On the existence and exponential attractivity of a unique positive almost periodic solution to an impulsive hematopoiesis model with delays

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by  Trinh Tuan Anh

ON THE EXISTENCE AND EXPONENTIAL ATTRACTIVITY OF A UNIQUE
POSITIVE ALMOST PERIODIC SOLUTION TO AN IMPULSIVE
HEMATOPOIESIS MODEL WITH DELAYS
5
1 TRINHTUANANH,TRANVANNHUNGANDLEVANHIEN
0
2

n
a
Abstract. Inthispaper,ageneralizedmodelofhematopoiesiswithdelaysandimpulsesisconsidered.
J
Byemployingthecontractionmappingprincipleandanoveltypeofimpulsivedelayinequality,weprove
3 theexistenceofauniquepositivealmostperiodicsolutionofthemodel. Itisalsoprovedthat,underthe
2 proposedconditionsinthispaper,theuniquepositivealmostperiodicsolutionisgloballyexponentially
attractive. Anumericalexampleisgiventoillustratetheeffectiveness oftheobtainedresults.
]
A Hematopoiesis model; almost periodic solution; impulsive systems.
C
1. Introduction
.
h
The nonlinear delay differential equation
t
a
m b
(1.1) x˙(t)= ax(t)+ , n>0,
− 1+xn(t τ)
[ −

where a,b,τ are positive constants,proposedby Mackeyand Glass [17], has been used as anappropriate
1
modelforthedynamicsofhematopoiesis(bloodcellsproduction)[5,7,17]. Inmedicalterms,x(t)denotes
v
8 the density of mature cells in blood circulationat time t and τ is the time delay between the production
8 of immature cells in the bone marrow and their maturation for release in circulating bloodstream.
8 As we may know, the periodic or almost periodic phenomena are popular in various naturalproblems
5
of real world applications [2,5,6,9,10,16,19,22,23]. In comparing with periodicity, almost periodicity is
0
more frequent in nature and much more complicated in studying for such model [20,21]. On the other
.
1 hand, many dynamical systems describe the real phenomena depend on the history as well as undergo
0
abrupt changes in their states. This kind of models are best described by impulsive delay differential
5
equations [3,18,20]. A great deal of effort from researchers has been devoted to study the existence and
1
: asymptotic behavior of almost periodic solutions of (1.1) and its generalizationsdue to their extensively
v
realistic significance. We refer the reader to [8,13,15,21,24,25] and the references therein. Particularly,
i
X in [21], Wang and Zhang investigated the existence, nonexistence and uniqueness of positive almost
r periodic solution of the following model
a
b(t)x(t τ(t))
(1.2) x˙(t)= a(t)x(t)+ − , n>1,
− 1+xn(t τ(t))
−
by using a new fixed point theorem in cone. Very recently, using a fixed point theorem for contraction
mappingcombiningwiththeLyapunovfunctionalmethod,Zhangetal.[25]obtainedsufficientconditions
for the existence and exponential stability of a positive almost periodic solution to a generalized model
of (1.1)
m
b (t)
i
(1.3) x˙(t)= a(t)x(t)+ , n>0.
− 1+xn(t τ (t))
i
i=1 −
X
By employing a novel argument, a delay-independent criteria was established in [13] ensuring the ex-
istence, uniqueness, and global exponential stability of positive almost periodic solutions of a non-
autonomousdelayedmodelof hematopoiesiswith almostperiodic coefficients anddelays. In [1], Alzabut
AcceptedforpublicationinAMV.
1
2 T.T.ANH,T.V.NHUNG&L.V.HIEN
et al. considered the following model of hematopoiesis with impulses
b(t)
x˙(t)= a(t)x(t)+ , t 0, t=t ,
(1.4) − 1+xn(t τ) ≥ 6 k
−
∆x(t )=γ x t− +δ , k N,
k k k k ∈
where tk represents the instant at which(cid:0)the(cid:1) density suffers an increment of δk unit and ∆x(tk) =
x t+ x t− . The density of mature cells in blood circulation decreases at prescribed instant t by
k − k k
some medication and it is proportional to the density at that time t−. By employing the contraction
(cid:0) (cid:1) (cid:0) (cid:1) k
mapping principle andapplying Gronwall-Bellman’sinequality,sufficient conditions whichguaranteethe
existence andexponentialstability ofa positive almostperiodic solutionofsystem(1.4)were givenin [1]
as follows.
Theorem 1.1 ( [1]). Assume that
(C1) The function a C(R+,R+) is almost periodic in the sense of Bohr and there exists a positive
∈
constant µ such that a(t) µ.
