# Unique Positive Almost Periodic Solution for Discrete Nonlinear Delay Survival Red Blood  Cells Model

Authors: Xitao Yang; Siping Tang
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

We obtain sufficient conditions which guarantee the global attractivity
of solutions for nonlinear delay survival red blood cells model. Then, some criteria are established
for the existence, uniqueness and global attractivity of positive almost periodic solutions of the
almost periodic system.


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## Body

## 1. Introduction

Recently, Saker [1] studied the existence and global attractivity of positive periodic solution for the following discrete nonlinear delay survival red blood cells model:
(1.1)x(n+1)-x(n)=-δ(n)x(n)+P(n)e-q(n)x(n-τ(n)),
where δ(n), P(n), q(n), τ(n) are nonnegative bounded sequence, τ(n)∈N for all n∈Z and 0<δ(n)<1. The dynamic behavios of  (1.1) was investigated by many authors (see [1–18]) because of its biological and ecological significance. When δ(n),  P(n),q(n) and τ(n) are positive constants, the global attractivity of the positive equilibrium for equation
(1.2)x(n+1)-x(n)=-δx(n)+Pe-qx(n-τ)
has been investigated by some authors (see [3, 15–18] and reference therein). Recently, Ma and Yu [17] obtained the following Theorem A.Theorem 1 A.
Suppose thatδ∈(0,1), P,q∈[0,∞) and τ are nonnegative integer. If
(1.3)qx¯(1-(1-δ)τ+1)≤1,
then the unique positive equilibrium x¯ of (1.2) is a global attractor of all positive solutions of (1.2).Equation (1.2) is the discrete analogue of equation
(1.4)x'(t)=-δx(t)+pexp(-qx(t-τ)),
which was first used by Ważewska-Czyżewska and Lasota as a model for the survival of red blood cells in an animal in [19], see also [20]. Here, x(t) denotes the number of red blood cells at time t, δ is the probability of death of a red blood cell, p and q are positive constants related to the production of red blood cells per unit time and τ is the time required to produce a red blood cell. Researching the behavior of the  solution of  (1.4) and its analogue was posed as open problems by Kocic and Ladas [10] as well as Györi and Ladas [6].Because of seasonal variation   (1.1) needs not to be exactly periodic but almost periodic instead. It is natural to ask if the results in  [1] hold for the almost periodic case. It is a difficult problem in which significant difference appears in comparison with the periodic case, for example, contrary to periodic functions, there exists an almost periodic function x(t) such that x(t)>0 for all t∈ℝ  and  inft∈ℝx(t)=0.One purpose of the present paper is to extend Theorem A to   (1.1). The other purpose is to extend some results in [1] to the almost periodic case. In [1], the author use a fixed theorem to obtain the existence of positive periodic solution. The operator used in [1] depends on the period of system (1.1), therefore we cannot apply these topological tools to the almost periodic case. An important notion in almost periodic differential theory is the hull (cf. [21–23]). In this paper, we use essentially this notion to establish our results. We begin with some notations.Letτ=maxn∈ℤτ(n), I={-τ,-τ+1,…,-1,0}, C={φ:I→ℝ} and C+={φ∈C:φ≥0,φ(0)>0}. For each φ∈C, we define the norm of φ as ∥φ∥=maxs∈I|φ(s)|. Denote xn the element of C with xn=x(n+s) for all s∈I. For any bounded sequence e(n), denote e*=lim infn→∞e(n) and e*=lim supn→∞e(n), ∏j=lk‍e(j)=e(l)e(l+1)⋯e(k) if k≥l and ∏j=lk‍e(j)=1 if k<l. It is easy to see that, for any φ∈C+, there is a unique solution x(n,0,φ) of (1.1) with x0=φ and x(n,0,φ)>0 for all n∈ℕ. Now, we give some definitions.Definition 1.1 (see [22, 23]).
A sequencef(n) is said to be almost periodic, if for any ϵ>0, there is a constant l(ϵ)>0 such that in any interval of length l(ϵ) there exists τ¯∈ℤ such that the inequality
(1.5)|f(n+τ¯)-f(n)|<ϵ
is satisfied for all n∈ℤ.Denote set(1.6)H(f)={g:thereis{nk}⊂ℤsuchthatlimn→∞f(n+nk)=g(n)∀n∈ℤ}
the hull of f. It is easy to see that if f(n) is almost periodic, then for all f1∈H(f),
(1.7)lim infn→∞f(n)=infn∈ℤf(n)=infn∈ℤf1(n),lim supn→∞f(n)=supn∈ℤf(n)=supn∈ℤf1(n).Definition 1.2.
Letf:ℤ×C→ℝ. Then f is said to be almost periodic in n∈ℤ uniformly on compact set of C, if f(n,·) is continuous for each n∈ℤ, and for any ϵ>0 and every compact set K⊂C, there is a constant l(ϵ,K)>0 such that in any interval of length l(ϵ,K) there exists τ¯ such that the inequality
(1.8)|f(n+τ¯,φ)-f(n,φ)|<ϵ
is satisfied for all n∈ℤ and φ∈K.Definition 1.3.
The positive solutionN(n) to (1.1), in the sense that N(n)>0 for all n≥0, is said to be global attractive, if for any φ∈C+,
(1.9)limn→∞(x(n,0,φ)-N(n))=0.The remainder of this paper is organized as follows. The main results are given in Section2 while the proofs are left to Section 4. In Section 3, we will give some lemmas needed in the proofs of main results.


