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
1 Introduction
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),
Trang 2[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)
2 Preliminaries
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» -
Trang 32.2 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
[eutdt ReA >0,
#(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
Trang 42.3 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)
Trang 53.4 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:
s9(ƒ) C sp(z)U 2x2, (3.1)
sp(+) C ơa(A) Ò sp() (3.2)
Trang 6Hence, aa) = [ -Xz0)w = [ e*{r( + 1) = z()|Œ — Ít) + allt 0 0 = f eMe(e+ 1) — alee tears fe zctehat 1 e“[a(k + 1) — x(k)](t — k)dt + g(A) > i! Me oO — rr k+1 [x(k + 1) ~ 2( vf e*(t — k)dt + g(A) iM Oo Ly =3 r&+1)~z(9)-1e” ự — BI" + s eˆS4] k=0 + g0) =Ÿ e+1)- ere an + 9() xa xc yy c— = gee x(k + 1) -Sie™* k)| + g( À) k=0 k=0 — œ-À œ
= — " Ile Senta r(k+1)— Sle *a(k)) +9(A) (36) k=0
Trang 7In the case of ReA < 0, by computating, as abow, we wes the same result So, in all cases the following estunate holds true: \ c^ ~ |r (cˆ | - Ayr) P(A) + FO) (3.8) Where Ạ ¬ ` ¬" p(A)= ————Az(0) + ——+(0) Since êÀ —À=~=T 1 1 nee at Eee a 2181 cÀ — ] ome Le eh boss l A ro
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 eđ 1 â a(A) if and only of e& € 1 + o( A) In fact, since e on the
unit circle 1 we have
Trang 83.2 <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|| li | / [Az(f) + /()]#i < | / 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
Trang 9Proposition 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 + oo +00 JA)= | *f0w= [ */t)e 0 0 ` k+1 = 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 = 1 = ƒ(r)c* x r=0 k=0 te n6 cố
sarge | pte 2 f(rje
Trang 101=¢ SỐ = EEE ryan som r=0 k:=0 I-e*^ 1 & “~ TỶ 1= /{r~ Đet9 r=0 1 sai e> 1 p-1 À =i = À gà l—e^» = TỰ = r— lỤe (—p+r+1) 1—e*^ i & as ore Lie r=0 Thus, ^ 1 — e> 1 p-1 FO) = — y2 „ƒ(r)e"*" for all Red # 0 (3.9) 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
¿ = Az(fl) + củủa, (3.10
Trang 11Obvously, 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 %)(ƒ) C sp(z)LI 2x2 (3.11) Dy tsing Fourier-Carlemanu transform, in the case of Red > 0 we have +00 te<© f(r) = / e™ F(t)dt = | eo eSadt 0 0 Z8 k+l = we fat k=0 | Le Naik = ae’ A k=0 1-e* a À I-er^ Similarly, in the case of ReA < 0 we also have +œ +00 f(A) = - J e*ƒ(~t)dt = — i eteil-ladt 0 0 a k+l = ae k=0 J at : le (A~i)(k+1) = eS ` ae k=0 {+e a À (1+ 1= a) ¬ e a wư 1-e Heace, ““ 1 - a f(A) = ~~ » ——— for all Red # 0 À 1 ame er À
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,
Trang 12Ôn 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, ¢ = ella (3.12) 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 rar l1-e 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
Trang 13ease A= 0 and if equation (3.14) has a periodic solution with period p then equation (1.1) also has a periodic solution with the same period It follows from
Theorem 3.4 in [16] that if {f(n}} is a periodic sequence with period 7, then Eq (3 14) has a periodic solution 7 if it has a bounded solution Therfore, if f is periodic
with rational period 7 =; a then by Proposition 3.6 we have the sequence {f(n)} is periodic with period p So we have the following result
Theorem 3.9 Suppose thut { is a pernodic function with rational period tT = ee Then equation (1.1) has a priodic solution with period + = p if and only if it has a
bounded solution
