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On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument
Nonlinear Analysis 59 (2004) 1189 – 1205 www.elsevier.com/locate/na
On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument夡 Rong Yuan Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China Received 10 April 2003; accepted 21 July 2004
Abstract We study the spectrum containment of almost periodic solution to second order differential equations with piecewise constant argument. Some known (periodic, quasi-periodic) results would be expanded. As a corollary, it is shown that such equations with periodic perturbations possess a quasiperiodic solution and no periodic solution. This phenomena is due to the piecewise constant argument. The results are extended to nonlinear equations. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 34K14; 34K13 Keywords: Almost periodic solution; Spectrum; Module containment; Delay equations; Piecewise constant argument
1. Introduction Differential equations with piecewise constant argument (EPCA, for short) have received extensive investigations (see, e.g., [1,3,8–15] and references therein). In these equations, the derivatives of the unknown functions depend on not just the time t at which they are determined, but on constant values of the unknown functions in certain intervals of the time t before t. These equations combine the properties of differential equations and difference 夡 This
work was supported by NSFC(10371010) and RFDP. E-mail address: [email protected] (R. Yuan).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.07.031
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
equations, which are the properties of hybrid systems. The hybrid dynamical systems have widely been studied (see [16] and its references therewith). Some of these systems may be described by EPCA. The present paper deals with the following differential equation of the form x (t) + p(t)x(t) = qx([t]) + f (t),
(1)
where p : R → R, t → p(t), is 1-periodic and continuous, q(= 0) is a real constant, [·] is the greatest integer function, and f : R → R is a continuous function. Eq. (1) under f (t)=0 has been discussed by Wiener and Lakshmikantham in [11]. Let AP be the set of almost periodic functions and PT be the set of T -periodic functions. The existence of almost periodic solution and quasi-periodic solution of Eq. (1) has been shown in [14]. Furthermore, if f ∈ PT and T is rational, the existence of periodic solution for Eq. (1) has been shown in [14]. The main purpose of this paper is to investigate the spectrum containment of almost periodic solution to the second-order differential equations with piecewise constant argument (1). It should be explained that there is no discussion about the spectrum or module containment of almost periodic solutions of Eq. (1). The main question we address in the present paper is what is the spectrum or module of almost periodic solution for Eq. (1) if f ∈ AP. If g(t) is another almost periodic function then the module containment property mod(g) ⊂ mod(f ) can be characterized in several ways (see [4–6]). For periodic functions this inclusion just means that the minimal period of g(t) is a multiple of the minimal period of f (t). Therefore the discussion about module containment is very important. For doing those, we will discuss the Bohr spectrum for almost periodic sequence and compare the spectrum’s relations between almost periodic sequence (solution) and almost periodic function (solution). We also note that there was a discussion about the spectrum containment of almost periodic solution to ordinary differential equations in [2], but no paper to EPCA. Definition 1 (Wiener and Lakshmikantham [10,11]). A solution of Eq. (1) on R is a function x(t) that satisfies the conditions: (i) x(t) is continuously differentiable on R; (ii) x (t) exists at each point t ∈ R, with the possible exception of the points [t] ∈ R where it has one-sided limits; (iii) Eq. (1) is satisfied on each interval [n, n + 1) with integral endpoints. 