simplified form which comes from making two transformations on (1). Moreover, the oscillations of the numerical solutions are discussed and conditions under which the numerical solutions oscillate are obtained. In Section 4, we study the asymptotic behavior of nonoscillatory solutions. In Section 5, we present numerical examples that illustrate the theoretical results for the numerical methods. Finally, Section 6 gives conclusions and issues for future research.


## 2. Preliminaries

In this section, we start by introducing some definitions, lemmas and theorems that will be employed throughout the work.Definition 1.
A functionx(t) of (1) is said to oscillate about K* if x(t)-K* has arbitrarily large zeros. Otherwise, x(t) is called nonoscillatory. When K*=0, we say that x(t) oscillates about zero or simply oscillates.Definition 2.
A sequence{xn} is said to oscillate about {yn} if {xn-yn} is neither eventually positive nor eventually negative. Otherwise, {xn} is called nonoscillatory. If {yn}={y} is a constant sequence, we simply say that {xn} oscillates about {y}. When {yn}={0}, we say that {xn} oscillates about zero or simply oscillates.Definition 3.
We say (1) oscillates if all of its solutions are oscillatory.Theorem 4 (see [16]).
Consider the difference equation(3)an+1-an+∑j=-klqjan+j=0,
assume that k,l∈N and qj∈R for j=-k,…,l. Then the following statements are equivalent:(i)
every solution of (3) oscillates;(ii)
the characteristic equationλ-1+∑j=-klqjλj=0 has no positive roots.Theorem 5 (see [16]).
Consider the difference equation(4)an+1-an+pan-k+qan=0,
where k>0,  p>0 and q>0. Then the necessary and sufficient conditions for the oscillation of all solutions of (4) are q∈(0,1) and
(5)p(k+1)k+1kk>(1-q)k+1.Lemma 6.
The inequalityln(1+x)>x/(1+x) holds for x>-1 and x≠0.Lemma 7.
The inequalityex<1/(1-x) holds for x<-1 and x≠0.Lemma 8 (see [17]).
For allm≥M,(i)
(1+a/(m-θa))m≥ea if and only if 1/2≤θ≤1 for a>0,  φ(-1)≤θ≤1 for a<0;(ii)
(1+a/(m-θa))m<ea if and only if 0≤θ<1/2 for a<0,  0≤θ≤φ(1) for a>0,where φ(x)=1/x-1/(ex-1) and M is a positive constant.


## 3. Oscillations of Numerical Solutions

### 3.1. Transformation

For (1), we take an initial condition of the form
(6)x(t)=ψ(t),-τ≤t≤0,
where ψ∈C([-τ,0],(0,∞)),ψ(0)>0.In order to simplify (1), we introduce a similar method in [13]. The change of variables
(7)x(t)=ωy(t)
transforms (1) to the delay differential equation
(8)y′(t)=a1+yμ(t-τ)-γy(t),
where a=β0/ω. One can see that (8) has a unique equilibrium K and that
(9)a1+Kμ=γK.
The following result concerning oscillations of the analytic solution of (8) is given in [13].Theorem 9.
Assume that(10)aμKμ-1τ(1+Kμ)2eaτ/K(1+Kμ)>1e,
then every positive solution of (8) oscillates about its positive equilibrium K.Therefore, we obtain the following corollary naturally.Corollary 10.
Assume that the condition(11)aμKμ-1τ(1+Kμ)2eaτ/K(1+Kμ)>1e
holds, then every positive solution of (1) oscillates about its positive equilibrium K*=ωK.Next, we also introduce an invariant oscillation transformationy(t)=Kez(t), then (8) can be written as
(12)z′(t)+aK(1+Kμ)[f1(z(t))f2(z(t-τ))-f1(z(t))z′(t)+aK(1+Kμ)-f2(z(t-τ))+2]=0,
where
(13)f1(u)=1-e-u,f2(u)=1+1+Kμ1+Kμeμu.
Then y(t) oscillates about K if and only if z(t) oscillates about zero. Moreover, for the sake of brevity, let
(14)T=aK(1+Kμ),
then (12) becomes
(15)z′(t)=-Tf1(z(t))f2(z(t-τ))+Tf1(z(t))+Tf2(z(t-τ))-2T.
For convenience, denote
(16)L=aμKμ-1(1+Kμ)2,
then the inequality (11) yields
(17)LτeTτ>1e.


