The difference equation yn+1−yn=−αyn+∑j=1mβje−γjyn−kj is studied and some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable, are obtained.

1. Introduction

The delay differential equation
(1)dN(t)dt=-αN(t)+βe-γN(t-τ),fort≥0,
was first proposed by Wazewska-Czyzewska and Lasota [1] as a model for the survival of red blood cell in an animal. Here, N(t) denotes the number of red blood cells at time t, α is the probability of death of red blood cells, β and γ are positive constants which are related to the production of red blood cells, and τ is the time which is required to produce a red blood cell. The oscillation and global attractivity of (1) were studied by Győri and Ladas [2] and Li and Cheng [3], while the bifurcation and the direction of the stability were investigated by Song et al. [4]. Xu and Li [5] and Liu [6] considered its generalization with several delays and obtained sufficient conditions for the global stability of survival blood cells model with several delays and piecewise constant argument.

Research on the oscillation and global stability of the discrete analogue of (1), that is, for the equation
(2)xn+1-xn=-αxn+βe-γxn-k,n=0,1,2,…,
where
(3)α∈(0,1),β,γ∈(0,∞),k∈{1,2,…},
was proposed by Kocić and Ladas [7] as an open problem.

Kubiaczyk and Saker [8] investigated the oscillation of (2) about its positive equilibrium point x¯, where x¯ is the unique solution of the equation
(4)αx¯=βe-γx¯,
and showed that every solution of (2) oscillates about x¯ if
(5)βγe-γx¯>(kk+1)k+1(1-α)k+1.

Meng and Yan [9] investigated the global attractivity of the positive equilibrium point x¯ and showed that x¯ is a global attractor of all positive solutions of (2) if
(6)β2γ2α2eγ(Q1+x¯)<1,
where Q1=(β/α)e-βγ/α.

Zeng and Shi [10] established another condition for global attractivity of x¯ and showed that x¯ is a global attractor of all positive solutions of (2) if
(7)βγα≤e.

Obviously, the condition (7) improves (6).

Kubiaczyk and Saker [8] also considered (2) when k=1 and proved that x¯ is a global attractor of all positive solutions of (2) provided that
(7′)βγe-γx¯<α.

Ma and Yu [11] proved that x¯ is a global attractor of all solutions of (2) if
(8)γx¯(1-(1-α)k+1)≤1.

By (2), we have
(9)x¯eγx¯=βα.
So, if (7) holds, then we have γx¯eγx¯=βγ/α≤e, which implies that γx¯≤1. Hence, (8) is satisfied. But, the converse is not true. So, the condition (8) improves (7).

In addition, we can also easily see that the conditions (7) and (7′) are equivalent to the condition γx¯≤1.

For the system with delay, many authors deemed that arbitrary finite number of discrete delays is more appropriate than the single discrete delay; see [12–14] and the references cited therein.

Stemming from the above discussion, the difference equation in the following form will be studied in this paper:
(10)yn+1-yn=-αyn+∑j=1mβje-γjyn-kj,
where
(11)α∈(0,1),βj,γj∈(0,∞),kj∈{1,2,…},j=1,2,…,m;∑j=1mβj=β.
Besides, we denote that k=max1≤j≤m{kj}, l=min1≤j≤m{kj}, γ*=min1≤j≤m{γj}.

Obviously, the case m=1 is the form of (2). Besides, if y-k,y-k+1,…,y-1, y0∈[0,∞), then, the corresponding solution of (10) is positive, and (10) has a unique positive equilibrium point y¯, which satisfies
(12)αy¯=∑j=1mβje-γjy¯.

The aim of this paper is to investigate the oscillation and the global asymptotic stability of (10).

2. Some LemmasLemma 1 (see [<xref ref-type="bibr" rid="B1">7</xref>, page 6]).

Assume that pi∈(0,∞) and ki∈{0,1,…} with ∑i=1m(pi+ki)≠1, i=1,2,…,m. Let {pi(n)} be sequences of positive numbers such that
(13)liminfn→∞pi(n)⩾pifori=1,2,…,m.
Suppose that the linear difference inequality
(14)zn+1-zn+∑i=1mpi(n)zn-ki⩽0forn=0,1,…
has an eventually positive solution. Then, the difference equation
(15)xn+1-xn+∑i=1mpixn-ki=0
has a positive solution.