2. Elements of almost periodic function and sequence Definition 2 (Corduneanu [4], Fink [5], Levitan and Zhikov [6]). A function f : R → R is called an almost periodic function introduced by Bohr, if it is continuous and for any > 0, the -translation set of f T (f, ) = { ∈ R; |f (t + ) − f (t)| < , t ∈ R} is a relative dense set on R.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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It is well-known that for every almost periodic function f (t), T (f, )nZ is relatively dense (see [7]) and the mean value T +a T 1 1 Mt {f } = lim f (t) dt f (t) dt = lim T →∞ 2T −T T →∞ 2T −T +a exists uniformly with respect to a. Let {j } denote the set of all real numbers such that T 1 a(j ; f ) := lim f (t) exp(−i j t) dt = 0. T →∞ 2T −T It is well-known that the set of numbers {j } in the above formula is countable. With each almost periodic function f we associate the Fourier series ak exp(i k t), f (t) ∼ k
where ak = a(k ; f ). The elements ak are called the Fourier coefficients and the numbers {k } the Fourier exponents of f . Denoted by f the set of all Fourier exponents {k } of f , which is called the spectrum of f . The set { N 1 nj j } for all integers N and integers nj is called the module of f (t), denoted by mod(f ), which is the least additive subgroup of the integral numbers containing the Fourier exponents of f (t). A finite or countable set of real numbers 1 , 2 , . . . , n , . . . is said to be rationally independent, if the equality r1 1 + r2 2 + · · · + rn n = 0 (r1 , . . . , rn are rational and n is an arbitrary natural number) implies that all of r1 , r2 , . . . , rn are zero. A finite or countable set of rationally independent real numbers 1 , 2 , . . . , n , . . . is called a rational basis of a countable set of real numbers 1 , 2 , . . . , n , . . . if every n is representable as a finite linear combination of the j with rational coefficients, that is,
n = r1(n) 1 + r2(n) 2 + · · · + rm(n)k mk
(n = 1, 2, . . .),
(n)
(n)
where the rj are rational numbers. If all the rj are integers, then the basis is called an integer basis. If a basis consists of a finite number of terms, then it is called a finite basis. When a basis in mod(f ) is both integer and finite, f is called to be quasi-periodic. If we denote by P, QP, AP the set of periodic functions, quasi-periodic functions, almost periodic functions, respectively, it is well-known that the following relation holds:
P ⊂ QP ⊂ AP, and every element of the module for QP is linear combinations with integer coefficients of only finitely many frequencies 1 , . . . , r , whereas in Bohr’s theory any denumerable set of real numbers is admitted for the frequencies. Set i2m,t ¯ QP() = f (t) = fm e |fm | < + ∞, f−m = fm . m
m
It is easy to see that every function in QP() is quasi-periodic.
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Definition 3 (Corduneanu [4], Fink [5]). A sequence g := {g(n)}n∈Z is called an almost periodic sequence, if for any > 0, there exists an l = l() ∈ N with the property that for any a ∈ Z there is a ∈ Z satisfying a < a + l and |g(n + ) − g(n)| < , ∀n ∈ Z. Suppose that g is an almost periodic sequence. Then the continuous function G : R → R defined by G(n) = g(n), n ∈ Z, and G(n + s) = g(n) + s[g(n + 1) − g(n)] for s ∈ [0, 1] is an almost periodic function (see [4,5] ). By the approximation theorem (see [4–6]) for almost periodic functions, G can be approximated by some trigonometric polynomials Pk (t), k = 1, 2, . . . , of the form Pk (t) = nk x exp(i k,m t), where xk,m ∈ C and k,m ∈ R, such that m=1 k,m 1 sup |G(t) − Pk (t)| < , k t∈R
k = 1, 2, . . . .