### 3.2. The Difference Scheme

In this subsection we consider the adaptation of theθ-methods. Let h=τ/m be a given step size with integer m>1. The adaptation of the linear θ-method and the one-leg θ-method to (15) leads to the same numerical process of the following type:
(18)zn+1=zn-Th(θf1(zn+1)f2(zn+1-m)zn-Th+(1-θ)f1(zn)f2(zn-m))+Th(θf1(zn+1)+(1-θ)f1(zn))+Th(θf2(zn+1-m)+(1-θ)f2(zn-m))-2Th,
where 0≤θ≤1,  zn+1 and zn+1-m are approximations to z(t) and z(t-τ) of (15) at tn+1, respectively.Letzn=ln(xn/Kω), and take into account of the expressions of f1 and f2 we have
(19)xn+1=xnexp(+1-θxn(ωμ+xn-mμ))-hThTKωμ+1(1+Kμ)xn+1xnexp×(θxn+1(ωμ+xn+1-mμ)+1-θxn(ωμ+xn-mμ))-hT1-θxn(ωμ+xn-mμ)).Definition 11.
We call the iteration formula (19) as the exponential θ-method for (1), where xn+1 and xn+1-m are approximations to x(t) and x(t-τ) of (1) at tn+1, respectively.The following theorem gives the convergence of exponentialθ-method. We can easily prove it by the method of steps which is used in [18].Theorem 12.
The exponentialθ-method (19) is convergent with order
(20)1,ifθ≠12,2,ifθ=12.


