ON THE LASOTA-WAZEWSKA MODEL WITH PIECEWISE CONSTANT ARGUMENT

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Acta Mathematica Scientia 2006,26B(2):371-378

BFB@YQ www.wipm.ac.cn/publish/

O N THE LASOTA-WAZEWSKA MODEL WITH PIECEWISE CONSTANT ARGUMENT* Feng Q i k a n g ( 4$i* ) College of Statistics, S h a d University of Finance and Economics, Sham5 030006, China E-mail: [email protected] m

Yuan Rong ( tiff ) Department of Mathematic, Beijing Normal University, Beijing 100875, China

Abstract In this article, a delay differential equation with piecewise constant argument is considered; the existence and global attractivity condition of‘almost periodic solution and quasi-periodic solution are obtained. K e y words Almost periodic solution, quasi-periodic solution, piecewise constant argument 2000 MR Subject Classification 34K14, 34K20

1

Introduction

This article considers the following nonlinear delay differential equation with piecewise constant argument k(t>= - S ( W t ) +p(t)f(z([tI))

(1)

with initial condition s(0) = 20 > 0, here 6, p E C(R,R+) satisfy b(t) f 0, p(tj f 0; f : [0,+oo) + (0,too) is a real analytic function with Lipschitz constant L. This equation stems from the following autonomous equation

which was used by Wazewska-Czyzewska & Lasota [l]as a model for the survial of red blood cells in an animal, here y ( t ) denotes the number of red blood cells at time t , 6 > 0 is the probability of death of a red blood cell, p and y axe positive constants related to the production of red blood cells per unit time, and T is the time required to produce a red blood cell. The oscillation and attractivity of Eq. (2) have been extensively studied (see Ref. [2-51). Nonautonomous delay differential equation

*Received November 8, 2004; revised June 27, 2005

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is investigated in [6,7]. When G ( t ) , p ( t )are nonnegative w-periodic functions and T = mu (m E Z+), the existence and global attractivity of w-periodic solution as well as the oscillation of solutions for Eq.(3) with f(x) = e-" are discussed in [6]; when G(t) and p ( t ) are nonnegative almost periodic functions, the existence and attractivity of almost periodic solution for Eq.(3) are studied in [7]. Differential equations with piecewise constant argument (EPCA) were first considered by Cooke & Wiener [8]and Shah & Wiener [9] (also see [lo]). These equations have the structure of continuous dynamical systems within intervals of unit length. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the values of the solution at such points. Therefore, they combine the properties of differential equations and difference equations. Yuan & Hong [ll]first study the almost periodic solution of EPCA, after that, there have been many works in this field (see [12-161). The purpose of this paper is to investigate the existence and global attractivity of almost periodic solution and quasi-periodic solution of Eq.(l). Our motivation to study (1) comes from the recent works [12,15]on the following logistic delay differential equations

2

Preliminary and Main Theorem

For the reader's convenience, we shall introduce a few concepts from [17,18]. Definition 1 Assume that w1, w2, . . . ,w, E R are rationally independent. A continuous function z : R --+ R", t + z(t) is said to be quasi-periodic with frequencies (w1, w2, - . . ,w,) if there exists a periodic function F(ul,u2, . . . , u,)of u l , u2, . . . , u, with period 1, such that

.

~ (= t )F(wlt, ~ 2 t- ,.. ,~ , t )Vt, E R.

For convenience, we denote w mlwl+ m2w2

+ . . . + m,w,.

= (w1, wp,

. . . , w,), m = (ml, m 2 , .. . , m,) E Z', and (m, w)

=

Set

It is easy to see that every function in QP(w) is quasi-periodic with frequencies w. Setting 1. = C , (urn\,it is easily shown that (QP(w), ((.II)is a Banach space. Set

I

m

m

every sequence in QP(w; Z) can be seen as a function in QP(w) taking values at integer point t = n E Z. We define

I I ~ I =I C IUmI,vu E

QP(W;

m

then (QP(w;Z), 11 . 11) is a Banach space (See [15]). Set

z),

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obviously, QP(w) c GQP(l,w), Q P ( u , Z ) c GQP(1,w). Definition 2 A continuous function z(t) is said to be almost periodic if for any the etranslation set of z