k n − g(n)|) = 0, where z xk,m zk,m In particular, it follows that limk→∞ (supn∈Z | nm=1 k,m := exp(ik,m ), which shows that f is a uniform limit of sequences of the form: finite m=1
n xm z m ,
xm ∈ C, zm ∈ S 1 := {z ∈ C; |z| = 1}.
It follows from [4] that the limit: a(z; g) = lim
N→∞
N 1 −k 1 z g(k) = lim N→∞ 2N 2N k=−N
N−m
z−k g(k)
k=−N−m
exists uniformly for m ∈ Z when z ∈ S 1 . a(z; g) is called the Bohr transform of g. In case that g is trigonometric polynomial, that is, g is of the following form: g(n) =
p k=1
xk zkn ,
xk ∈ C, zk ∈ S 1 (zk = zl (k = l))
it is easy to check that a(z; g) = xk if z = zk for some k = 1, 2, . . . , p, and a(z; g) = 0 if z = zk for any k = 1, . . . , p. We denote by b (g) the set which consists of all numbers z ∈ S 1 satisfying a(z; g) = 0, and call b (g) the Bohr spectrum of g. It is easy to see that b (g) is at most countable. With each almost periodic sequence {g(n)} we also associate the Fourier series xk zkn , zk ∈ S 1 , g(n) ∼ k
where xk = a(zk ; g). The elements xk are called the Fourier coefficients and the numbers zk the Fourier exponents of {g(n)}. We can formulate the approximation theorem for an almost periodic sequence as follows.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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Lemma 1. Suppose that g is an almost periodic sequence. For any > 0 there is a sequence of approximating polynomials P (n) of the form P (n) =
n k=1
n xk, zk, ,
zk, ∈ b (g)
such that sup |g() − P ()| < , ∈Z
where xk, is the product of a(zk, ; g) and certain positive number. Proof. This Lemma can be established by modifying the arguments in [6, pp. 14–21]. We omit its proof. It follows from this Lemma that if f and g are almost periodic sequences and a(z; f ) ≡ a(z; g), then f ≡ g. 3. Statement of main theorems Suppose that u(t) and v(t) are the solutions of system x (t) + p(t)x(t) = 0
(2)
satisfying the conditions u(0) = 1, u (0) = 0
and v(0) = 0, v (0) = 1, u(t) v(t) = 1, for t ∈ R by using respectively. The Wronskian w(t) = w[u, v](t) = u (t) v (t) the Liouville’s formula. Since the concept of almost periodic functions is in terms of functions on R, it is suggested that one should find a solution of Eq. (1) on R. If x(t) is a solution of Eq. (1) on R, then x(t) satisfies the following relations: x (t) + p(t)x(t) = qcn + f (t),
n t < n + 1,
(3)
where cn = x(n). The general solution of the Eq. (3) on the interval n t < n + 1 can be written as xn (t) = An (t)u(t − n) + Bn (t)v(t − n). The variation of constants method leads to the system An (t)u(t − n) + Bn (t)v(t − n) = 0, An (t)u (t − n) + Bn (t)v (t − n) = qcn + f (t),
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whence
An (t) = −qcn · Bn (t) = qcn ·
t n
t
v(s − n) ds −
u(s − n) ds +
n
n t
n
t
f (s)v(s − n) ds + cn ,
f (s)u(s − n) ds + dn ,
where n t < n + 1 and dn = x (n). If we set u(t) v(t) W (t) = t t , 0 u(s) ds 0 v(s) ds u(t) v(t) t Un (t) = t 0 f (s + n)u(s) ds 0 f (s + n)v(s) ds
(4)
it follows that x(t) could be written as x(t) = cn u(t − n) + dn v(t − n) − qcn W (t − n) − Un (t − n),
n t < n + 1 (5)
and x (t) = cn u (t − n) + dn v (t − n) − qcn W (t − n) − Un (t − n), n t < n + 1.
(6)
At t = n + 1, we will arrive at cn+1 = (u(1) − qW (1))cn + dn v(1) − Un (1),
(7)
dn+1 = (u (1) − qW (1))cn + dn v (1) − Un (1).
(8)
We can eliminate dn and dn+1 from (7) and (8) and derive the equation cn+2 − (u(1) + v (1) − qW (1))cn+1 + [1 + q(v(1)W (1) − v (1)W (1))]cn = Un (1)v (1) − v(1)Un (1) − Un+1 (1).
(9)
It is easy to calculate v(1)W (1) − v (1)W (1) = −
1
v(s) ds,
0
Un (1)v
(1) − v(1)Un (1) =
1
f (s + n)v(s) ds.
0
Thus, Eq. (9) can be rewritten as
cn+2 − (u(1) + v (1) − qW (1))cn+1 + 1 − q 1 = f (s + n)v(s) ds − Un+1 (1)hn . 0
1 0
v(s) ds cn (10)
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
The corresponding homogeneous equation of Eq. (10) is 1 cn+2 − (u(1) + v (1) − qW (1))cn+1 + 1 − q v(s) ds cn = 0.
