Massera criterion for periodic solutions of differential equations with piecewise constant argument

DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT NGUYEN TRUNG THANH Department of Mathematics, Hanoi University of Science Abstract In this note, we prove the almost periodicity

DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT

ARGUMENT

NGUYEN TRUNG THANH Department of Mathematics, Hanoi University of Science

Abstract

In this note, we prove the almost periodicity of bounded solutions and a

so-called Massera criterion for the existence of periodic solutions to differential

equation with piecewise constant argument

In this note, we are concernded with differential equations with piecewise constant argument of the form

# = Ar(H) + ƒ(f]), z) € C", (1.1)

where A is a linear operator on C”, f is a bounded continuous function from R to

C”, |.| is the largest integer function Differential equations with piecewise constant argument have been considered in many works since they are found appropriate to

various applications (see, for example {20, 21, 23] and the references therein)

The main purpose of this note is show a spectral condition for almost periodicity

of bounded solutions and the existence of periodic solutions to equation (1.1) via the so-called Massera criterion Massera criterrion (13) was first introduced by Massera

in 1950 to ordinary differential equations, saying that the linear differential equation

of the form

b= Alt)e(t) + f(t),

where A, f is continuous, periodic with the same period 7, has a periodic solution with petiod 7 if and only if it has a bounded solution on R* Subsequently, it has been extended to ordinary functional ditferential equations (OFDE) of delay type in

[1], to OFDE with advance and delay in [5, 11, 12], to abstract functional differentia equations in [22] Recently, it has been extended to almost periodic solutions o evolution equations in (15, 16, 17, 18] For a more complete introduction to thi

topic we refer the reader to any introduction of these papers and counterexample

for almost periodic equations in [7, 8]

However, as will be shown in this note, in general Massera criterion does no hold true for Eq (1.1) For instance, if f is periodic with irrational period, ther

equation (1.1) has no periodic solutions If in addition, the period of f is assume‹

to be rational, we we can show that Massera criterion for (1.1) holds true

The main technique of this note is to use the notion of spectrum of a functioi

which has been widespreadly employed in recent researches such as (4, 17, 18, 19]

We will estimate the spectrum of a bounded function And based on the obtainec estimates we will consider the almost periodicity or periodicity of solutions Thi main results of the note are Theorem 3.2, 3.5 and 3.9 An estimate of the spectrun

of a bounded solution to Eq (1.1) is obtained in Theorem 3.2 Theorem 3.5 gives : spectral condition for almost periodicity of bounded solutions to Eq (1.1) Basec

on [16], Theorem 3.9 shows the existence of periodic solutions to Eq (1.1)

In this section we recall the notion of a spectrum of a bounded function and som: important properties For more details we refer the reader to [9, 19]

2.1 Notations

Throughout the note we will use the following notations: Z, R, C stand for the sets o

integers, real and complex numbers, respectively X denotes a given complex Banac!

space If A is a linear operator, then the notations (4), ø(4) and R(A, A) stan

for the spectrum, resolvent set, and resolvent of the operator A We also denot

the spectrum of a function f by sp(f) The notation £j,.(R, X) means the Banac

space of measurable, local integrable functions from R to X In this space, th

subspace BM(R, X) consists of f € Li,.(R,X) such that sup,cr fr ||ƒ(s)ll đ» -

+oo As usual, BC(R,X), BUC(R,X).AP(R,X) siand for the spaces of all X vallued bounded continuous and bounded uniformly continuous functions on Roan their subspace of almost periodic functions, respectively

2.2 The spectruun of a bounded function

In this note, we will use the notion of Carlemann spectrum of a function u € BAI R,X), denoted by sp(u)j, consisting of all real numbers € such that the Fourier- Carlemann transform of u

#(A) :=

— fr eMu(—t)dt if Rea <0

has 10 holomorphic extension to any neighborhood of 1€ (see, e.g., [9, 19]) Some

basic properties of the Fourier-Carlemann transform and the spectrum of a function and their relation to the behavior of the function are listed below for the reader’s

convenience We refer the readers to [10, 19] for the proof

Proposition 2.1 Let u, vu€ BM(R,X) anda,GeC Then

(i, (aut Bv)(d) = aii(d) + Ø80),

(ii If there exists u € BM(R, X) then

>=

u(A) = Au(A) ~ u(0),

(iti Let A be a continuous linear operator from X to X Put (Au)(t) := Au(t), Vt €