### 3.3. Oscillation Analysis

It is not difficult to know thatxn oscillates about K* if and only if zn is oscillatory. In order to study oscillations of (19), we only need to consider the oscillations of (18). The following conditions which are taken from [13] will be used in the next analysis:
(21)uf1(u)>0,foru≠0,limu→0f1(u)u=1,f2(u)>0,anyu,limu→0f2(u)=2,f1(u)≤uforu>0,f1(0)=0,f2(u)≤2u≥0,f2(0)=2.
For (18), its linearized form is given by
(22)zn+1=zn-hθTzn+1-h(1-θ)Tzn-hθLzn+1-m-h(1-θ)zn-m,
which is equivalent to
(23)zn+1=1-h(1-θ)T1+hθTzn-hθL1+hθTzn+1-m-h(1-θ)L1+hθTzn-m.
It follows from [16] that (18) oscillates if (23) oscillates under the condition (21).Definition 13.
Equation (19) is said to be oscillatory if all of its solutions are oscillatory.Definition 14.
We say that the exponentialθ-method preserves the oscillations of (1) if (1) oscillates, then there is a h0>0 or h0=∞, such that (19) oscillates for h<h0. Similarly, we say that the exponential θ-method preserves the nonoscillations of (1) if (1) nonoscillates, then there is a h0>0 or h0=∞, such that (19) nonoscillates for h<h0.In the following, we will study whether the exponentialθ-method preserves the oscillations of (1). That is, when Corollary 10 holds, we will investigate the conditions under which (19) is oscillatory.Lemma 15.
The characteristic equation of (22) is given by
(24)ξ=R(-h(T+Lξ-m)).Proof.
Letzn=ξnz0 in (22), we have
(25)ξn+1z0=ξnz0-hθTξn+1z0-h(1-θ)Tξnz0-hθLξn+1-mz0-h(1-θ)Lξn-mz0,
that is
(26)ξ=1-hθ(T+Lξ-m)ξ-h(1-θ)(T+Lξ-m),
which is equivalent to
(27)ξ=1-h(1-θ)(T+Lξ-m)1+hθ(T+Lξ-m)=1-h(T+Lξ-m)1+hθ(T+Lξ-m).
In view of [19], we know that the stability function of the θ-method is
(28)R(x)=1+x1-θx,
thus the characteristic equation of (22) is given by (24). This completes the proof of the lemma.Lemma 16.
IfLτeTτ>1/e, then the characteristic equation (24) has no positive roots for 0≤θ≤1/2.Proof.
LetV(ξ)=ξ-R(-h(T+Lξ-m)). By Lemma 8, we have
(29)R(-h(T+Lξ-m))≤exp(-h(T+Lξ-m)),ξ>0,0≤θ≤1/2.
Now we will prove that W(ξ)=ξ-exp(-h(T+Lξ-m))>0 for ξ>0. Suppose the opposite, that is, there exists a ξ0>0 such that W(ξ0)≤0, then we have ξ0≤exp(-hT-hLξ0-m), and
(30)ξ0m≤exp(-Tτ-Lτξ0-m).
Multiplying both sides of the inequality (30) by LτeTτeξ0-m, we obtain
(31)LτeTτeξ0-mξ0m≤LτeTτeξ0-mexp(-Tτ-Lτξ0-m),
which gives
(32)LτeTτe≤Lτξ0-mexp(1-Lτξ0-m),
therefore we have the following two cases.
Case1. If 1-Lτξ0-m=0, then LτeTτe≤1, which contradicts the condition LτeTτ>1/e.
Case2. If 1-Lτξ0-m≠0, then in view of Lemma 7, we get
(33)exp(1-Lτξ0-m)<11-(1-Lτξ0-m)=1Lτξ0-m,
that is,
(34)Lτξ0-mexp(1-Lτξ0-m)<1,
so LτeTτe<1, which is also a contradiction to LτeTτ>1/e.
Combining both the cases, by (29) we obtain that for ξ>0(35)V(ξ)=ξ-R(-h(T+Lξ-m))≥ξ-exp(-h(T+Lξ-m))=W(ξ)>0,
which implies that the characteristic equation (24) has no positive roots. The proof of the lemma is complete.Without loss of generality, in the case of1/2<θ≤1, we assume that m>1.Lemma 17.
IfLτeTτ>1/e and 1/2<θ≤1, then the characteristic equation (24) has no positive roots for h<h0, where
(36)h0={∞,forLτ≥1,τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),forLτ<1.Proof.
SinceR(-h(T+Lξ-m)) is an increasing function of θ when ξ>0, then for ξ>0 and 1/2<θ≤1(37)R(-h(T+Lξ-m))=1-h(1-θ)(T+Lξ-m)1+hθ(T+Lξ-m)≤11+h(T+Lξ-m).
Next, we will prove that the inequality
(38)ξ-11+h(T+Lξ-m)>0,ξ>0
holds under certain conditions.
From (38), it follows that
(39)ξ-11+h(T+Lξ-m)=(1+hT)ξ1-m1+h(T+Lξ-m)λ(ξ),
where
(40)λ(ξ)=ξm-11+hTξm-1+hL1+hT,
so we only need to prove λ(ξ)>0 for ξ>0. It is not difficult to know that λ(ξ) is the characteristic polynomial of the following difference scheme
(41)zn+1-zn+hL1+hTzn+1-m+hT1+hTzn=0.
In view of Theorems 4 and 5, we have that λ(ξ) has no positive roots if and only if
(42)hL1+hTmm(m-1)m-1>(1-hT1+hT)m,
which is equivalent to
(43)lnLτ+(m-1)ln(1+1+Tτm-1)>0.
We examine two cases depending on the position of Lτ: Either Lτ≥1 or Lτ<1.
Case1. If Lτ≥1, by m>1, then (43) holds true.
Case2. If Lτ<1 and
(44)h<τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),
then according to Lemma 6 we have
(45)lnLτ+(m-1)ln(1+1+Tτm-1)>lnLτ+(m-1)(1+Tτ)/(m-1)1+(1+Tτ)/(m-1)=lnLτ+(m-1)(1+Tτ)m+Tτ>0.
Therefore the inequality (38) holds for h<h0, where
(46)h0={∞,forLτ≥1,τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),forLτ<1.
So we get that the following inequality:
(47)V(ξ)=ξ-R(-h(T+Lξ-m))≥ξ-11+h(T+Lξ-m)>0,
holds for h<h0 and ξ>0, which implies that the characteristic equation (24) has no positive roots. This completes the proof.Remark 18.
By inequality (43) and condition Lτ<1, we have that
(48)τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ)>0,
thus h0 is meaningful.In view of (21), Lemmas 16 and 17, and Theorem 4, we have the first main theorem of this paper.Theorem 19.
IfLτeTτ>1/e, then (19) is oscillatory for
(49)h<{∞,when0≤θ≤1/2,h0,when1/2<θ≤1,
where h0 is defined in Lemma 17.