Lemma 2 (see [<xref ref-type="bibr" rid="B1">7</xref>, page 5]).

Consider the linear homogeneous difference equation
(16)xn+k+∑i=1kqixn+k-i=0forn=0,1,…,
where k is a nonnegative integer and qi∈R, i=1,2,…,k. Then, the following statements are equivalent:

every solution of (16) oscillates;

the characteristic equation of (16)
(17)λk+∑i=1kqiλk-i=0

has no positive roots.
Lemma 3 (see [<xref ref-type="bibr" rid="B1">7</xref>, page 12]).

Assume that p1,p2,…,pk∈R and k is a nonnegative integer. Then, ∑i=1k|pi|<1 is a sufficient condition for the asymptotic stability of the difference equation
(18)xn+k+p1xn+k-1+⋯+pkxn=0forn=0,1,….

Lemma 4.

Assume that (11) holds, and {yn} is a solution of (10) with positive initial conditions y-k,…,y0. Then,
(19)limsupn→∞yn≤βα.

Proof.

Clearly, we have yn>0, for n=-k,-k+1,…,0,1,2,…. So by (10), we can find that
(20)yn+1=(1-α)yn+∑j=1mβje-γjyn-kj≤(1-α)yn+β.
Define a sequence {ωn} by
(21)ωn+1=(1-α)ωn+β,ω0=y0.
Obviously,
(22)yn≤ωn=(1-α)nω0+βα[1-(1-α)n].
So, we have
(23)limsupn→∞yn≤βα.

Lemma 5.

Assume that (11) holds, and
(24)∑j=1mβjγje-rjy¯<α.

Then, the positive equilibrium y¯ of (10) is locally asymptotically stable.

Proof.

To prove that the positive equilibrium y¯ is locally asymptotically stable, it suffices to prove that the zero solution of the linear equation of (10) is locally asymptotically stable. The linearized equation associated with (10) about positive equilibrium y¯ is
(25)yn+1=(1-α)yn-∑j=1mβjγje-γjy¯yn-kj,
which satisfies
(26)|1-α|+∑j=1m|βjγje-γjy¯|≤1.
Then, by Lemma 3, the positive equilibrium solution y¯ of (10) is locally asymptotically stable.

Lemma 6 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

The following system of inequalities,
(27)μ≤e-λ-1,λ≥e-μ-1,
with λ,μ being real numbers, have exactly one solution λ=μ=0.

3. Main ResultsTheorem 7.

Assume that (11) holds, and
(28)∑j=1mβjγje-γjy¯(1-α)-l-1>1.
Then, every positive solution of (10) oscillates about the positive equilibrium y¯.

Proof.

Assume for the sake of contradiction that (10) has a positive solution {yn} which does not oscillate about y¯. We assume that yn>y¯ eventually. If yn<y¯ eventually, the proof is similar and will be omitted. So, there exists an n0⩾0 such that yn>y¯ for n⩾n0, and consequently yn-k>y¯ for n⩾n1, where n1=n0+k.

From Lemma 4, we have {yn} as a bounded sequence. In the following, we will claim that
(29)limn→∞yn=y¯.
Otherwise, let
(30)μ=limsupn→∞yn.
Then, μ>y¯ and there exists a subsequence {yni} such that
(31)limi→∞yni+1=μ,yni+1-yni>0forni≥n1,i=1,2,….
Equation (10) can be reformulated in the form
(32)yn+1=(1-α)yn+∑j=1mβje-γjyn-kj.
Then, from (31) and (32), we find that
(33)αyni+1≤∑j=1mβje-γjyni-kj.
So, we obtain
(34)αμ⩽limsupi→∞∑j=1mβje-γjyni-kj≤∑j=1mβje-γjy¯=αy¯,
which is a contradiction. Accordingly, (29) holds.