R then

Au(A) = A8(A)

Proposition 2.2 (/19, p 20]) Letu, v€ BM(R,X),a€ C\ {0} Then

(i) sp(u) ts closed,

(an) splu( +h) = sp(u),

(0u) sp(au) = sp(u),

(it) sp(u+v) © sp(w) U spr),

((Ì sp(u)C sp(u) if € BM(R,X),

(vu) If A is a continuous linear operator from X to X then sp(Au) C sp(u),

(ov) If uy € BƯƠ(R X), ty converge to u uniformly and sp(u,) C A, then

sp(u) CA

2.3 Almost periodic and periodic functions

There are close relations between spectra of functions and their behaviors at infinity Infact, we have

Proposition 2.3 (see, e.g [6], page 29 ) Let f € BUC(R,X) Then f is peri- odic with period r if and only if sp(f) C 2nZ/r

Recall that a subset E Cc R is said to be relatively dense if there exists a number

1 > 0 (inclusion length) such that every interval {a,a +) contains at least one point

of E Let f be a continuous function from R to X Recall also that f is said to

be almost periodic (in the sense of Bohr) if for every ¢ > 0 there exists a relatively dense set T(e, f) such that

sup ||/ + z) — ƒ()|| < e, vr € TŒ, /)

Proposition 2.4 Let ue BUC(R,X) Then

(i) If sp(u) ts countable and X does not contain any subspace isomorphic to the

space of sequences co, then u is almost periodic,

(it) If sp(u) is discrete, then u is almost periodic

For the proof we refer the readers to [10]

Remark 2.5 If dimX < oo, then it never contains any subspaces isomorphic to co

So, this condition is automatically fulfilled in the finite dimensional case

3 Almost periodic and periodic solutions

In this section, we will deal with almost periodicity of bounded solutions and the

existence of periodic solutions to Eq (1.1) First, we make precise the notion of solutions to Eq (1.1)

Definition 3.1 A function z(-): R + C” is said to be a solution of Eq (1.1) af

it is continuous on R differentiable on R except at most of integers and satisfies

Eq (1.1) on every inteval [njn + 1),n € 2, where at t == n the derivative of a ts the right one

3.4 The spectrum of a bounded solution

For a matrix A we denote a.(A) - {£e Re eS! < of A)} The notation f stands

for the fiction, defined by the formula

f(t) = f({t|), Vie R

Theorem 3.2 Let x(-) is a@ bounded solution of Eq (1.1) Then the following

estimates hold true:

Proof We first consider the case of Re\ > 0 By taking the Fourier-Carlemann transforms of functions and by Proposition 2.1 we have

~

Since x(-) is a solution to Eq (1.1),

Ẩ(A) = (Az(]))(A) + (2)

+œ

= Ƒ£*(A() + f0}

0

-fe MAri(e)aes fe ™ F(t) dt

+00

0

Set

(A) = / e*z([fÌ)dt = ` / eMa(k)dt = S7 AE ea (hy,

We have

Laval “A l = cTA

k=O

On all intervals (n,n + 1),1 © Z we have z(t) — Ar(n) + f(n), so x(t) is linear on

(njn +1), Le

Hence,

aa) = [ -Xz0)w = [ e*{r( + 1) = z()|Œ — Ít) + allt

1

e“[a(k + 1) — x(k)](t — k)dt + g(A)

> i! Me oO

k+1

[x(k + 1) ~ 2( vf e*(t — k)dt + g(A)

iM Oo

=3 r&+1)~z(9)-1e” ự — BI" + s eˆS4] k=0 + g0)

= — " Ile Senta r(k+1)— Sle *a(k)) +9(A) (36) k=0

From (3.5) and (3.6) we have

A) ={ Y Ife cm A 9(2) ~ "0 Ì~ 1—gz90)} + ø0)