## 3.1. Transformation

For (1), we take an initial condition of the form
(6)x(t)=ψ(t),-τ≤t≤0,
where ψ∈C([-τ,0],(0,∞)),ψ(0)>0.In order to simplify (1), we introduce a similar method in [13]. The change of variables
(7)x(t)=ωy(t)
transforms (1) to the delay differential equation
(8)y′(t)=a1+yμ(t-τ)-γy(t),
where a=β0/ω. One can see that (8) has a unique equilibrium K and that
(9)a1+Kμ=γK.
The following result concerning oscillations of the analytic solution of (8) is given in [13].Theorem 9.
Assume that(10)aμKμ-1τ(1+Kμ)2eaτ/K(1+Kμ)>1e,
then every positive solution of (8) oscillates about its positive equilibrium K.Therefore, we obtain the following corollary naturally.Corollary 10.
Assume that the condition(11)aμKμ-1τ(1+Kμ)2eaτ/K(1+Kμ)>1e
holds, then every positive solution of (1) oscillates about its positive equilibrium K*=ωK.Next, we also introduce an invariant oscillation transformationy(t)=Kez(t), then (8) can be written as
(12)z′(t)+aK(1+Kμ)[f1(z(t))f2(z(t-τ))-f1(z(t))z′(t)+aK(1+Kμ)-f2(z(t-τ))+2]=0,
where
(13)f1(u)=1-e-u,f2(u)=1+1+Kμ1+Kμeμu.
Then y(t) oscillates about K if and only if z(t) oscillates about zero. Moreover, for the sake of brevity, let
(14)T=aK(1+Kμ),
then (12) becomes
(15)z′(t)=-Tf1(z(t))f2(z(t-τ))+Tf1(z(t))+Tf2(z(t-τ))-2T.
For convenience, denote
(16)L=aμKμ-1(1+Kμ)2,
then the inequality (11) yields
(17)LτeTτ>1e.


## 3.2. The Difference Scheme

In this subsection we consider the adaptation of theθ-methods. Let h=τ/m be a given step size with integer m>1. The adaptation of the linear θ-method and the one-leg θ-method to (15) leads to the same numerical process of the following type:
(18)zn+1=zn-Th(θf1(zn+1)f2(zn+1-m)zn-Th+(1-θ)f1(zn)f2(zn-m))+Th(θf1(zn+1)+(1-θ)f1(zn))+Th(θf2(zn+1-m)+(1-θ)f2(zn-m))-2Th,
where 0≤θ≤1,  zn+1 and zn+1-m are approximations to z(t) and z(t-τ) of (15) at tn+1, respectively.Letzn=ln(xn/Kω), and take into account of the expressions of f1 and f2 we have
(19)xn+1=xnexp(+1-θxn(ωμ+xn-mμ))-hThTKωμ+1(1+Kμ)xn+1xnexp×(θxn+1(ωμ+xn+1-mμ)+1-θxn(ωμ+xn-mμ))-hT1-θxn(ωμ+xn-mμ)).Definition 11.
We call the iteration formula (19) as the exponential θ-method for (1), where xn+1 and xn+1-m are approximations to x(t) and x(t-τ) of (1) at tn+1, respectively.The following theorem gives the convergence of exponentialθ-method. We can easily prove it by the method of steps which is used in [18].Theorem 12.
The exponentialθ-method (19) is convergent with order
(20)1,ifθ≠12,2,ifθ=12.


