具有逐段常量的三階微分方程解的周期性和概周期性

張 寶, 李洪旭

(四川大學數學學院, 成都 610064)

1 Introduction

Differential equations with piecewise constant arguments, introduced by Cooke, Wiener and Shah[1-2], have been studied intensively over the past few decades. Periodicity and almost periodicity of this kind of differential equations are attractive topics in the qualitative theory of differential equations due to their significance and applications in physicology, control theory and others.Therefore many works appeared in this field (see e.g. Refs.[3-10]).

In 1994, Papaschinopoulos and Schinas[11]studied the existence, uniqueness and asymptotic stability of solutions for the equation

where [·] is the greatest integer function. In 2012, Zhuang[12]presented some results on the existence and uniqueness of almost periodic solutions of the followingNth-order neutral differential equations:

(x(t)+px(t-1))(N)=qx([t])+f(t).

In 2013, Zhuang and Wu[13]studied the almost periodicity for the third-order neutral delay-differential equations of the form

(x(t)+px(t-1))?=qx([t])+f(t).

Recently, Bereketoglu, Lafci and Oztepe[14]considered the oscillation, nonoscillation and periodicity of a third-order equation

x?(t)-a2x′(t)=bx([t-1]).

Motivated by the above-mentioned results, in this paper we consider the following system

(1)

witha,b∈Rsuch that

(a3+ab-absinha)(a3-ab+bsinha)≠0

(2)

2 Preliminaries

LetZ,NandRdenote the sets of all integers, positive integers and real numbers, respectively, and forn∈Nandp∈N,Z[n,n+p]=Z∩[n,n+p]. We denote byBC(R,R) the Banach space of all bounded continuous functionsf:R→Rwith supremum norm and byB(Z,R) the Banach space of all bounded sequences {cn}n∈Zwith supremum norm. Now, we give some definitions and lemmas, which can be found(or simply deduced from the theory) in any book, say Ref.[15], on almost periodic functions.

Definition2.1A functionx:R→Ris said to be a solution of problem (1) if it satisfies the following conditions:

(i)x? exists onRwith the possible exception of the points (2n-1), where the one-sided 3rd derivatives exist;

(ii)xsatisfies (1) on each interval[2n-1,2n+1),n∈Z.

Definition2.2A subsetSofRis called relatively dense inRif there exists a positive numberLsuch that [a,a+L]∩S≠? for alla∈R. A functionf∈BC(R,R) is said to be almost periodic if for everyε>0 the set

T(f,ε)={τ:|f(t+τ)-f(t)|<εfor allt∈R}

is relatively dense inR. We denote the set of such functions byAP(R).

Definition2.3A setP∈Zis said to be relatively dence inZif there exists a positive integerpsuch thatZ[n,n+p]∩P≠? for alln∈Z. A sequencex∈B(Z,R) is said to be almost periodic if for everyε>0 the set

T(x,ε)={τ∈Z:|x(n+τ)-x(n)|

is relatively dense inZ. We denote the set of sequences byAPS(R).

Lemma2.4x∈APS(R) if and only if there existsf∈AP(R) such thatf(n)=x(n) forn∈Z.

3 Main results

3.1 The form of the solution

Because of the piecewise constant arguments, by the Picard theorem for the classical ordinary differential equations, we can get easily the existence and uniqueness of the solution for system (1). Now we induce the difference equation corresponding to (1).

Letx(t) be a solution of (1), and

x(n)=cn,x′(n)=dn,x″(n)=en,n∈Z.

Then (1) reduces to

x?(t)-a2x′(t)=bc2n,t∈[2n-1,2n+1),n∈Z.

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Fort∈[2n-1,2n+1),n∈Z, it is well known that the solution of the above equation is given as

x(t)=Kn+Lncosha(t-2n+1)+

(3)

with constantsKn,LnandMn.Lettingt=2n-1 in (3), we have

(4)

Takingt=2n-1 in the first and second derivatives of (3),respectively, we get

(5)

From (4) and (5), we derive

(6)

Substituting (5) and (6) in (3), we have

(7)

and then

(8)

By the continuity ofx(t), settingt=2n+1 in (7) and (8), it follows that

(9)

Meanwhile, settingt=2nin (7) and (8), we get

(10)

Letvn=(cn,dn,en)Tand

Then (9) and (10) become

v2n+1=Av2n-1+Bv2n,Cv2n=Dv2n-1.