= Ir + + 1}g(A) + 3 ©" 2(0)

Thus

From (3.3), (3.4) and (3.7) we have

A2(A) - z(0)= A-r—rlf(A) _— pee) + fd)

Therefore,

In the case of ReA < 0, by computating, as abow, we wes the same result So, in all cases the following estunate holds true:

(cˆ | - Ayr) P(A) + FO) (3.8)

p(A)= ————Az(0) + ——+(0)

Since

êÀ —À=~=T 1 1

cÀ — ] ome Le eh boss l

P(X) is holomorphic in C From (3.8) we see that if € ¢ sp(x) and € ¢ 27Z then

L(A) has a holomorphic extension to a neighborhood of i€ Moreover e“* — 1 # 0, so

we have

À ek]

[(e* — 1 ~ A)@(A) — a(A)]

This shows that f has a holomorphic extension to a neighborhood of if, hence,

€ ¢ sp(f) We then get the first estimate

On the other hand, if € ¢ sp(f) then Ff) has a holomorphic extension to a

neighborhood of ig; if e — 1 ¢ o(A) then there exists the bounded inverse of

ef -]-A

R(A) := (e* —1 — A), and R(A) is holomorphic in a neighborhood of 7£ Therefore, Z(A) = R(A)[p(A) +

A=) F(A)] has a holomorphic extension to a neighborhood of i€ Hence, € ¢ sp(x)

And we have the second estimate

Remark 3.3 e® — 1 © a(A) if and only of e& € 1 + o( A) In fact, since e on the

unit circle 1 we have

eel tal(A)ee™® € (14 af A) ONL

As the set [L + o(A)) OL ts finite, o4( A) 1s discrete

3.2 <A sufficient condition for almost periodicity of bounded

solutions

In this subsection we will give a spectral condition for almost periodicity of bounded solutions to Eq (1.1) We have

Lemma 3.4 Let z(-) be a bounded solution to Eq (1.1) Then, r(-) is uniformly

continuous

Proof Since A is a linear operator on C", A is bounded Moreover, since x(t) is

bounded, there exist constants M,,Mo > 0 such that

| Ax(t)|] < Mi llzŒ)|| < My llz|[ < Mẹ, Vý c R

On the other hand, by the boundedness of ƒ, there exists a positive num»er À⁄; such that

lƒ/)J| < M: Y+ c R

Hence,

sup ||Áz(f|) + ƒ(W])| S Ma + M; =: M te

For every e > 0, we have

tạ

Iz(a)-z()| = | i i(t)de||

< | / I(4z()) + /(Ir))IIới

€

< M |t¿ạ— tị| < £ for all |tạ — tị| < i

This shows the uniform continuity of x(t)

Theorem 3.5 In addition to the assumptions of the above theorem, tf sp(f) 1s dis- crete, then any bounded solution of (1.1) is almost periodic

Proof Since the set o,:(A) is discrete, by Theorem 3.2, sp(a) is discrete Moreover,

by Lemma 3.4 x(t) is uniformly continuous Therefore, 2(t) is almost periodic by

Proposition 2.4

Proposition 3.6 Let ƒ(L) be a periodic function with rational peoriod T = i Then, sp(f) ws discrete Morcover, the following estimate holds true

sp(ƒ) + 3x2 = sp(f)

Procf By the definition of f we have

f(t +p) = f(lt+ pl) = f(lt] +p) = f(lt)) = ft) vt e R

Hence

f(t+p) = f(t) VteR

By vsing Fourier-Carlemann transform of f in the case of Re(A) > 0 we have

JA)= | *f0w= [ */t)e

= S> f(k) i edt

k=0 k

L-¢" he

=o Tie

k=0

Le eae

TH ST 3 yy f(kp + rje Art")

k==0 r=0

_ =+Ä p-1 oO

r=0 k=0

sarge | pte 2 f(rje

Sivilarly, in the case of ReX < 0 we have

fay=~ fe a-nar=~ fe A-eyat

=~ » kaa f eva

[eh

I-e*^ 1 &

=i = À gà l—e^» = TỰ = r— lỤe (—p+r+1)