(C2) The sequence γ is almo≥st periodic and 1 γ 0, k N.
k k
(C3) The sequences{ tp} are uniformly almost p−erio≤dic an≤d there∈exists a positive constant η such that
(C4) iTnhfke∈fNutn1kct=ionη,bw{hkeC}re(R0+<,Rσ+≤)tiks<almtko+s1t,∀pekri∈odNic, liinmtkh→e∞setnks=e o∞f Baonhdr,tpkb(=0)tk=+p0,−atnkd,kt,hper∈eNex.ists
∈
a positive constant ν such that sup b(t) <ν.
t∈R+| |
(C5) The sequence δ is almost periodic and there exists a constant δ >0 such that sup δ <δ.
{ k} k∈N| k|
If ν <µ, then equation (1.4) has a unique positive almost periodic solution.
Unfortunately, the above theorem is incorrect. For this, let us consider the following example.
Example 1.1. Consider the following equation
(1.5) x˙(t)= x(t), t 0,t=k N, ∆x(k)= 1, k N.
− ≥ 6 ∈ − ∈
Note that (1.5) is a special case of (1.4). Moreover, we can easily see that equation (1.5) satisfies
conditions (C1)-(C5), where t =k,γ =0 and δ = 1.
k k k
−
Suppose that system (1.5) has a positive almost periodic solution x∗(t). It is obvious that
x∗(t)=e−tx∗(0) e−(t−k), t>0.
−
k∈N,k≤t
X
For any positive integer n, we have
1 e−n e−1
0<x∗(n)=e−nx∗(0) e−1 − − <0 as n
− 1 e−1 −→ 1 e−1 →∞
(cid:18) − (cid:19) −
which yields a contradiction. This shows that (1.5) has no positive almost periodic solution. Thus,
Theorem 1.1 is incorrect, and Theorem 3.2 in [1] is also incorrect.
Motivatedbytheaforementioneddiscussions,inthispaperweconsiderageneralizedmodelofhematopoiesis
with delays, harvesting terms [4,12,14] and impulses of the form
m
b (t)
i
x˙(t)= a(t)x(t)+
− i=1(cid:20)1+xαi(t−τi(t))
X
(1.6) T v (s)
i
+c (t) ds H (t,x(t σ (t))) , t=t ,
i Z0 1+xβi(t−s) − i − i (cid:21) 6 k
∆x(t )=x t+ x t− =γ x t− +δ , k Z,
k k − k k k k ∈
where m is given positive(cid:0)int(cid:1)eger,(cid:0)a,b(cid:1)i,ci : R(cid:0) (cid:1)R,i m := 1,2,...,m , are nonnegative functions;
H : R R R ,i m, are nonnegative func→tions re∈present{harvesting t}erms; τ (t),σ (t) 0,i m,
i + i i
are tim×e dela→ys; α ,β∈,i m, are positive numbers and T >0 is a constant; γ ,δ ,k Z, are≥const∈ants;
i i k k
∈ ∈
IMPULSIVE HEMATOPOIESIS MODELS WITH DELAYS 3
v (t),i m, are nonnegative integrable functions on [0,T] with T v (s)ds = 1; t ,k Z, is an
i ∈ 0 i { k} ∈
increasing sequence involving the fixed impulsive points with lim t = .
k→±∞ k
R ±∞
The main goal of the present paper is to establish conditions for the existence of a unique positive
almostperiodic solutionofmodel(1.6). Itis alsoprovedthat, under the proposedconditions,the unique
positive almost periodic solution of (1.6) is globally exponentially attractive.
The rest of this paper is organized as follows. Section 2 introduces some notations, basic definitions
and technical lemmas. Main results on the existence and exponential attractivity of a unique positive
almost periodic solutionof (1.6) are presented in section 3. An illustrative example is given in section 4.
The paper ends with the conclusion and cited references.
2. Preliminaries
Let tk k∈Z be a fixed sequenceof realnumbers satisfying tk <tk+1 for all k Z, limk→±∞tk = .
Let X{be}an interval of R, denoted by PLC(X,R) the space of all piecewise l∈eft continuous funct±io∞ns
φ:X R with points of discontinuity of the first kind at t=t , k Z.
k
The→followingnotationswillbe usedinthispaper. Forboundedfu∈nctionsf :R R, F :R R R
+
→ × →
and a bounded sequence z , we set
k
{ }
f = inf f(t), f =supf(t),
L M
t∈R t∈R
F = inf F(t,x), F = sup F(t,x),
L M
(t,x)∈R×R+ (t,x)∈R×R+
z = inf z , z =supz .
L k M k
k∈Z k∈Z
The following definitions are borrowed from [18].
Definition 2.1 ( [18,20]). The set of sequences tp , where tp = t t ,p,k Z, is said to be
{ k} k k+p − k ∈
uniformly almost periodic if for any positive number ǫ, there exists a relatively dense set of ǫ-almost
periods common for all sequences.