## 2. Main Results

Theorem 2.1.
Assume that0<δ*<1 and P*>0. Then, for any φ∈C+, the solution x(n,0,φ) of (1.1) satisfies
(2.1)u¯≤lim infn→∞x(n,0,φ)≤lim supn→∞x(n,0,φ)≤v¯,
where (u¯,v¯) is the limit of {(un,vn)} with u0=0 and
(2.2)vk=P*δ*e-q*uk-1,k=1,2,…,uk=P*δ*e-q*vk,k=1,2,….The following corollary follows from  [24, Theorems  1 and  2].Corollary 2.2.
Assume that the conditions of Theorem2.1 are satisfied, and δ(n), P(n), q(n) and τ(n) are ω-periodic with ω>0. Then (1.1) admits a positive ω-periodic solution.Remark 2.3.
Whenτ(n)≡ω, under the conditions of Corollary 2.2, Saker in [1, Theorem  2.1] proved that the conclusion of Corollary 2.2. Thus Corollary 2.2 extends  Theorem  2.1  in  [1].Theorem 2.4.
Assume that the conditions of Theorem2.1 are satisfied and
(2.3)aq*≤1,
where
(2.4)a=lim supn→∞∑k=nn+τ‍P(k)∏i=k+1n+τ‍(1-δ(i))e-q(k)x¯(k-τ(k)),
and x¯(n) is a positive solution of (1.1). Then every positive solution x(n) of (1.1) satisfies
(2.5)limn→∞[x(n)-x¯(n)]=0.Corollary 2.5.
Assume that the conditions of Theorem2.1 are satisfied, and
(2.6)q*lim supn→∞∑k=nn+τ‍P(k)∏i=k+1n+τ‍(1-δ(i))e-q(k)u¯≤1,
where u¯ is the constant given in Theorem 2.1. Then every positive of (1.1) is globally attractive.Remark 2.6.
Whenδ(n)≡δ, P(n)≡P and q(n)≡q, here δ, P and q are positive constants, we consider the global attractivity of the positive equilibrium x¯ for  (1.2). In this case, δx¯=Pe-qx¯ and
(2.7)a=limsupn→∞∑k=nn+τ‍P(k)∏i=k+1n+τ‍(1-δ(i))e-q(k)x¯(k-τ(k))=Pe-qx¯1-(1-δ)τ+1δ=x¯(1-(1-δ)τ+1).
We see that (2.3) changes to (1.3)  and Theorem 2.4  reproduces Theorem A.Theorem 2.7.
Assume that system (1.1) is almost periodic, that is, δ(t), P(n), q(n) and τ(n) are almost periodic, and the conditions of Corollary 2.5 are satisfied. Then there is a unique globally attractive positive almost periodic solution p(n) for  (1.1).Remark 2.8.
LetP(n)=1/10+(1/20)sinn, q(n)=τ(n)=2  and δ(n)=δ be a positive constant such that
(2.8)310e-(1/10δ)e-3/10δ1-(1-δ)3δ≤1,3(1-δ)δ>1.
Since P*=3/20, P*=1/20, q*=q*=2, v1=P*/δ=3/20δ, u¯>u1=(P*/δ)e-q*v1=(1/20δ)e-3/10δ,, τ=supn∈Zτ(n)=2 and
(2.9)q*limsupn→∞∑k=nn+τ‍P(k)∏i=k+1n+τ‍(1-δ(i))e-q(k)u¯≤310e-(1/10δ)e-3/10δ1-(1-δ)3δ≤1,
we can see that there is unique globally attractive positive almost periodic solution p(n) for   (1.1). Since
(2.10)P*q*(1-δ)δ=3(1-δ)δ>1,
we cannot obtain the existence of positive almost periodic solution of  (1.1) by [25, Theorem  4.1]. Moreover, we should point out that  (4.1) in [25] is not correct. To see this, we consider  (1.2) with P(1-δ)/δ<lnP/q<1. Let f(x)=-δx+Pe-qx. It is easy to see that the positive equilibrium x¯ of (1.2) is the unique zero point of f. Since
(2.11)f(lnPq)=1-δlnPq>1-δ>0,
we see that x¯>lnP/q>P(1-δ)/δ. Thus we cannot conclude that for each positive solution x(n) of (1.2),
(2.12)lim supn→∞x(n)≤α:=P(1-δ)δ.
Therefore the conclusions of  [25, Theorem  4.1] may not hold.


