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

-MASSERA CRITERION DIFFERENTIAL FOR EQUATIONS PERIODIC WITH SOLUTIONS PIECEWISE OF 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 Introduction In this note, we are concernded with differential equations with piecewise constant argument of the form (1.1) # = Ar(H) + ƒ(f]), z) € C", 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 where A, b= Alt)e(t) + f(t), f is continuous, periodic with the same period 7, has a periodic solution with petiod 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 124 [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) Preliminaries In this section we recall the notion of a spectrum of a bounded function and som: important properties 2.1 For more details we refer the reader to [9, 19] 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), for the spectrum, ø(4) and R(A, A) stan 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 125 2.2 The spectruun of a bounded In this note, we will function 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 FourierCarlemann transform of u #(A) := [eutdt — fr ReA >0, eMu(—t)dt if Rea = 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 (i) (/19, p 20]) Letu, v€ BM(R,X),a€ C\ {0} Then 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 € sp(u) CA BƯƠ(R X), ty converge to u uniformly and sp(u,) C A, then 126 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) odic with period r if and only if sp(f) C 2nZ/r Then f is peri- Recall that a subset E Cc R is said to be relatively dense if there exists a number > (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 ¢ > 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) (i) Then 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 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” it is continuous on R is said to be a solution of Eq differentiable on R except at most Eq (1.1) on every inteval [njn + 1),n € 2, where right one (1.1) af of integers and satisfies att == n the derivative of a ts the 3> 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 Vie R f(t) = f({t|), Theorem 3.2 Let x(-) is a@ bounded solution of Eq (1.1) Then the following estimates hold true: Proof We s9(ƒ) C sp(z)U 2x2, (3.1) sp(+)C (3.2) ơa(A) Ò sp() first consider the case of Re\ > By taking the Fourier-Carlemann transforms of functions and by Proposition 2.1 we have ~ #(2A) = À7(^) - z(0) (ou) Since x(-) is a solution to Eq (1.1), Ẩ(A) = (Az(]))(A) + (2) +œ = Ƒ£*(A() + f0} +00 +00 -fe MAri(e)aes fe ™ F(t) dt 0 +00 = afc At r((t})dt + f(A) (3.4) Set +00 g = k+1 ae (A) == / e*z([fÌ)dt == ` / eMa(k)dt = S7 fe Lo AE x k=0 k X ea (hy,/ : k=0 We have Vy Laval AK (k) “A E A l = cTA g(A) (3.5) 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 128 Hence, aa) = [ -Xz0)w = [ e*{r( + 1) = z()|Œ — Ít) + allt 0 — = f eMe(e+1) — alee tears fe zctehat e“[a(k + 1) — x(k)](t — k)dt + g(A) > oO iMe ! rr [x(k + 1) ~ 2( vf e*(t — k)dt + g(A) Oo Ly iM k+1 =3 r&+1)~z(9)-1e” ự — BI"+ s eˆS4+ ]g0) k=0 =Ÿ e+1)- ere xa = xc gee yy k=0 — œ-À an+ 9() c— x(k + 1) -Sie™* k=0 œ Sle *a(k)) +9(A) Ile Senta r(k+1)— " = — k)| + g( À) k=0 (36) From (3.5) and (3.6) we have Ife cm A 9(2) ~ "0 Ì~ 1—gz90)} + ø0) Y A) ={ — e7* — Ae~* = Ir = e—1 + 1}g(A) + 1+A- ©" 2(0) 9) + A+1-e = *20) Thus XN Ww g(A) = 2122) — A+i—e —~ (0)] (3.7 From (3.3), (3.4) and (3.7) we have À ^ A2(A) - z(0)= A-r—rlf(A) Therefore, À+Íl1—t _— x pee) + fd) - 129 In the case of ReA < 0, by computating, as abow, we wes the same result So, in all cases the following estunate holds true: \ c^ (cˆ | Where - Ayr) P(A) ~ + Ạ ¬ ` |r FO) (3.8) ¬" p(A)= ————Az(0) + ——+(0) Since êÀ —À=~=T at1 nee a 2181 cÀ — ] ome Le eh l A ro Eee boss 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“* — # 0, so we have À ek] [(e* — ~ 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 — ¢ 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 â a(A) if and only of e& € + o( A) unit circle we have eel As the set [L + o(A)) OL tal(A)ee™® € (14 af A) ONL ts finite, o4( A) 1s discrete In fact, since e on the 130 3.2 we have +00 te