Definition 2.2 ( [10,18]). A function φ PLC(R,R) is said to be almost periodic if the following
∈
conditions hold
(i) The set of sequences tp is uniformly almost periodic.
{ k}
(ii) For any ǫ>0, there exists δ =δ(ǫ)>0 suchthat, if t,t¯belong to the same interval ofcontinuity
of φ(t), t t¯ <δ, then φ(t) φ(t¯) <ǫ.
| − | | − |
(iii) For any ǫ > 0, there exists a relatively dense set Ω of ǫ-almost periods such that, if ω Ω then
φ(t+ω) φ(t) <ǫ for all t R, k Z satisfying t t >ǫ. ∈
k
| − | ∈ ∈ | − |
For equation (1.6), we introduce the following assumptions
(A1) The function a(t) is almost periodic in the sense of Bohr and a >0.
L
(A2) The functions b (t),c (t),i m, are nonnegative and almost periodic in the sense of Bohr.
i i
∈
(A3) The function H (t,x),i m, are bounded nonnegative and almost periodic in the sense of Bohr
i
in t R uniformly in x∈R and there exist positive constants L such that
+ i
∈ ∈
H (t,x) H (t,y) L x y , (t,x),(t,y) R R .
i i i +
| − |≤ | − | ∀ ∈ ×
(A4) Thefunctionsτ (t),σ (t),i m,arealmostperiodicinthesenseofBohr,τ˙ (t),σ˙ (t)arebounded,
i i i i
∈
inft∈R(1 τ˙i(t))>0, inft∈R(1 σ˙i(t))>0.
− −
(A5) The sequence δ is almost periodic.
k
{ }
(A6) The sequence γ is almost periodic satisfying
k
{ }
γ > 1, Γ = sup Γ(q,p)< , Γ = inf Γ(q,p)>0,
L M L
− p,q∈Z,p≥q ∞ p,q∈Z,p≥q
p
where Γ(q,p)= (1+γ ),p q.
i
≥
i=q
(A7) The set of sequenQces {tpk} is uniformly almost periodic, η =infk∈Zt1k >0.
4 T.T.ANH,T.V.NHUNG&L.V.HIEN
Remark 2.1. It should be noted that model (1.6) includes (1.4) as a special case. For that model,
assumptions (A3), (A4) obviously be removed. Furthermore, we make assumption (A6) in order to
correct condition (C2) in [1].
The following lemmas will be used in the proof of our main results.
Lemma 2.1 ([18]). Let Assumption (A7) holds. Assume that functions g (t),i m, are almost periodic
i
∈
in the sense of Bohr, a function φ(t) and sequences δ , γ are almost periodic. Then for any ǫ>0,
k k
there exist ǫ (0,ǫ), relatively dense sets Ω R, { Z} s{uch}that
1
∈ ⊂ P ⊂
(a1) φ(t+ω) φ(t) <ǫ,t R, t t >ǫ,k Z;
k
(a2) |g (t+ω)− g (t|) <ǫ,t∈ R,|i− m|; ∈
i i
(a3) |γ γ−<ǫ, |δ ∈δ <ǫ∈, tp ω <ǫ ,ω Ω,p ,k Z.
| k+p− k| | k+p− k| | k− | 1 ∈ ∈P ∈
Lemma 2.2. Let Assumption (A7) holds. Assume that functions f (t,x),i m, are almost periodic in
i
t R uniformly in x R in the sense of Bohr, a function φ(t) and seque∈nces δ , γ are almost
k k
pe∈riodic. Then for any∈compact set R and positive number ǫ, there exist ǫ {(0,}ǫ),{rel}atively dense
1
sets Ω R, Z such that M⊂ ∈
⊂ P ⊂
(b1) φ(t+ω) φ(t) <ǫ,t R, t t >ǫ,ω Ω;
k
(b2) |f (t+ω,−x) f|(t,x) <∈ ǫ,t| −R,x| ,ω∈ Ω,i m;
i i
(b3) |γ γ <−ǫ, δ | δ <∈ǫ, tp ∈ωM<ǫ∈,ω ∈Ω,p ,k Z.
| k+p− k| | k+p− k| | k− | 1 ∈ ∈P ∈
Proof. The proof of this lemma is similar to the proof of Lemma 2.1 in [20] so let us omit it here. (cid:3)
Lemma 2.3. For given ǫ>ǫ >0, real number ω and integers k,p such that tp ω <ǫ , if t t >ǫ
for all i Z and t <t<t1 then t <t+ω <t . | k− | 1 | − i|
k−1 k k+p−1 k+p
∈
Proof. The proof is straight forward, so let us omit it here. (cid:3)
Lemma 2.4. Let Assumptions (A6) and (A7) hold. If p Z satisfies γ γ ǫ for all i Z, then
i+p i
∈ | − |≤ ∈
Γ
Γ(n+p,k+p) Γ(n,k) M (k n+1)ǫ, k,n Z,k n.