Remark 3.10 Jn the statement of the above theorem, the boundedness means that u(t) ts defined and bounded on the whole real line A less restricted condition on the boundedness on the positive half line as in the usual Massera criterion may not yield the existence of a bounded solution on the whole line
Acknowledegement The author is grateful to Nguyen Van Minh for pointing out
the problem and his constant interest in the work
References
[1] S N Chow and J K Hale, Strongly limit-compact maps Funke Ekvac., 17
(1974), 31-38
|2| D Daners and P K Medina,” Abstract Evolution Equations, Periodic Problems
and Applications’ , Pitman Research Notes in Math Ser volume 279, Longman, New York, 1992
13] A.M Fink, ” Almost periodic Differential Equation” , Springer-Verlag, 1974
4) T Furumochi, T Naito and Nguyen Van Minh, Boundedness and almost pe-
riodicity of solutions of partial fictional! differential equations, J Differential
Equations 180 (2002), 125-152
(51 Y Hino and S Murakami, Periodic sohitions of a linear Volterra system Dif- ferential equations (Xanthi, 1987), 319 326, Lecture Notes in Pure and Appl
Trang 14[6] [7] l8] [9] [10] [11] [12] [13] [14] [16]
Y Hino, T Naito, Nguyen Van Minh, Jong Son Shin, Almost periodic solu- tions of Differential Equations in Banach spaces , Taylor and Francis, |.oadon- NewYork, 2002
Z 5 Hu and A B Mingarelli, On a question in the theory of almost periodic differential equations, Proc Amer Math Soc., 127 (1999), 2665-2670
R Johnson, A linear almost periodic equation with an almost automorphic
solution, Proc Amer Math Soc 82 (1981), no 2, 199-205
Y Katznelson, ” An Introduction to Harmonic Analysis” , Dover Publications, New York , 1968
B.M Levitan, V.V Zhikov, ” Almost Periodic Functions and Differential Equa- tions”, Moscow Univ Publ House 1978 English translation by Cambridge University Press, 1982
Y Li, Z Lin and Z Li, A Massera type criterion for linear functional differen-
tial equations with advanced and delay, Journal of Mathematical Analysis and
Applications, 200 (1996), 715-725
Y Li, F Cong, Z Lin and W Liu, Periodic solutions for evolution equations, Nonlinear Anal., 36 (1999), 275-293
J L Massera, The existence of periodic solutions of systems of differential
equations, Duke Math J., 17 (1952), 457-475
S Murakami T Naito, Nguyen Van Minh, Evolution Semigroups and Sums
of Commuting Operators: A New Approach to the Admissibility Theory of
Function Spaces, Journal of Differential Equations 164 (2000), 240-285 S Murakimi, T Naito, and Nguyen Van Minh , Massera theorem for almost periodicity of Solutions of Functional Differential Equations, Journal of Math- ematical Society of Japan To appear
T Naito, Nguyen Van Minh, R Miyazaki, Y Hamaya, Boundedness and al- most periodicity in dynamical systems, Journal of Difference Equations anv Applications, 7 (2001), 507-527
Trang 15odic Solution of Evolution Equatious, Studia Mathematica 145 (2001), 97-111 J Priss, ” Buolutionary Integral Equations and Applications”, Birkhauser,
Basel, 1993
—G Seifert, Almost solutions of certaint Differential Equations with piecewise constant delays and almost periodic time dependence, Journal of Differential Equations 164, (2001), 451-458
| J H Shen and [ P Stavroulakis, Oscillatory and nonoscillatory delay equa- tion with piecewise constant argument, Journal of Mathematical Analysis and Applications, 248 (2000), 385-401
J S Shin and 'T Naito, Semi-Fredholm operators and periodic solutions for linear functional differential equations, Journal of Differential Equations, 153
(1999), 407-441