E

> 0,

+

is a relative dense set on R (that is, there is 1 > 0, such that [a,a 11 n T ( z ,E ) # 0 for any a E R). T is called an €-almost period of z. Definition 3 A sequence z : Z 4 R" is called an almost periodic sequence if the E-

translation set of z

T ( z ,E ) = {T E

z;15(n-k T ) - X ( n ) I

< E , n E z}

is a relative dense set on Z for all E > 0. Denote by dP the set of all almost periodic functions with norm IIzlI = suptER1z(t)l, and by d P ( Z ) the set of all almost periodic sequences with norm 1 1 ~ 1 1= supnEz Iz(n)I,then ( A P , 11 . 11) and ( d P ( Z ) , 11 11) are both Banach spaces. Noticing that the concept of almost periodic functions is in terms of functions on R, we should find a solution of Eq.(l) on R. Since the standard definition of sequence is that it is a function on Z+, we will use the term "extended sequence" to mean a function on Z as in [12,15], and denote by ES the set of extended sequences, by BES the set of bounded extended sequences, respectively. Following [lo], a function z : R + R is called a solution of Eq.(l) if z ( t )satisfies the following conditions: (i) z is continuous on R; (ii) The derivative k ( t ) of z(t) exists everywhere except for the points n E Z, where the right derivatives exist; (iii) z satisfies Eq.(l) for t E (n,n l ) , n E Z. By direct integration for Eq.(l) we obtain that

+

and by the continuity requirement for solutions of Eq.(l) we get easily that

z(k + 1) = z ( k ) exp

(-Lk+' ik+'

+f(z(k))

d(s)ds)

p ( s )exp (1[+'

(5) 6(q)dq) ds,

k E Z.

If z ( t ) solves Eq.(l), clearly the difference equation (5) defines an extended sequence

{dk)} k E Z . Lemma 1 ([19]) Assume u(t) E dP, { ~ ( n )E }d P ( Z ) , then for any and T ( u ,E ) n T ( v ;E ) are both relatively dense. Lemma 2 (i) If a ( t ) E dP, then J[+' a(t)dt} E d P ( Z ) ;

E

> 0, T ( u ,E ) n Z

{

(ii) If a ( t ) E QP(w),then

{J;"

a(t)dt} E QP(w,Z) (see [15]);

(iii) If a(t) E GQP(l,w), then {JL+'a(t)dt} E QP(w,Z), and

a(s)ds E GQP(1,w).

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Proof (i) For any given e Choose T E T ( a ,e) n 2,we have

> 0,

by Lemma 1, the set T ( a , e ) n Z is relatively dense.

lL1+7 ik+' l+'+ o(t)dt/ 5

a(t)dt -

This shows

la(t

T)

- a(t)ldt

< E.

{ SF' a(t)dt} E d P ( Z ) .

(ii) Set a ( t ) = C , am({t})eizT(m+')[t],we have

This shows

{ Jkk+'

a ( t ) d t } E &P(w,Z). And we have

Lemma 3 Suppose h : R -, R is a real analytic function, then we have (i) {h(u(n))}E QP(w;Z ) , for any { ~ ( n E) }QP(u;Z) (see [151); (ii) h(v(t))E BQ'P(l,w),for any v ( t ) E OQP(1,w)(we omit the proof). For any q E BC(R.,R+), denote qm = kinf EZ

(Lk+'

Lemma.4 Define @ : B E S y(k

,

q(s)ds)

+

+ 1 ) = z(k)exp +f(z(k))

(l"gods)

qM =m ;

.

BES as y = @z,where

(-ik+' J"" (- lk+' C(s)ds)

6(q)clv) ds, k E Z .

p ( s )exp

k

(i) Assume 6 , p E d P , then y E d P ( Z ) when z E d P ( Z ) ; (ii) Assume 6 , p E QP(w)),then y E QP(u,Z ) ) when z E QP(u,Z ) ) . Proof (i) Suppose 6 , p E d P and z E d P ( Z ) . Set

A(k) = exp

(-L"

6(s)ds)

, B ( k ) = L + ' p ( s ) exp

(-lk+'

6(q)dv) ds.

By the uniform continuity of e-z, for any given E > 0, there exists a > 0 such that le-* - e-"l

Take T E T ( p ,;)

E

< - when 2PM

(J*"" p ( s )exp (-lk+'+T d(q)dr() ds k+r

- z'l < a.

n T(6,a),we have

IB(k + T ) - W)I =

Iz

-

lk+' (-lk+' p ( s )exp

6(q)dq) ds

Feng & Yuan: LASOTA-WAZEWSKA MODEL WITH PIECEWISE CONSTANT ARGUMENT

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375

ds

From this and Lemma 1, it follows that { B ( k ) } E d P ( Z ) . Lemma 2 and Lemma 3 lead to { A ( k ) } , {f(z(k))} E d P ( Z ) . Hence y E d P ( Z ) . (ii) Let b , p E QP(w). Rewrite

By Lemma 2 and Lemma 3, we know

thus

{

z E

QP(w,Z), clearly f(z) E QP(w,Z). So we have {y(k)} E QP(w,Z). This completes the

J;+l

p ( s )exp

(Jil6(v)dv) ds} and {exp (- J;+l b(s)ds)} are both in &P(w,Z). When

proof. Theorem 1

dP satisfy e-6m + L ~ M < 1, then Eq.(l) has a unique positive almost periodic solution u* (t)which is globally attractive; . . ,w,,), (ii) Assume that 1,wl,u2,. . . ,wT are rationally independent, and denote w = (w1,w2,. if 6 , p E QP(w) satisfy e-6-+LpM < 1,then Eq.(l) has a unique positive quasi-periodic solution (i) Assume that 6 , p E

u*(t) with frequencies (1,w1,w2,

.. . ,wr),and u*(t)is globally attractive.