| − |≤ 1+γ − ∀ ∈ ≥
L
Proof. Using the facts that eu ev u v max eu,ev , u,v R and ln(1+u) ln(1+v)
1 | − | ≤ | − | { } ∀ ∈ | − | ≤
u v , u,v > 1, from (A6), we have
1+min u,v | − | ∀ −
{ }
k+p k
Γ(n+p,k+p) Γ(n,k) exp ln(1+γ ) exp ln(1+γ )
i i
| − |≤(cid:12) − (cid:12)
(cid:12) (cid:18)i=n+p (cid:19) (cid:18)i=n (cid:19)(cid:12)
(cid:12) X X (cid:12)
(cid:12) k (cid:12)
(cid:12) Γ Γ (cid:12)
(cid:12) M γ γ M (k n+1)ǫ, k (cid:12) n.
i+p i
≤ 1+γ | − |≤ 1+γ − ≥
L L
i=n
X
The proof is completed. (cid:3)
Lemma 2.5. Let Assumption (A7) holds. For any α>0, 0<ǫ<η/2, we have
e−α(t−tk) 1 , tk+ǫe−α(t−s)ds 2e21αη ǫ .
≤ 1 e−αη ≤ 1 e−αη
tXk<t − tXk<tZtk−ǫ −
Proof. The proof follows by some direct estimates and, thus, is omitted here. (cid:3)
Now,letσ =max T,τ ,σ ,then0<σ <+ . Frombiomedicalsignificance,weonlyconsider
i∈m iM iM
{ } ∞
the initial condition
(2.1) x(s)=ξ(s) 0, s [α σ,α), ξ(α)>0, ξ PLC([α σ,α],R).
≥ ∈ − ∈ −
IMPULSIVE HEMATOPOIESIS MODELS WITH DELAYS 5
It should be noted that problem (1.6) and (2.1) has a unique solution x(t) = x(t;α,ξ) defined on
[α σ, ) which is piecewise continuous with points of discontinuity of the first kind, namely t ,k Z,
k
− ∞ ∈
at which it is left continuous and the following relations are satisfied [18]
x t− =x(t ), ∆x(t ):=x t+ x t− =γ x t− +δ .
k k k k − k k k k
Related to (1.6), we c(cid:0)onsi(cid:1)der the following linear(cid:0)eq(cid:1)uation(cid:0) (cid:1) (cid:0) (cid:1)
(2.2) y˙(t)= a(t)y(t), t=t , ∆y(t )=γ y(t−), k Z.
− 6 k k k k ∈
Lemma 2.6. Let Assumptions (A1), (A6) and (A7) hold. Then
Be−aM(t−s) H(t,s) Ae−aL(t−s), s t,
≤ ≤ ≤
where
t
exp a(r)dr , if t <s t t ,
(2.3) H(t,s)= − s k−1 ≤ ≤ k
Γ(n,(cid:16)k)eRxp − st(cid:17)a(r)dr , if tn−1 <s≤tn ≤tk <t≤tk+1
(cid:16) (cid:17)
is the Cauchy matrix of (2.2), A=maxRΓ ,1 and B =min Γ ,1 .
 { M } { L }
Proof. The proof is straight forward from (2.3), so let us omit it here. (cid:3)
Similar to Lemma 36 in [18] and Lemma 2.6 in [20] we have the following lemma.
Lemma 2.7. Let Assumptions (A1), (A6) and (A7) hold. Then, for given 0 < ǫ < ǫ, relatively dense
1
sets Ω R, Z, satisfying
⊂ P ⊂
(c1) a(t+ω) a(t) <ǫ, t R,ω Ω;
(c2) |γ γ−<ǫ,| tp ω∈<ǫ ,∈ω Ω, p , k Z,
| k+p− k| | k− | 1 ∈ ∈P ∈
the following estimate holds
H(t+ω,s+ω) H(t,s) ǫMe−21aL(t−s),
| − |≤
for any ω Ω, t,s R satisfying t s, t t >ǫ, s t >ǫ,k Z, where
k k
∈ ∈ ≥ | − | | − | ∈
2 2 1 2
(2.4) M =max ,Γ + 1+ .
M
a a 1+γ a η
(cid:26) L (cid:20) L L (cid:18) L (cid:19)(cid:21)(cid:27)
Proof. We divide the proof into two possible cases as follows.