1—e*^ i & as

Thus,

^ 1 — e> 1 p-1

r=0

This shows that if € € sp(f), then e~*? = 14 € € 2nZ/p so sp(f) C 2nZ/p Moreover, if £ € sp(f) + e~*? = 1 then Vn € Z we have

e~1€12mm)P — TP = 1 = £ + 2mn € sp(ƒ)

Therefore,

sp(f) = sp(f) + 2nZ

Remark 3.7 By Proposition 3.6, if f is a periodic function with a rational perio

Ts 5 then for all bounded solution to Eq (1.1) is almost periodic Specifically, i

o(A) = @, i.e, all agenvalues of A are not on the unit circle with center (-1,0) then for every bounded solution to Eq (1.1) we have sp(x) C sp(f) C 2nZ/p

Therefore, it is periodic with periodic tT = p

3.3 The existence of periodic solutions to Eq (1.1)

In this subsection we will prove the Massera criterion for periodic solutions of Eq (1.1), where f is periodic with rational period In the case where f is periodic wit! irrational period, we will give a counter-example asserting that the Massera criterion

does not hold true

Example 3.8 Consider the equation

whereae C",a #0)

Obvously, f(t) := ea is periodic with period 27 We are going to prove that Eq

(3 1) has no periodic solutions In fact, suppose that it has a periodic solution x(t), then it follows from Theorem 3.2 that

Dy tsing Fourier-Carlemanu transform, in the case of Red > 0 we have

f(r) = / e™ F(t)dt = | eo eSadt

Z8 k+l

k=0 |

Le Naik

A

k=0

Similarly, in the case of ReA < 0 we also have

f(A) = - J e*ƒ(~t)dt = — i eteil-ladt

= ae k=0 J at :

= eS ` ae

k=0

À (1+ 1= a)

Heace,

““ 1 - a

f(A) = ~~ » ——— for all Red # 0

Obviously, w(A) ss S— is holomorphie on C Moreover, if 1 - e = 0 then

w A) 4 0 This shows that f(A) has a holomorphic extension to a neighborhood of

i€, except at the set {€ € R} such that 1 -e' “= 0, or, {€:1-€ € 27Z} = {E €

272 + {1}} Therefore,

sp(f) = 2nZ-r {1}

Ôn the other hand, by Proposition 2.3 if z is periodic with period 7, then sp(7) C 2z2Z/r Hence,

2xZ + {1} C 2mZ/r U2nZ

Since 27Z + {1} 2x2 = Ú, we have 2zZ + {1} C 2zZ/r Hence, there exists a

number rn € Z such that 1 = 2zm/7 = r = 2mm Therefore, 2rZ + {1} C 2Z/m This is a contradiction showing that Eq (3.10) has no periodic solutions

To prove that the Massera criterion does not hold true, we consider to a simple

case of equation (3.10) with A = 0,

Thus,

x(t) = ea(t—n)+ a(n), Vt € [n,n + 1) (3.13)

Since z(t) is contionuous,

a(n +1) =2(n) + ea

Hence,

ne oo i(n+1)

z(n +1) = 2(0) + Svea = 2(0) + aS a Wn > 0

Since ||1 — cltn+ĐÌ| < 2 we have

lz(ø + 1)| < IIe(0)ll+ r— — fla — l|I — e'||

For n < 0 we also have the boundedness of the sequence {z(n)}„«o So {z(7)}:nez k

bounded It follows from (3.13) that z(t) is bounded So Eq (3.12) has a sonndec

solution However, as shown above, it has not any periodic solution

In the case where f is a periodic function with rational period Supposse thai

z(-) is a solution to Eq (1.1) Then

x(t) = [Ar(n) + f(n)|(t — n) + x(n), Vt € [n,n +1), Vn € Z

Together with (1.1) we deal with the difference equation

a(n +1) = (A+ I)z(n) + f(n), n € Z (3.14

Obviously, the existence of bounded solutions to equation (1.1) is equivaleit to th y

existence of bounded solutions to equation (3.14) We deal with Eq (3.1)) bì th