## 3. Some Lemmas

Lemma 3.1 (see [26]).
The following system of inequalities,(3.1)y≤e-x-1,x≥e-y-1,
where x≤y are real numbers, have exactly one solution x=y=0.Lemma 3.2.
Assume thatf:ℤ×C→ℝ, is almost periodic sequence in n uniformly on compact set of C. If there is a solution x(n) (n≥0) to the equation
(3.2)x(n+1)=f(n,xn),
such that A≤lim infn→∞x(t)≤lim supn→∞x(t)≤B. then for each g∈H(f), equation
(3.3)x(n+1)=g(n,xn)
has a solution p(n) which is defined on ℤ such that
(3.4)A≤p(n)≤B,n∈ℤ.Proof.
LetD=supn∈ℕ|x(n)| and S={φ∈C:∥φ∥≤D}. Then S is a compact subset of C. For each g∈H(f), there is {nk}⊂ℕ such that nk→∞ and f(n+nk,φ)→g(n,φ) uniformly on ℤ×S. By the diagonal process, we can choose {nkj}⊂{nk} such that x(n+nkj)→p(n) as j→∞ and p(n) satisfies (3.3) on ℤ. We want to prove that (3.4) holds.
For eachϵ>0, there is a T>0 such that
(3.5)A-ϵ≤x(n)≤B+ϵ,n≥T.
For any fixed n∈ℤ, there is J>0 such that n+nkj>T for all j>J. It follows from (3.5) that
(3.6)A-ϵ≤x(n+nkj)≤B+ϵ,j≥J.
Setting j→∞ and ϵ→0, we see that (3.4) holds. This completes the proof.Lemma 3.3.
Assume thatf:ℤ×C→ℝ, is almost periodic sequence in n uniformly on compact set of C. Let S be a compact set of ℝ. If for all g∈H(f), equation
(3.7)x(n+1)=g(n,xn)
admits a unique solution pg(n), defined on ℤ, whose range is in S, then all these solutions pg(n) are almost periodic.The proof is similar to that of   [21, Theorem 2.10.1]. We omit it here.