Case 1: t < s t t . By Lemma 2.3, t < s+ω t+ω < t . Since a(t+ω) a(t) ǫ,
k−1 k k+p−1 k+p
≤ ≤ ≤ | − | ≤
∀t∈R, ǫ<η/2 and 21aL(t−s)e−12aL(t−s) <1, it follows from (2.3), (2.4) that
t t
H(t+ω,s+ω) H(t,s) = exp a(r+ω)dr exp a(r)dr
| − | − − −
(cid:12) (cid:18) Zs (cid:19) (cid:18) Zs (cid:19)(cid:12)
(cid:12) t (cid:12)
(2.5) (cid:12)(cid:12)e−aL(t−s) a(r+ω) a(r)dr (cid:12)(cid:12)
≤ | − |
Zs
2
ǫe−12aL(t−s) ǫMe−21aL(t−s).
≤ a ≤
L
Case 2: t <s t t <t t . Similarly, we have
n−1 n k k+1
≤ ≤ ≤
t <s+ω <t t+ω <t .
n+p−1 n+p k+p+1
≤
By Lemma 2.4, from (2.3)-(2.5) we obtain
t+ω t
H(t+ω,s+ω) H(t,s) =Γ(n+p,k+p) exp a(r)dr exp a(r)dr
| − | − − −
(cid:12) (cid:18) Zs+ω (cid:19) (cid:18) Zs (cid:19)(cid:12)
(cid:12) t (cid:12)
(cid:12) (cid:12)
+ Γ(n+p,k+(cid:12)p) Γ(n,k) exp a(r)dr (cid:12)
| − | −
(cid:18) Zs (cid:19)
2Γ ǫ Γ ǫ
M e−21aL(t−s)+ M (k n+1)e−aL(t−s)
≤ a 1+γ −
L L
6 T.T.ANH,T.V.NHUNG&L.V.HIEN
2Γ ǫ Γ ǫ t s
M e−21aL(t−s)+ M − +1 e−aL(t−s)
≤ a 1+γ η
L L (cid:18) (cid:19)
2Γ ǫ Γ ǫ 2
M e−21aL(t−s)+ M 1+ e−21aL(t−s)
≤ a 1+γ a η
L L (cid:18) L (cid:19)
ǫMe−21aL(t−s).
≤
The proof is completed. (cid:3)
It is worth noting that, the proof of Lemma 2.7 is different from those in [1,18,20]. By employing
Lemma 2.4, we obtain a new bound for constant M given in (1.4).
Lemma 2.8 ( [11]). Assume that there exist constants R,S > 0,τ 0, T R and a function y
0
PLC([T τ, ),R+) satisfying ≥ ∈ ∈
0
− ∞
R
(d1) ∆y(t ) γ y(t−) for t T , where γ > 1 and max (γ +1)−1,1 < ;
k ≤ k k k ≥ 0 k − tk≥T0 k S
(d2) D+y(t) Ry(t)+Sy(t) for t T ,t = t , where y(t) = sup y(s) and D+ denotes the
≤ − ≥ 0 6 k (cid:8) t−τ≤s≤t (cid:9)
upper-right Dini derivative;
(d3) τ t t for all k Z satisfies t T .
k k−1 k 0
≤ − ∈ ≥
Then
y(t) y(T ) (γ +1) e−λ(t−T0), t T ,
0 k 0
≤ ∀ ≥
(cid:18)T0<Ytk≤t (cid:19)
where 0<λ R Smax (γ +1)−1,1 eλτ.
≤ − tk≥T0 k
(cid:8) (cid:9)
3. Main results
Letusset = φ PLC(R,R):φ is almost periodic,φ(t) 0 for all t R and φ =sup φ(t).
D1 ∈ ≥ ∈ k k t∈R| |
We define an operantor F : 1 PLC(R,R) as follows o
D →
t m b (s) T v (r)
i i
Fφ(t)= H(t,s) +c (s) dr
(3.1) Z−∞ Xi=1(cid:26)1+φαi(s−τi(s)) i Z0 1+φβi(s−r)
H (s,φ(s σ (s))) ds+ H(t,t )δ .
i i k k
− −
(cid:27) tXk<t
Itcanbe verifiedthat x∗(t)=φ(t) is analmostperiodic solutionon of (1.6) ifandonly if Fφ=φ.
1
D
We define the following constants
δ = inf δ , δ =sup δ , η =supt1,
k∈Z| k| k∈Z| k| k∈Z k
m
A Aδ
M
M = (b +c H )+ ,
1 aL iM iM − iL 1 e−aLη
i=1 −
(3.2) X
B m biL ciL BδLe−aMη
+ H + if δ 0,
M2 =aaBMM Xi=mi1=(cid:18)1 1+1+MbiML1αiαi +1+1+cMMiLβ1βii−−HiMiM(cid:19)+ 11−Ae−−δLaeL−ηaMiηf¯δL <L0.≥
(cid:18) 1 1 (cid:19)
Lemma 3.1. Let AssumptPions (A1)-(A7) hold. If φ then
1
∈D
M Fφ(t) M , t R.