## 3.3. Oscillation Analysis

It is not difficult to know thatxn oscillates about K* if and only if zn is oscillatory. In order to study oscillations of (19), we only need to consider the oscillations of (18). The following conditions which are taken from [13] will be used in the next analysis:
(21)uf1(u)>0,foru≠0,limu→0f1(u)u=1,f2(u)>0,anyu,limu→0f2(u)=2,f1(u)≤uforu>0,f1(0)=0,f2(u)≤2u≥0,f2(0)=2.
For (18), its linearized form is given by
(22)zn+1=zn-hθTzn+1-h(1-θ)Tzn-hθLzn+1-m-h(1-θ)zn-m,
which is equivalent to
(23)zn+1=1-h(1-θ)T1+hθTzn-hθL1+hθTzn+1-m-h(1-θ)L1+hθTzn-m.
It follows from [16] that (18) oscillates if (23) oscillates under the condition (21).Definition 13.
Equation (19) is said to be oscillatory if all of its solutions are oscillatory.Definition 14.
We say that the exponentialθ-method preserves the oscillations of (1) if (1) oscillates, then there is a h0>0 or h0=∞, such that (19) oscillates for h<h0. Similarly, we say that the exponential θ-method preserves the nonoscillations of (1) if (1) nonoscillates, then there is a h0>0 or h0=∞, such that (19) nonoscillates for h<h0.In the following, we will study whether the exponentialθ-method preserves the oscillations of (1). That is, when Corollary 10 holds, we will investigate the conditions under which (19) is oscillatory.Lemma 15.
The characteristic equation of (22) is given by
(24)ξ=R(-h(T+Lξ-m)).Proof.
Letzn=ξnz0 in (22), we have
(25)ξn+1z0=ξnz0-hθTξn+1z0-h(1-θ)Tξnz0-hθLξn+1-mz0-h(1-θ)Lξn-mz0,
that is
(26)ξ=1-hθ(T+Lξ-m)ξ-h(1-θ)(T+Lξ-m),
which is equivalent to
(27)ξ=1-h(1-θ)(T+Lξ-m)1+hθ(T+Lξ-m)=1-h(T+Lξ-m)1+hθ(T+Lξ-m).
In view of [19], we know that the stability function of the θ-method is
(28)R(x)=1+x1-θx,
thus the characteristic equation of (22) is given by (24). This completes the proof of the lemma.Lemma 16.
IfLτeTτ>1/e, then the characteristic equation (24) has no positive roots for 0≤θ≤1/2.Proof.
LetV(ξ)=ξ-R(-h(T+Lξ-m)). By Lemma 8, we have
(29)R(-h(T+Lξ-m))≤exp(-h(T+Lξ-m)),ξ>0,0≤θ≤1/2.
Now we will prove that W(ξ)=ξ-exp(-h(T+Lξ-m))>0 for ξ>0. Suppose the opposite, that is, there exists a ξ0>0 such that W(ξ0)≤0, then we have ξ0≤exp(-hT-hLξ0-m), and
(30)ξ0m≤exp(-Tτ-Lτξ0-m).
Multiplying both sides of the inequality (30) by LτeTτeξ0-m, we obtain
(31)LτeTτeξ0-mξ0m≤LτeTτeξ0-mexp(-Tτ-Lτξ0-m),
which gives
(32)LτeTτe≤Lτξ0-mexp(1-Lτξ0-m),
therefore we have the following two cases.
Case1. If 1-Lτξ0-m=0, then LτeTτe≤1, which contradicts the condition LτeTτ>1/e.
Case2. If 1-Lτξ0-m≠0, then in view of Lemma 7, we get
(33)exp(1-Lτξ0-m)<11-(1-Lτξ0-m)=1Lτξ0-m,
that is,
(34)Lτξ0-mexp(1-Lτξ0-m)<1,
so LτeTτe<1, which is also a contradiction to LτeTτ>1/e.
Combining both the cases, by (29) we obtain that for ξ>0(35)V(ξ)=ξ-R(-h(T+Lξ-m))≥ξ-exp(-h(T+Lξ-m))=W(ξ)>0,
which implies that the characteristic equation (24) has no positive roots. The proof of the lemma is complete.Without loss of generality, in the case of1/2<θ≤1, we assume that m>1.Lemma 17.
IfLτeTτ>1/e and 1/2<θ≤1, then the characteristic equation (24) has no positive roots for h<h0, where
(36)h0={∞,forLτ≥1,τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),forLτ<1.Proof.
SinceR(-h(T+Lξ-m)) is an increasing function of θ when ξ>0, then for ξ>0 and 1/2<θ≤1(37)R(-h(T+Lξ-m))=1-h(1-θ)(T+Lξ-m)1+hθ(T+Lξ-m)≤11+h(T+Lξ-m).
Next, we will prove that the inequality
(38)ξ-11+h(T+Lξ-m)>0,ξ>0
holds under certain conditions.
From (38), it follows that
(39)ξ-11+h(T+Lξ-m)=(1+hT)ξ1-m1+h(T+Lξ-m)λ(ξ),
where
(40)λ(ξ)=ξm-11+hTξm-1+hL1+hT,
so we only need to prove λ(ξ)>0 for ξ>0. It is not difficult to know that λ(ξ) is the characteristic polynomial of the following difference scheme
(41)zn+1-zn+hL1+hTzn+1-m+hT1+hTzn=0.
In view of Theorems 4 and 5, we have that λ(ξ) has no positive roots if and only if
(42)hL1+hTmm(m-1)m-1>(1-hT1+hT)m,
which is equivalent to
(43)lnLτ+(m-1)ln(1+1+Tτm-1)>0.
We examine two cases depending on the position of Lτ: Either Lτ≥1 or Lτ<1.
Case1. If Lτ≥1, by m>1, then (43) holds true.
Case2. If Lτ<1 and
(44)h<τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),
then according to Lemma 6 we have
(45)lnLτ+(m-1)ln(1+1+Tτm-1)>lnLτ+(m-1)(1+Tτ)/(m-1)1+(1+Tτ)/(m-1)=lnLτ+(m-1)(1+Tτ)m+Tτ>0.
Therefore the inequality (38) holds for h<h0, where
(46)h0={∞,forLτ≥1,τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ),forLτ<1.
So we get that the following inequality:
(47)V(ξ)=ξ-R(-h(T+Lξ-m))≥ξ-11+h(T+Lξ-m)>0,
holds for h<h0 and ξ>0, which implies that the characteristic equation (24) has no positive roots. This completes the proof.Remark 18.
By inequality (43) and condition Lτ<1, we have that
(48)τ(1+Tτ+lnLτ)1+Tτ(1-lnLτ)>0,
thus h0 is meaningful.In view of (21), Lemmas 16 and 17, and Theorem 4, we have the first main theorem of this paper.Theorem 19.
IfLτeTτ>1/e, then (19) is oscillatory for
(49)h<{∞,when0≤θ≤1/2,h0,when1/2<θ≤1,
where h0 is defined in Lemma 17.