## 4. The Proofs of Main Results

Proof of Theorem2.1.
Letx(n)=x(n,0,φ) for each φ∈C+. From (1.1), we have
(4.1)x(1)=(1-δ(0))x(0)+P(0)e-q(0)x(-τ(0))≥(1-δ(0))x(0)>0,
which prove that x(n)>0 (n=1,2,…) by induction. For any ϵ∈(0,δ*), there is N>0 such that for n>N,
(4.2)δ*-ϵ<δ(n)<δ*+ϵ,P*-ϵ<P(n)<P*+ϵ,q*-ϵ<q(n)<q*+ϵ.
By the fact that x(n)>0 we deduce that
(4.3)x(n+1)≤(1-δ*+ϵ)x(n)+P*+ϵ,n>N.
This implies
(4.4)x(n)≤(1-δ*+ϵ)n-N-1x(N+1)+P*+ϵδ*-ϵ[(1-(1-δ*+ϵ)n-N-1],n>N.
By the fact that ϵ is arbitrary, we obtain that
(4.5)lim supn→∞x(n)≤P*δ*=v1.
For each ϵ∈(0,δ*), there is N1>N such that
(4.6)x(n)<v1+ϵ,n≥N1.
By (1.1), (4.2), we obtain that
(4.7)x(n+1)≥(1-δ*-ϵ)x(n)+(P*-ϵ)e-(q*+ϵ)(v1+ϵ),n≥N1+τ.
It follows that for n≥N1+τ(4.8)x(n)≥(1-δ*-ϵ)n-N1-τx(N1+τ)+P*-ϵδ*+ϵe-(q*+ϵ)(v1+ϵ)[(1-(1-δ*-ϵ)n-N1-τ].
This implies that
(4.9)lim infn→∞x(n)≥P*δ*e-q*v1=u1.
By induction we can see that
(4.10)uk≤lim infn→∞x(n)≤lim supn→∞x(n)≤vk,k=1,2,….
It is easy to see that {uk} increase and {vk} decrease. Thus the limit of {(un,vn)} exists. Therefore we have
(4.11)u¯≤lim infn→∞x(n)≤lim supn→∞x(n)≤v¯.
This completes the proof of Theorem 2.1.Proof of Theorem2.4.
Setz(n)=x(n)-x¯(n). Then z(n) satisfies
(4.12)z(n+1)-z(n)=-δ(n)z(n)+P(n)e-q(n)x¯(n-τ(n))(e-q(n)z(n-τ(n))-1).
The proof will be accomplished by showing that
(4.13)limn→∞z(n)=0.
We will prove that (4.13) holds in each of the following two cases.
Case 1.
{z(n)} is nonoscillatory. Suppose that {z(n)} is eventually nonnegative. The case that {z(n)} is eventually nonpositive is similar and will be omitted. By (4.12) we see that {z(n)} is eventually decreasing. Thus the limit of {z(n)} exists. Let b=limn→∞z(n). Then b≥0. We claim that b=0. Otherwise, there would exist N>0 such that
(4.14)b2<z(n)<3b2,n≥N.
It follows from (4.12) that
(4.15)z(n+1)-z(n)<-b2δ(n),n≥N+τ.
This implies that
(4.16)z(n)<-b2∑k=N+τn‍δ(k)+z(N+τ),n≥N+τ.
By the fact that δ*>0 we see that limn→∞z(n)=-∞, which contradicts the fact that {z(n)} is eventually nonnegative. Thus b=0. Therefore (4.13) holds in this case.Case 2.
{z(n)} is oscillatory. Let λ=lim infn→∞z(n) and μ=lim supn→∞z(n). Then
(4.17)-∞<λ≤0≤μ<∞.
There exist positive sequences {pi} and {qi} such that limi→∞pi=∞,
(4.18)pi<qi<pi+1,z(pi)<0 and z(qi)>0 for i∈ℕ, z(k)≥0 for pi<k≤qi and z(k)≤0 for qi<k≤pi+1 and
(4.19)lim supi→∞Ai=μ,lim infi→∞ai=λ,
where Ai=max{z(j)∣pi≤j≤qi}  and ai=min{z(j)∣qi≤j≤pi+1}. Let Mi=min{j∣pi≤j≤qi,z(j)=Ai} and mi=min{j∣qi≤j≤pi+1,z(j)=ai}. Since z(Mi)>z(Mi-1), by (4.12) we have