2 1
≤ ≤ ∀ ∈
IMPULSIVE HEMATOPOIESIS MODELS WITH DELAYS 7
Proof. Let φ . By Lemma 2.5 and Lemma 2.6, from (3.1) we have
1
∈D
t m
Fφ(t) Ae−aL(t−s) (biM +ciM HiL)ds+AδM e−aL(t−tk)
≤ −
(3.3) Z−∞ Xi=1 tXk<t
m
A Aδ
(b +c H )+ M =M , t R.
≤ aL iM iM − iL 1 e−aLη 1 ∀ ∈
i=1 −
X
For each t R, let n be an integer such that t < t t . If δ 0 then, by Lemma 2.6, it
∈ 0 n0 ≤ n0+1 L ≥
follows from the fact (n k)η t t (n k)η¯, k n that
n k
− ≤ − ≤ − ∀ ≤
H(t,tk)δk BδLe−aM(t−tk) BδLe−aM(tn0+1−tk)
≥ ≥
tXk<t tXk<t tkX≤tn0
(3.4)
= BδLe−aM(tn0+1−tk) ≥ ∞ BδLe−aMη¯q = B1δLee−−aaMMηη¯¯.
kX≤n0 Xq=1 −
If δ <0 then from (A7) and Lemma 2.6, we have
L
H(t,tk)δk AδLe−aL(t−tk) AδLe−aL(tn0−tk)
≥ ≥
tXk<t tXk<t tkX≤tn0
(3.5)
∞
Aδ
Aδ e−aLqη = .
≥ L 1 e−aLη
q=0 −
X
From (3.4) and (3.5) we obtain
m
B b c
iL iL
(3.6) Fφ(t)≥ aM Xi=1(cid:18)1+M1αi + 1+M1βi −HiM(cid:19)+tXk<tH(t,tk)δk ≥M2.
The proof is completed. (cid:3)
Now we are in position to introduce our main results as follows.
Theorem 3.1. Under the Assumptions (A1)-(A7), if φ then Fφ(t) is almost periodic.
1
∈D
Proof. Let φ . For given ǫ (0,η/2), there exists 0 < δ < ǫ/2 such that, if t,t¯belong to the same
1
∈ D ∈
interval of continuity of φ(t) then
(3.7) φ(t) φ(t¯) <ǫ, t t¯ <δ.
| − | | − |
By Lemma 2.2 and Lemma 2.7, there exist 0<ǫ <δ, relatively dense sets Ω R, Z such that,
1
⊂ P ⊂
for all ω Ω, we have
∈
H(t+ω,s+ω) H(t,s) δMe−21aL(t−s), t s, t tk >δ, s tk >δ;
| − |≤ ≥ | − | | − |
φ(t+ω) φ(t) <δ, t R, t t >δ,k Z;
k
| − | ∈ | − | ∈
H (t+ω,x) H (t,x) <δ, t R,x [φ , φ ],i m;
i i L
| − | ∈ ∈ k k ∈
(3.8) a(t+ω) a(t) <δ, t R;
| − | ∈
b (t+ω) b (t) <δ, c (t+ω) c (t) <δ, t R,i m;
i i i i
| − | | − | ∈ ∈
τ (t+ω) τ (t) <δ, σ (t+ω) σ (t) <δ, t R,i m;
i i i i
| − | | − | ∈ ∈
γ γ <δ, δ δ <δ, tp ω <ǫ ,p ,k Z.
| k+p− k| | k+p− k| | k− | 1 ∈P ∈
8 T.T.ANH,T.V.NHUNG&L.V.HIEN
Let ω Ω, p . One can easily see that
∈ ∈P
t m b (s+ω)
i
Fφ(t+ω)= H(t+ω,s+ω)
Z−∞ i=1(cid:26)1+φαi(s+ω−τi(s+ω))
X
T c (s+ω)v (r)
(3.9) i i
+ dr H (s+ω,φ(s+ω σ (s+ω))) ds
Z0 1+φβi(s+ω−r) − i − i (cid:27)
+ H(t+ω,t )δ .
k+p k+p
tXk<t
We define E ( t )= t R: t t >ǫ, k Z . For t E ( t ), i m, let us set
ǫ k k ǫ k
{ } { ∈ | − | ∀ ∈ } ∈ { } ∈
t H(t+ω,s+ω)b (s+ω) H(t,s)b (s)
i i
C = ds,
i Z−∞(cid:12)1+φαi(s+ω−τi(s+ω)) − 1+φαi(s−τi(s))(cid:12)
(cid:12) (cid:12)
t (cid:12) T c (s+ω)v (r) (cid:12) T c (s)v (r)
D = (cid:12)H(t+ω,s+ω) i i dr H(t,(cid:12)s) i i dr ds,
(3.10) i Z−∞(cid:12)(cid:12) Z0 1+φβi(s+ω−r) − Z0 1+φβi(s−r) (cid:12)(cid:12)
t (cid:12) (cid:12)
(cid:12) (cid:12)
Ei = (cid:12)H(t+ω,s+ω)Hi(s+ω,φ(s+ω σi(s+ω))) H(t,s)Hi(s,φ(s σi(s(cid:12)))) ds,
| − − − |
Z−∞
G= H(t+ω,t )δ H(t,t )δ ,
k+p k+p k k
| − |
tXk<t
then we have
m
(3.11) Fφ(t+ω) Fφ(t) (C +D +E )+G, t E ( t ).