Letwn=v2n-1, we obtain the difference equation corresponding to (1):

(11)

The matrixCis invertible because of condition (2). Then it is easy to get the characteristic equation of (11):

We notice that the eigenvaluesλ≠0 by (2). Now we have the following theorem.

Theorem3.1The form of the solution of system (1) with condition (2) is as follows.

x(t)=(l1(t)+l2(t))wn,

t∈[2n-1,2n+1),n∈Z

(12)

where

andwn=(w0,Pw0,P2w0)Qn(λ) is the solution of (11). HereP=A+BC-1DandQn(λ) is a vector determined by the eigenvaluesλ1,λ2,λ3in the following 3 cases:

(i) if all the eigenvalues are simple,

(13)

(ii) ifλ1=λ2≠λ3, then

(iii) ifλ1=λ2=λ3, then

ProofIt is obvious that (12) holds from (6) and (11). So we need only to prove that the solutionwnof (11) have the required form.

(i) If all the eigenvalues are simple, the solution of (11) can be given as

(14)

wherekj=(k1j,k2j,k3j)T,j=1,2,3. Then we have

(w0,Pw0,P2w0)=(w0,w1,w2)=

which implies that

and

That is (13) holds.

(ii) Ifλ1=λ2≠λ3, the solution of (11) can be expressed by

wherekj=(k1j,k2j,k3j)T,j=1,2,3. Then (ii) can be proved by the same arguments of (i).

(iii) Ifλ1=λ2=λ3, the solution of (11) can be written as

wherekj=(k1j,k2j,k3j)T,j=1,2,3. Then, similarly to (i), we can prove (iii).

Lemma 3.2

From (11), one have

(15)

Here the constantsrij,i=1,2,j=1,2,3 only depend ona,b.

ProofAccording to (11), we have

This together with (11) forms a system of 4 equations with 4 variablesc2n-1,d2n-1,c2n+1,d2n+1. Then, by a fundamental calculation, we can get (15) with

r11=-a/(2sinh2a),

r12=a(-2a3-2a3cosh2a+bsinha-

r13=a(a3+2a3cosh2a-ab+bsinh2a-

r21=a2/(4sinh2a)

r22=a2(-2a3cosh2a+2ab-bsinha-

r23=a2(-a3+2a3cosh2a+ab+2bsinha-

3.2 Periodicity and almost periodicity

We first consider the periodicity of the solution of problem (1).

(i)x(t) is 2k-periodic if and only if {wn}n∈Zisk-periodic;

(ii)x(t) is 2k-periodic if and only if {c2n-1}n∈Zisk-periodic.

Proof(i)Ifx(t+2k)=x(t) fort∈R, it is easy to get from the definition of the solution {wn}n∈Zof (12) that {wn}n∈Zisk-periodic. Conversely, by Theorem 3.1, it is easy to see thatl1(t) andl2(t) are 2-periodic. Suppose that {wn}n∈Zisk-periodic. Then it follows from (12) thatx(t) is 2k-periodic.

(ii) Ifx(t) is 2k-periodic, by (i), we can see that {c2n-1}n∈Zisk-periodic. Conversely, suppose that {c2n-1}n∈Zisk-periodic. Then {d2n-1}n∈Zand {e2n-1}n∈Zarek-periodic by (15). Thus {wn}n∈Zisk-periodic. Sox(t) is 2k-periodic by (i).

For the almost periodicity of the solution of problem (1), we have the following result.

(i)x∈AP(R) if {wn}n∈Z∈VAPS(R);

(ii)x∈AP(R) if and only if {c2n-1}n∈Z∈APS(R).

Proof(i) Assume that {wn}n∈Z∈APS(R).By Theorem 3.1, it is easy to see thatl1(t) andl2(t) are periodic. Thusl1,l2∈APS(R), and then we getx∈AP(R) from (12).

(ii) Ifx∈AP(R), it is easy to see thatx(2t-1) is also almost periodic int. Then {c2n-1}n∈Z={x(2n-1)}n∈Z∈APS(R). Conversely, if {c2n-1}n∈Z∈APS(R), we have {d2n-1}n∈Z,{e2n-1}n∈Z∈APS(R) by (15), and then {wn}n∈Z∈APS(R). By (ⅰ), we havex∈AP(R).