(4.20)δ(Mi-1)z(Mi-1)<P(Mi-1)e-q(Mi-1)x¯(Mi-1-τ(Mi-1))(e-q(n)z(Mi-1-τ(Mi-1))-1).
We claim that
(4.21)Mi-1-pi≤τ.
We first assume that z(Mi-1)<0. Therefore we have Mi-1≤pi. Thus (4.21) holds. Assume now that z(Mi-1)≥0. By (4.20) we obtain that z(Mi-1-τ(Mi-1))<0. This implies that Mi-1-pi≤τ(Mi-1)≤τ. Thus (4.21) also holds.
Similarly, we can obtain that(4.22)mi-1-qi≤τ.
For each ϵ>0, there is N>0 such that for n>N,
(4.23)q*-ϵ<q(n)<q*+ϵ,λ-ϵ<z(n)<μ+ϵ,∑k=nn+τ‍P(k)∏i=k+1n+τ‍(1-δ(i))e-q(k)x¯(k-τ(k))<a+ϵ,
Let z(n)=y(n)∏i=0n-1‍(1-δ(i)). By (4.12) we obtain that
(4.24)y(n+1)=y(n)+(∏i=0n‍(1-δ(i)))-1P(n)e-q(n)x¯(n-τ(n))(e-q(n)z(n-τ(n))-1).
This implies that
(4.25)y(Mi)=y(pi)+∑n=piMi-1‍(∏j=0n‍(1-δ(j)))-1P(n)e-q(n)x¯(n-τ(n))(e-q(n)z(n-τ(n))-1).
It follows from (4.21) and (4.23)–(4.24) that for i large enough
(4.26)z(Mi)=∏j=0Mi-1‍(1-δ(j))×{y(pi)+∑n=piMi-1‍(∏j=0n‍(1-δ(j)))-1P(n)e-q(n)x¯(n-τ(n))(e-q(n)z(n-τ(n))-1)∑n=piMi-1‍(∏j=0n‍(1-δ(j)))-1}≤(e-(q*+ϵ)(λ-ϵ)-1)∑n=piMi-1‍P(n)∏j=n+1Mi-1‍(1-δ(j))e-q(n)x¯(n-τ(n))≤(a+ϵ)(e-(q*+ϵ)(λ-ϵ)-1).
Using (4.19), then by ϵ being  arbitrary, this implies that
(4.27)μ≤a(e-q*λ-1).
Now we will prove that
(4.28)λ≥a(e-q*μ-1).
In fact, by (4.22)–(4.24), for i large enough we have
(4.29)z(mi)=∏j=0mi-1‍(1-δ(j)){y(qi)+∑n=qimi-1‍(∏j=0n‍(1-δ(j)))-1P(n)e-q(n)x¯(n-τ(n))(e-q(n)z(n-τ(n))-1)}≥(e-(q*+ϵ)(μ+ϵ)-1)∑n=qimi-1‍P(n)∏j=nmi-1‍(1-δ(j))e-q(n)x¯(n-τ(n))≥(a+ϵ)(e-(q*+ϵ)(μ+ϵ)-1).
Note that ϵ is arbitrary. By (4.19) we can see that (4.28) holds. By Lemma 3.1 we see that λ=μ=0. Thus, (4.13) also holds in this case. This completes the proof.Proof of Theorem2.7.
We will prove our conclusion by applying Lemmas3.1 and 3.2. For any (δ1,P1,q1,τ1)∈H((δ,P,q,τ)), consider equation
(4.30)x(n+1)-x(n)=-δ1(n)x(n)+P1(n)e-q1(n)x(n-τ1(n)).
It follows from Theorem 2.1 that for any φ∈C+,
(4.31)u¯≤lim infn→∞x(n,0,φ)≤lim supn→∞x(n,0,φ)≤v¯,
where u¯, v¯ are the constants given in Theorem 2.1, and x(n,0,φ) is the solution of (1.1) with x0=φ. By the almost periodicity of   (1.1) and Lemma 3.2, we can find a solution N(n) of (4.30) defined on ℤ whose range is in [u¯,v¯]. Now we will prove that the solution which possesses the above properties is unique. Suppose that N¯(n) is another solution of (4.30) defined on ℤ whose range is in [u¯,v¯]. Let z(n)=N(n)-N¯(n). Then z(n) satisfies
(4.32)z(n+1)-z(n)=-δ1(n)z(n)+P1(n)e-q1(n)N¯(n-τ(n))(e-q1(n)z(n-τ1(n))-1).
We claim that z(n)≡0. Otherwise, we would have
(4.33)z(ζ)≠0
for some ζ∈ℤ. Consider the following two cases.
Case 1.