i i i ǫ k
| − |≤ ∈ { }
i=1
X
We also define
t b (s+ω)
i
C = H(t+ω,s+ω) H(t,s) | | ds,
i1 Z−∞| − |1+φαi(s+ω−τi(s+ω))
t b (s+ω) b (s)
i i
C = H(t,s) | − | ds,
(3.12) i2 Z−∞ 1+φαi(s+ω−τi(s+ω))
t
C = H(t,s)φ(s+ω τ (s+ω)) φ(s τ (s+ω))ds,
i3 i i
| − − − |
Z−∞
t
C = H(t,s)φ(s τ (s+ω)) φ(s τ (s))ds
i4 i i
| − − − |
Z−∞
and Ki =supφL≤x≤kφkαixαi−1. It can be seen from (3.10) and (3.12) that
(3.13) C C +C +b K (C +C ), i m.
i i1 i2 iM i i3 i4
≤ ∈
ByLemma2.5andLemma2.6,from(3.8),(3.12)andthefactthat −t∞e−12aL(t−s)ds=2/aL,wehave
(3.14) Ci1 ≤ 2biaMLM +tXk<t2AbiMZtkt−k+ǫǫe−aL(t−s)ds≤biM(cid:18)2RaML + 14−Aee21−aaLLηη(cid:19)ǫ,
and
t A
C Aǫe−aL(t−s)ds= ǫ,
i2
≤ a
(3.15) Z−∞ L
t
C Aǫe−aL(t−s)ds+2A φ e−aL(t−s)ds.
i3
≤ k k
Z−∞ tXk<tZ{s:|s−τi(s+ω)−tk|<ǫ,s≤t}
It should be noted that, by (A4), t τ (t), i m, are strictly increasing functions, and thus, there
i
exist the inverse functions τ∗(t) of t τ−(t). For e∈ach t R, denote t¯=t ǫ τ (t+ω) then
i − i ∈ − − i
(3.16) t+ω =τ∗(t¯+ω+ǫ).
i
IMPULSIVE HEMATOPOIESIS MODELS WITH DELAYS 9
Let λi =infs∈Rτ˙i∗(s), λi =sups∈Rτ˙i∗(s),i∈m, then, by (A4), 0<λi,λi <∞. Therefore,
(3.17) τ∗(t¯+ω+ǫ) τ∗(t +ω+ǫ) λ (t¯ t ), t <t¯,
i − i k ≥ i − k k
and hence, from equations (3.15)-(3.17), we have
Aǫ τi∗(tk+ω+ǫ)−ω
Ci3 +2A φ e−aL(t−s)ds
≤ aL k ktXk<t¯Zτi∗(tk+ω−ǫ)−ω
(3.18) ≤ Aaǫ +2Akφk e−aL[τi∗(t¯+ω+ǫ)−τi∗(tk+ω+ǫ)][τi∗(tk+ω+ǫ)−τi∗(tk+ω−ǫ)]
L
tXk<t¯
Aǫ 1 4λ φ
≤ aL +4AkφkλiǫtXk<t¯e−aLλi(t¯−tk) ≤(cid:18)aL + 1−ei−kaLkλiη(cid:19)Aǫ.
By the same arguments used in deriving (3.18), we obtain
(3.19) Ci4 ≤AǫZ−t∞e−aL(t−s)ds+2AkφktXk<tZτiτ∗i(∗t(kt−k+ǫ)ǫ)e−aL(t−s)ds≤(cid:18)a1L + 1−4λei−kaφLkλiη(cid:19)Aǫ.