Set
(35)yn=y¯+xnforn=-k,-k+1,….
By the assumption yn>y¯, we have that xn is an eventually positive solution of the difference equation
(36)xn+1-(1-α)xn+αy¯-∑j=1mβje-γj(y¯+xn-kj)=0,n=0,1,…,
which can also be rewritten in the form
(37)xn+1-(1-α)xn+∑j=1mp(n-kj)xn-kj=0,n=0,1,…,
where p(n-kj)=(βje-γjy¯-βje-γj(y¯+xn-kj))/(xn-kj), j=0,1,…,m.

By some simple calculations and (29), we get
(38)limn→∞p(n-kj)=βjγje-γjy¯>0.

One can easily see that the hypothesis of Lemma 1 is satisfied and so the linear equation
(39)xn+1-(1-α)xn+∑j=1mβjγje-γjy¯xn-kj=0
has an eventually positive solution.

Let {xn} be an eventually positive solution of (39); then zn=(1-α)-nxn is an eventually positive solution of
(40)zn+1-zn+∑j=1mβjγje-γjy¯(1-α)-kj-1zn-kj=0,n=0,1,….
Let
(41)F(λ)=λn+1-λn+∑j=1mβjγje-γjy¯(1-α)-kj-1λn-kj
be the characteristic polynomial of (40). Now, we prove that F(λ)>0, for λ>0.

If λ⩾1, then obviously F(λ)>0. Else if 0<λ<1, we have
(42)F(λ)=λn(λ-1+∑j=1mβjγje-γjy¯(1-α)-kj-1λ-kj)⩾λn(λ-1+∑j=1mβjγje-γjy¯(1-α)-l-1λ-kj)⩾λn(λ-1+∑j=1mβjγje-γjy¯(1-α)-l-1)≥λn+1>0.

Therefore, the characteristic equation of (40)
(43)λn+1-λn+∑j=1mβjγje-γjy¯(1-α)-kj-1λn-kj=0
has no positive roots.

According to Lemma 2, (40) has no nonoscillatory solution.

This is a contradiction. The proof is completed.

Theorem 8.

Assume that (11) holds, and
(44)y¯[1-(1-α)k+1]≤1.
Then, the positive equilibrium y¯ of (10) is a global attractor of all positive solutions of (10).

Proof.

To prove that the positive equilibrium y¯ is a global attractor of all positive solutions of (10), it suffices to show that (29) holds.

We will prove that (29) holds in each of the following two cases.

Case 1 ({yn} is nonoscillatory). Let {yn} be eventually positive. The case that {yn} is eventually positive is similar and will be omitted. So,there exists an n0⩾0 such that yn>y¯ for n⩾n0, and consequently yn-k>y¯ for n⩾n1, where n1=n0+k.

From Lemma 4, we have {yn} as a bounded sequence. Assume for the sake of contradiction that (29) is not satisfied. Let
(45)μ=limsupn→∞yn.
Then, μ>y¯ and there exists a subsequence {yni} such that
(46)limi→∞yni+1=μ,yni+1-yni>0forni≥n1,i=1,2,….
It follows from (10) that
(47)αyni+1≤∑j=1mβje-γjyni-kj.
So, we obtain
(48)αμ⩽limsupi→∞∑j=1mβje-γjyni-kj≤∑j=1mβje-γjy¯=αy¯,
which is a contradiction. Accordingly, (29) holds.

Case 2 ({yn} is strictly oscillatory). To show that (29) holds, it suffices to prove that limn→∞xn=0 holds, when {xn} is a strictly oscillatory solution of (36).