There is aT<ζ such that z(n)≥0 (or ≤0) for all n≤T. Without loss of generality, we suppose that z(n)≥0 for all n≤T. Then z(n+1)≤z(n) for n≤T. Since z(n) is bounded on ℤ, we see that limn→-∞z(n) exists, denoted by l. We claim that l=0. Otherwise, there would exist N<T such that
(4.34)l2<z(n)<3l2,n<N.
Note that δ1*=δ*=infn∈ℤδ(n). It follows from (4.32) that
(4.35)z(n+1)-z(n)<-l2δ*,n<N.
This implies that
(4.36)z(n)<z(n-k)-kl2δ*∀k∈ℕ,n<N.
Hence limn→-∞z(n)=+∞, which contradicts the fact that z(n) is bounded on ℤ. Thus l=0. Therefore z(n)=0 for all n≤T. It follows from (4.32) that z(n)=0 for all n∈ℤ, which contradicts (4.33).Case 2.
There exist positive sequences{pi} and {qi} such that limi→∞pi=-∞,
(4.37)pi+1<qi<pi,z(pi)<0 and z(qi)>0 for i∈ℕ, z(k)≥0 for pi+1<k≤qi and z(k)≤0 for qi<k≤pi . Let d=supn≤p1z(n) and c=infn≤p1z(n). Then d>0 and c<0. For each ϵ∈(0,min(-c,d)), there are n0,n0≤p1 such
(4.38)z(n0)>d-ϵ,z(n0)<c+ϵ.
Suppose n0∈[pj,qj-1] and n0∈[qi,pi]. Let C=max{z(k)∣pj≤k≤qj-1}  and c=min{z(k)∣qi≤k≤pi}. Set M=min{k∣pj≤k≤qj-1,z(k)=C} and m=min{k∣qi≤k≤pi,z(k)=c}. Similar to the argument in the proof of Theorem 2.4, we can obtain that
(4.39)M-1-pj≤τ,m-1-qi≤τ.
Let a¯=limsupn→∞∑k=n+1n+τ‍P(k)∏i=kn+τ‍(1-δ(i))e-q(k)u¯. By the almost periodicity of δ(n), P(n) and q(n), we have for all n∈ℤ,
(4.40)q*≤q1(n)≤q*,∑k=nn+τ‍P1(k)∏i=k+1n+τ‍(1-δ1(i))e-q1(k)u¯≤a¯.
Let z(n)=y(n)∏i=0n-1‍(1-δ1(i)). By (4.32) we obtain that
(4.41)y(n+1)=y(n)+(∏i=0n‍(1-δ1(i)))-1P1(n)e-q1(n)N¯(n-τ1(n))(e-q1(n)z(n-τ1(n))-1).
This implies that
(4.42)y(M)=y(pj)+∑n=pjMi-1‍(∏k=0n‍(1-δ1(k)))-1P1(n)e-q1(n)N¯(n-τ1(n))(e-q1(n)z(n-τ1(n))-1).
Since c≤z(n)≤d for all n≤p1, it follows from (4.39)–(4.40) that
(4.43)d-ϵ<z(n0)<z(M)=∏k=0M-1‍(1-δ1(k))×{y(pj)+∑n=pjM-1‍(∏k=0n‍(1-δ1(k)))-1P1(n)e-q1(n)N¯(n-τ1(n))(e-q1(n)z(n-τ1(n))-1)∑n=pjM-1‍(∏k=0n‍(1-δ1(k)))-1}≤(e-q*c-1)∑n=pjMi-1‍P1(n)∏k=n+1M-1‍(1-δ1(k))e-q1(n)u¯≤a¯(e-q*c-1).
Thus, we have
(4.44)d-ϵ≤a¯(e-q*c-1).
By the fact that ϵ is arbitrary, we obtain that
(4.45)d≤a¯(e-q*c-1).
Similarly, we can prove that
(4.46)c≥a¯(e-q*d-1).
Equations (4.45) and (4.46) produce that c=d=0. Therefore z(n)=0 for all n≤p1. It follows from (4.32) that z(n)=0 for all n∈ℤ, which also contradicts (4.33). By Lemma 3.3 we see that (1.1) has a unique almost periodic solution p(n) whose range is in [u¯,v¯]. By Corollary 2.5 we can see that p(n) is global attractivity. This also implies  that the almost periodic solution of (1.1) is unique. This completes the proof.



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