Combining (3.13)-(3.15), (3.18) and (3.19), we readily obtain
A 2biM(M +AKi) 4e21aLη 8Kiλi φ
(3.20) C + +Ab + k k ǫ.
i ≤"aL aL iM(cid:18)1−e−aLη 1−e−aLλiη(cid:19)#
Next, let us set
t T v (r)
i
D = H(t+ω,s+ω) H(t,s) drds,
i1 Z−∞| − |Z0 1+φβi(s+ω−r)
t T v (r)
(3.21) D = H(t,s)c (s+ω) c (s) i drds,
i2 Z−∞ | i − i |Z0 1+φβi(s+ω−r)
t T 1 1
D = H(t,s) v (r) drds.
i3 Z−∞ Z0 i (cid:12)1+φβi(s+ω−r) − 1+φβi(s−r)(cid:12)
(cid:12) (cid:12)
It follows from (3.10) and (3.21) that (cid:12) (cid:12)
(cid:12) (cid:12)
(3.22) D c D +D +c D , i m.
i iM i1 i2 iM i3
≤ ∈
By Lemmas 2.5 and 2.6, from (3.8) we have
(3.23) Di1 ≤Z−t∞Mǫe−21aL(t−s)ds+tXk<t2AZtkt−k+ǫǫe−aL(t−s) ≤(cid:18)2aML + 14−Aee12−aaLLηη(cid:19)ǫ,
t A
D Aǫe−aL(t−s)ds= ǫ,
i2
≤ a
Z−∞ L
and
t T
D Ae−aL(t−s) G v (r)φ(s+ω r) φ(s r)drds
i3 i i
≤ | − − − |
Z−∞ Z0
t T tk+r+ǫ
AG ǫe−aL(t−s)ds+2AG φ v (r) e−aL(t−s)ds dr
i i i
≤ k k
(3.24) Z−∞ Z0 (cid:18)tkX+r<tZtk−r−ǫ (cid:19)
AG ǫ T
i +4AGi φ ǫ vi(r) e−aL(t−tk−r−ǫ) dr
≤ a k k
L Z0 (cid:18)tkX+r<t (cid:19)
1 4e21aLη φ
+ k k AG ǫ,
≤(cid:18)aL 1−e−aLη (cid:19) i
10 T.T.ANH,T.V.NHUNG&L.V.HIEN
where Gi =supφL≤x≤kφkβixβi−1. From (3.22)-(3.24), we readily obtain
A+2MciM +AciMGi 4AciMe21aLη(1+Gi φ )
(3.25) D + k k ǫ.
i ≤" aL 1 e−aLη #
−
Now, we define
t
E = H(t+ω,s+ω) H(t,s)H (s+ω,φ(s+ω σ (s+ω)))ds,
i1 i i
| − | −
Z−∞
t
E = H(t,s)H (s+ω,φ(s+ω σ (s+ω))) H (s+ω,φ(s σ (s+ω)))ds,
i2 i i i i
| − − − |
(3.26) Z−∞
t
E = H(t,s)H (s+ω,φ(s σ (s+ω))) H (s+ω,φ(s σ (s)))ds,
i3 i i i i
| − − − |
Z−∞
t
E = H(t,s)H (s+ω,φ(s σ (s))) H (s,φ(s σ (s)))ds,
i4 i i i i
| − − − |
Z−∞
then, from (3.10) and (3.26), we have
(3.27) E E +E +E +E , i m.
i i1 i2 i3 i4
≤ ∈
Alsousing Lemma2.5andLemma 2.6,from(3.8)andthe factthat −t∞e−12aL(t−s)ds=2/aL, weobtain
E 2HiMMǫ + 2AH tk+ǫe−aL(t−s)ds 2 RM + 2Ae21aLη H ǫ,
(3.28) i1 ≤ aL tXk<t iMZtk−ǫ ≤ aL 1−e−aLη! iM
t A
E Aǫe−aL(t−s)ds ǫ.
i4
≤ ≤ a
Z−∞ L
Let ξi =inft∈Rσ˙i∗(t),ξi =supt∈Rσ˙i∗(t),i∈m. Similarly to (3.18) and (3.19), we readily obtain
t
E L H(t,s)φ(s+ω σ (s+ω)) φ(s σ (s+ω))ds
i2 i i i
≤ | − − − |
Z−∞
1 4ξ φ
+ ik k AL ǫ,
(3.29) ≤(cid:18)aL 1−e−aLξiη(cid:19) i
t
E L H(t,s)φ(s σ (s+ω)) φ(s σ (s))ds
i3 i i i
≤ | − − − |
Z−∞
1 4ξ φ
+ ik k AL ǫ.
≤(cid:18)aL 1−e−aLξiη(cid:19) i
Inequalities (3.27)-(3.29) yield
(3.30) E A(2Li+1)+2HiMM + 4Ae21aLηHiM + 8ALiξikφk ǫ.
i ≤(cid:18) aL 1−e−aLη 1−e−aLξiη(cid:19)
Let us set
(3.31) G = H(t+ω,t ) H(t,t ), G = H(t,t )
1 k+p k 2 k
| − | | |
tXk<t tXk<t
then
(3.32) G H(t+ω,t ) H(t,t ) δ + H(t,t ) δ δ δ¯G +ǫG .
k+p k k+p k k+p k 1 2
≤ | − || | | || − |≤
tXk<t tXk<t