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(11)
0
Following [3], we can seek the particular solutions as cn = n for homogeneous difference equation (11). will satisfy the following equation: 1 2 − (u(1) + v (1) − qW (1)) + 1 − q v(s) ds = 0 (12) 0
that has two nontrivial solutions if 1 q v(s) ds = 1. 0
Now, we are in a position to formulate our main theorems. 1 Theorem 1. Suppose that q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. Then for any almost periodic function f (t), Eq. (1) possesses a unique almost periodic solution x(t). Furthermore, we have the following spectrum relation x = f + {2k|k ∈ Z}. Remark 1. The existence and uniqueness of almost periodic solution of Eq. (1) have been proved in [14] under the conditions of Theorem 1. It should be pointed out that no any explanation is given to spectrum and module in [14]. The study of the spectrum relation is the motivation of this paper. Corollary 1. Assume that 1, 1 , . . . , r are rationally independent. Suppose that 1 q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. Then for any f (t) ∈ QP(), Eq. (1) possesses a unique quasi-periodic solution x(t) with frequencies (1, ). Indeed, f = {2k1 1 + · · · + 2kr r |∀kj ∈ Z, 1 j r} is spanned by 21 , 22 , . . . , 2r with integer coefficients. 1 Corollary 2. Suppose that q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. If f (t) is T -periodic, then the following results hold: (1) If T = n0 , n0 ∈ Z+ , then Eq. (1) possesses a unique T -periodic solution; n0 , n0 , m0 ∈ Z+ , n0 and m0 are mutually prime, then Eq. (1) possesses a unique (2) If T = m 0 m0 T -periodic solution. (3) If T is irrational and f (t) ∈ PT , then Eq. (1) possesses a unique quasi-periodic solution with frequencies (1, T1 ), where i 2m t ¯ for T > 0. PT = f (t) = fm e T |fm | < + ∞, f−m = fm m
m
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Remark 2. The conclusion of corollary 1 is given in [14]. Thus, the present paper would improve the main results in [14]. Corollary 2 shows that EPCA with periodic perturbations could possess a quasi-periodic solution and no periodic solution. The appearance of quasiperiodic rather than periodic solutions is due to the piecewise constant argument. This new phenomenon illustrates a crucial difference between ODE and EPCA. Example 1. We consider the perturbation of the Mathieu equation with piecewise constants argument x (t) + (2 + 2 cos 2t)x(t) = qx([t]) + f (t).
(13)
Let u(t) = u(t; , ) and v(t) = v(t; , ) be the fundamental solutions of Mathieu equation x (t) + (2 + 2 cos 2t)x(t) = 0
(14)
satisfying the initial values u(0) = 1, u (0) = 0 and v(0) = 0, v (0) = 1, respectively. If = 0, Eq. (14) deduces to x (t) + 2 x(t) = 0
(15)
which has fundamental solutions u0 (t) = cos t and v0 (t) = 1 sin t, satisfying the initial values u0 (0)=1, u0 (0)=0 and v0 (0)=0, v0 (0)=1, respectively. The continuous dependence on parameters implies u(t) = cos t + o(), v(t) = 1 sin t + o(), and v (t) = cos t + o() as → 0, on 0 t 1. The characteristic Eq. (12) deduces to (1 − cos )q (1 − cos )q 2 − 2 cos + + o( ) + 1 − + o( ) = 0. (16) 2 2 If = 0, (16) deduces to (1 − cos )q (1 − cos )q =0 2 − 2 cos + + 1 − 2 2 which has nontrivial solutions if q = 0,
= k ,
(1−cos )q
2
(17)
= 1. It can be shown that if
q = 2
(18)
then the algebraic equation (17) does not possess roots of unity (see [14] for details). The roots of Eq. (16) are continuous on . So, there exists 0 > 0, such that if 0 || 0 , the norm of all roots of Eq. (16) is not equal to one. Therefore, if conditions (18) hold, there exists 0 > 0, such that if 0 || 0 , Eq. (13) possesses a unique almost periodic solution x(t) for any almost periodic function f (t) and x = f + {2k |k ∈ Z}. Note that Example 1 has been discussed in [14] except the spectrum that is a new result. In the present paper, we will also consider the following quasilinear differential equations with piecewise constant argument x (t) + p(t)x(t) = qx([t]) + f (t) + g(t, xt ),
(19)
where f : R1 → R1 is almost periodic, g : R1 × C → R1 is almost periodic in t uniformly on C, where C = C([−1, 0], R1 ). We define xt ∈ C as xt ( ) = x(t + ), −1 0.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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1 Theorem 2. Suppose that q 0 v(s) ds = 1. Assume that the norm of all roots of Eq. (12) is not equal to one. Then there exists a ∗ > 0, such that when 0 < | | < ∗ , Eq. (19) has a unique determined almost periodic solution x(t, ) with the module containment x ⊂ mod(f, g) + {2k}. The existence of almost periodic solution in Theorem 2 has been shown in [14]. But, the module containment is not discussed in [14]. Thus, Theorem 2 is an improvement of the results in [14].
4. Proofs of Theorems We begin with a lemma. Lemma 2. If u(t), v(t) are fundamental solutions of Eq. (2) and f : R → R is an almost (1) (2) periodic function with spectrum f , then the sequences {hn } and {hn } are almost periodic sequences with the Bohr spectrums b (h(1) ) = eif and b (h(1) ) = eif , where
1 } = f (n + s)u(s) ds , {h(1) n n∈Z n∈Z
0
{h(2) n }n∈Z =
1
f (n + s)v(s) ds
0
n∈Z
.