Proof (i) Suppose 6 , p E dP. Set S1 = {z E d P ( Z ) , z ( k ) 1 0,Vk E Z}. For any (Q!y)(k)> 0 for any k E Z, and

z,y E Sl, we have ( Q ! z ) ( k )> 0,

Then by Lemma 4, it follows that Q! is a contraction mapping S1 into S1. Hence CP has a unique fixed point x* = {z*(k); k E Z} in 1'5 satisfying z*(k)> 0 for any k E Z. Let u * ( t ) be a solution of Eq.(l) satisfying u*(k)= z*(k), k E Z. Then u*(t)can be written

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u * ( t ) = z*(k)exp

+f(z*(k))s,”p(s) exp

=

(- [d(V)dV)

(7) ds, k

I t < k + 1, k E Z.

lz*([t] + T)exp ( - L d ( s + i ) d s )

For any given E > 0, there exists u > 0 such that (e-’ - e-”I < & whenever 12 - z’l < u. Meanwile, {z*(lc)}E d P ( Z ) implies {f(z*(k))}E d P ( Z ) . Therefore the set T ( z * i, ) n

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T(f(z*),&) n T(p,&) n T(6,a) is a relative dense set. Take 7 E T ( z * ,B) n T ( f ( z * )&) , n T ( p , &) n T(6, u),from relation (8) we can arrive lu*(t+ T ) - u*(t)l < e . On the other hand, as a solution of Eq.(l), u*(t)is certainly continuous on R, so u*(t)E dP.That is, u*(t)is a unique positive almost periodic solution of Eq.(l). We claim that u * ( t ) is globally attractive. Let u(t)be any solution of Eq.(l), we can get !?om relation (5) that

+ ~ p M ) J u (-k 1) - u*(k- I ) / I (e-6m + ~ p M ) ~ l u-(u*(0)1, ~) k E z+,

lu(k) - u*(k)JI (e-6m

so lu(k)- u*(k)l -t 0 as k

-+

+m. From this, we instantly obtain

+ ~ p M ) l u ( l-~ u*(k)l )

1u(t)- u*(t)lI (e-'m

--f

0, as t

-t

+m.

Therefore the unique positive almost periodic solution u* ( t )is globally attractive. (ii) When 6 , p E QP(w), denote Sz = {z E Q P ( w , Z ) , z ( k )2 0,Vk E Z}, then the map CP defined by (6) is a contraction mapping SZ into S2, it has a unique fixed point in SZ still denoted by z* = {z*(k); k E Z}, and we have z*(k)> 0 for any k E Z. It follows from Lemma d(s)ds) and p ( s ) exp d(p)dp) ds 2 and Lemma 3 that z*([t]),f(z*([t])),exp are all in GQP(1,w). Therefore

(-

Atl

(&

is in GQP(1,w). It follows from the almost periodicity of u * ( t ) obtained above that u*(t)E QP(1,w ) . That is, u*(t)is the unique positive quasi-periodic solution of Eq.(l) with frequencies ( l , ~ )and , u * ( t ) is globally attractive. This completes the proof. M 1. Corollary Assume that 6, p are T-periodic functions satisfying e-6m + L ~ Q 120 (i) If T = -,no and mo E Z+ and mutually prime, then Eq.(l) possesses a unique m0 positive globally attractive no-periodic solution. (ii) If T is irrational, then Eq.(l) possesses a unique positive globally attractive quasiperiodic solution with frequencies (1, $). Example The following differential equation with piecewise constant delay

$(t)= -a(l +sinPt)z(t)

1 + -(1 +cos2t)exp(-z([t])) 2 1

has a unique globally attractive quasi-periodic solution z*@) with frequencies (1,-), when IT 2 a> In 1-sin1 I-sinl' 1 Indeed, b(t) = a(1 sin2t) and p ( t ) = -(1+ cos2t) are 7r-periodic functions, f(z) = e-" 2 is a real analytic function with Lipschitz constant L = 1. we have

+

rk+l

jk

fk+l

6(t)dt = Jk

.**:.

.I

a( 1+ sin 2t)dt = a( 1+ sin(2k + 1)sin 1) > a( 1 - sin 1)

and 1 (1 cos 2t)dt = - (1 2

+

+ cos(2k + 1) sin 1) < 1+2sin 1. ~

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2

2

In it follows that 6, > In From a > I-sinI l-sinl’ 1 - sin 1 LPM < 1.

PM

5

1

+ sin 1, thus ed6, + 2

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