To this end, let
(49){xpi+1,xpi+2,…,xqi}
be the ith positive semicycle of {xn} followed by the jth negative semicycle
(50){xqi+1,xqi+2,…,xs}.
Let xMi, xmi be the extreme values in these two semicycles with the smallest possible indices Mi and mi. Then, we claim that
(51)Mi-pi⩽k+1,mi-qi⩽k+1.
In the following, we will prove that (51) holds for positive semicycles, while for negative semicycles, the proof is similar and will be omitted. Assume for the sake of contradiction that the first inequality in (51) is not true. Then, Mi-pi>k+1 and the terms xMi-k-1,xMi-k,…,xMi-1 are in a positive semicycle. Because of xMi>xMi-1, (36) renders
(52)αxMi+αy¯≤∑j=1mβje-γj(y¯+xMi-kj-1).
So, we have
(53)xMi⩽1α∑j=1mβje-γj(y¯+xMi-kj-1)-y¯=1α[∑j=1mβje-γj(y¯+xMi-kj-1)-∑j=1mβje-γjy¯]=1α[∑j=1mβje-γjy¯(e-γjxMi-kj-1-1)].
So there exists at least a j s.t. xMi-kj-1<0, which contradicts that xMi-kj-1 is in the positive semicycle. So, (51) is true. Noting that {yn} is bounded from Lemma 4, we can let
(54)λ=liminfn→∞xn=liminfi→∞xmi,μ=limsupn→∞xn=limsupi→∞xMi.

To prove that limn→∞xn=0 holds, it is sufficient to show that λ=μ=0.

From (54), it follows that if ϵ∈(0,λ) is given, then there exists n2⩾0 such that
(55)λ-ϵ⩽xn⩽μ+ϵforn⩾n2+k.
Equation (36) can be reformulated in the form
(56)xn+1-(1-α)xn=-αy¯+∑j=1mβje-γj(y¯+xn-kj).
Multiplying (56) by (1-α)-n-1 and then summing up from n=pi to n=Mi-1 for i being sufficiently large, we get
(57)(1-α)-MixMi-(1-α)-pixpi=∑n=piMi-1(-αy¯)(1-α)-n-1+∑n=piMi-1∑j=1mβje-γj(y¯+xn-kj)(1-α)-n-1.
From (55) and xpi<0, we have
(58)(1-α)-MixMi⩽(-αy¯)∑n=piMi-1(1-α)-n-1+∑n=piMi-1∑j=1mβje-γj(y¯+xn-kj)(1-α)-n-1≤(-αy¯)∑n=piMi-1(1-α)-n-1+∑n=piMi-1∑j=1mβje-γjy¯e-γ*(λ-ɛ)(1-α)-n-1=(-αy¯)∑n=piMi-1(1-α)-n-1+(αy¯)e-γ*(λ-ɛ)∑n=piMi-1(1-α)-n-1=(-αy¯)(1-α)-Mi-(1-α)-Piα+αy¯e-γ*(λ-ɛ)(1-α)-Mi-(1-α)-Piα=y¯[(1-α)-Mi-(1-α)-Pi][e-γ*(λ-ɛ)-1].
So,
(59)xMi⩽y¯[e-γ*(λ-ɛ)-1][1-(1-α)Mi-Pi].
By using (54), ɛ is arbitrary and Mi-pi⩽k+1; we get
(60)μ⩽y¯[e-γ*λ-1][1-(1-α)k+1].
From the assumption of the theorem, we have
(61)μ⩽e-γ*λ-1.
By the same trick as in proving (61), we can prove that
(62)λ≥e-γ*μ-1.
Therefore, by Lemma 6, we can get λ=μ=0; that is, limn→∞xn=0, which implies that y¯ is a global attractor of all positive solutions of (10).

By Lemma 3 and Theorem 8, we can get the following result.

Theorem 9.

Suppose that (11) holds and that
(63)∑j=1mβjγje-rjy¯<α,y¯[1-(1-α)k+1]<1.
Then, the positive equilibrium y¯ is globally asymptotically stable.

Remark 10.

From Theorem 7, it is clear that if the condition (28) holds, then the oscillation condition for m=1 as established by Kubiaczyk and Saker [8] is already satisfied.

Remark 11.

When m=1, the condition of Theorem 8 is independent from the argument γ.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (G61074068, G61034007, G61174036, G61374065, and G61374002), the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province (ZR2010FM013), and the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01).

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