## 4. Asymptotic Behavior of Nonoscillatory Solutions

In this section, we will investigate the asymptotic behavior of nonoscillatory solutions of (19). The following lemma is an important result about asymptotic behavior of (8).Lemma 20 (see [13]).
Lety(t) be a positive solution of (8), which does not oscillate about K. Then limt→∞y(t)=K.From the relationship between (8) and (12), we know that the nonoscillatory solution of (12) satisfies limt→∞z(t)=0 if Lemma 20 holds. Furthermore, limt→∞x(t)=K* is also obtained. Next, we will prove that the numerical solution of (1) can inherit this property.Lemma 21.
Letzn be a nonoscillatory solution of (18), then limn→∞zn=0.Proof.
Without loss of generality, we may assume thatzn>0 for sufficiently large n. Then by condition (21) we know that the following inequalities holds true:
(50)f1(zi)>0,f2(zi)-1>0,f2(zi)-2<0
for sufficiently large i. Moreover, it is can be seen from (18) that
(51)zn+1-zn=-hθTf1(zn+1)[f2(zn+1-m)-1]-h(1-θ)Tf1(zn)[f2(zn-m)-1]+hθT[f2(zn+1-m)-2]+h(1-θ)T[f2(zn-m)-2],
which gives
(52)zn+1-zn-hθT[f2(zn+1-m)-2]-h(1-θ)T[f2(zn-m)-2]<0.
Thus
(53)zn+1-zn<hθT[f2(zn+1-m)-2]+h(1-θ)T[f2(zn-m)-2]<0,
then the sequence {zn} is decreasing, and therefore
(54)limn→∞zn=η∈[0,∞).
Now we will prove that η=0. If η>0, then there exists N∈N and ε>0 such that for n-m>N, 0<η-ε<zn<η+ε. Hence η-ε<zn-m and η-ε<zn-1+m. So inequality (52) yields
(55)zn+1-zn-hθT[1+Kμ1+Kμeμ(η-ε)-1]-h(1-θ)T[1+Kμ1+Kμeμ(η-ε)-1]<0,
which implies that zn+1-zn<B<0, where
(56)B=hTKμ(1-eμ(η-ε))1+Kμeμ(η-ε).
Thus zn→-∞ as n→∞, which is a contradiction to (54). This completes the proof.Therefore, the second main theorem of this paper is as follows.Theorem 22.
Letxn be a positive solution of (19), which does not oscillate about K*, then limn→∞xn=K*.


## 5. Numerical Experiments

In this section, we will give some numerical examples to illustrate our results.Firstly, we consider the equation(57)x′(t)=21+x7(t-2)-x(t),
with initial value x(t)=0.5 for t≤0. In (57), it is easy to see that condition (11) holds true and Lτ≈7>1. That is, the analytic solutions of (57) are oscillatory. In Figures 1–3, we draw the figures of the analytic solutions and the numerical solutions of (57), respectively. The parameters m=40,  θ=0.2 in Figure 2 and m=20, θ=0.8 in Figure 3. From the two figures, we can see that the numerical solutions of (57) oscillate about K*=1, which are in agreement with Theorem 19.Figure 1
The analytic solution of (57).Figure 2
The numerical solution of (57) with m=40 and θ=0.2.Figure 3
The numerical solution of (57) with m=20 and θ=0.8.Secondly, we consider(58)x′(t)=11+x6(t-1)-2x(t),
with initial value x(t)=0.6 for t≤0. In (58), it is not difficult to see that condition (11) is fulfilled. That is, the analytic solutions of (58) are oscillatory. In Figures 4–7, we draw the figures of the analytic solutions and the numerical solutions of (58), respectively. The parameters m=25, θ=0.3 in Figure 5, m=50,  θ=0.6 in Figure 6 and m=5,  θ=0.6 in Figure 7. We can see from the three figures that the numerical solutions of (58) oscillate about K*≈0.4929, which are consistent with Theorem 19. On the other hand, by direct calculation, we get h0≈0.1882. We notice that h=0.02<h0 and h=0.2>h0 in Figures 6 and 7, respectively, so the stepsize h0 is not optimal.Figure 4
The analytic solution of (58).Figure 5
The numerical solution of (58) with m=25 and θ=0.3.Figure 6
The numerical solution of (58) with m=50 and θ=0.6.Figure 7
The numerical solution of (58) with m=5 and θ=0.6.Thirdly, we consider another equation,(59)x′(t)=0.2×2.182.18+x8(t-0.15)-0.1x(t),
with initial value x(t)=39 for t≤0. For (59), it is easy to see that LτeTτ+1≈0.0505<1, so the condition (11) is not satisfied. That is, the analytic solutions of (59) are nonoscillatory. In Figures 8–10, we draw the figures of the analytic solutions and the numerical solutions of (59), respectively. In Figure 8, we can see that x(t)→K*≈1.6949 as t→∞. From Figures 9 and 10, we can also see that the numerical solutions of (59) satisfy xn→K*≈1.6949 as n→∞. That is, the numerical method preserves the asymptotic behavior of nonoscillatory solutions of (59), which coincides with Theorem 22.Figure 8
The analytic solution of (59).Figure 9
The numerical solution of (59) with m=10 and θ=0.4.Figure 10
The numerical solution of (59) with m=15 and θ=0.75.Finally, according to Definition14, we can see from these figures that the exponential θ-method preserves the oscillations of (57) and (58) and the nonoscillations of (59), respectively.All the above numerical examples confirm our theoretical findings.


## 6. Conclusions

In this paper, we discuss the oscillations of the numerical solutions of a nonlinear delay differential equation in the control of erythropoiesis. The convergent exponentialθ-method, namely the linear θ-method and the one-leg θ-method in exponential form, is constructed. We establish some conditions under which the numerical solutions oscillate in the case of oscillations of the analytic solutions. We also prove that nonoscillatory numerical solutions can inherit the corresponding properties of analytic solutions. It is pointed out that the stepsize h0 in Lemma 17 is not optimal. Therefore, our future work will be devoted to investigating this problem.



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