(20) (1)
(1)
Proof. Without loss of generality, it suffices to consider hn . The almost periodicity of {hn } has been proved in [14]. Since f (t) is an almost periodic function, it follows from the approxk pk,m eim t , m ∈ f imation theorem (see [4–6]) that for any k > 0 there exists Pk (t)= nm=1 such that 1 |f (t) − Pk (t)| < , k
k = 1, 2, . . . ,
where we can assume that pk,m eim t and p¯ k,m e−im t appear together in the trigonometric polynomial Pk (t). Define 1 1 (1) (2) Pk (n + s)u(s) ds, Qk (n) := Pk (n + s)v(s) ds, Qk (n) := 0
∀n ∈ Z, k = 1, . . . . (1)
A direct calculation shows Qk (n) =
0
(21) 1 (1) im n (1) , where qk,m = pk,m 0 eim s u(s) ds, m=1 qk,m e
nk
(1)
which is the Fourier coefficient corresponding to the Fourier exponent eim . Qk (n) is a (1) trigonometric polynomial, b (Qk ) = eiPk ⊂ eif := {eim t , m ∈ f }. It is easy to see that 1 (1) 0 |u(s)| ds sup |Qk (n) − h(1) | < , k = 1, 2, . . . . n k n∈Z
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
This implies that b (h(1) ) ⊂ eif . For any fixed j0 ∈ f , we can assume nk > j0 . So, we can rewrite nk
Pk (t) =
pk,m eim t + pk,j0 eij0 t .
m=1,m=j0
It is easy to see that a(j0 ; f ) = limk→∞ a(j0 ; Pk ) = limk→∞ pk,j0 . Thus, we can obtain 1 (1) (1) eij0 s u(s) ds. a(eij0 ; h(1) ) = lim a(eij0 ; Qk ) = lim qk,j0 = a(j0 ; f ) k→∞
It follows that e
ij0
∈ b
(h(1) ).
k→∞
Thus, b
(h(1) ) = eif .
0
Corollary 3. If f (t) is almost periodic and u, v are fundamental solutions of Eq. (2), then the sequence {Un (1)}n∈Z defined in (4) is almost periodic sequence with the Bohr spectrum b (Un (1)) ⊂ eif . Proof of Theorem 1. The existence and uniqueness of almost periodic solution of Eq. (1) have been proved in [14]. It is only needed to study the spectrum containment. Let 1 hn f (s + n)v(s) ds − Un+1 (1). (22) 0
We would like to distinguish several cases. Case 1: All roots of Eq. (12) are simple (Denoted by 1 , 2 , respectively). Without loss of generality, we assume that | 1 | < 1, | 2 | > 1. We define a sequence {cˆn } by n−(m+1) n−(m+1) 1 hm + k2∗ 2 hm , n ∈ Z , (23) cˆn = k1∗ m n−1
mn
where the constants k1∗ , k2∗ are defined by
k1 − k2 = 0, k1 1 − k2 2 = 1.
(24)
It is easy to know that the sequence {cˆn } satisfies the difference equation (10) and is an almost periodic sequence (e.g., see [14]). We define a sequence cˆn+1 − (u(1) − qW (1))cˆn + Un (1) dˆn v(1)
(25)
which is an almost periodic sequence. The above mentioned almost periodic sequences {cˆn } and {dˆn } satisfy Eqs. (7) and (8). For such almost periodic sequences {cˆn } and {dˆn } defined by (23) and (25), respectively, we can construct a solution x(t) of Eq. (1) by using (5) as x(t) = cˆn u(t − n) + dˆn v(t − n) − q cˆn W (t − n) − Un (t − n), n t < n + 1.
(26)
Since {cˆn } is a solution of difference equation (10), we know that the x(t) is continuous on R. It is easily shown that the solution x(t) defined by (26) is an almost periodic solution of Eq. (1) (e.g., see [14]).
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Set
(2) Qk (n)Qk (n) −
u(1) v(1) (1) (2) Qk (n + 1) Qk (n + 1)
(2)
(2)
(1)
= Qk (n) − u(1)Qk (n + 1) + v(1)Qk (n + 1) := (1)
nk
1199
qk,m eim n ,
(27)
m=1
(2)
where Qk (n) and Qk (n) are defined as in (21) and (1)
(2)
(2)
qk,m v(1)qk,m eim + qk,m − u(1)qk,m eim . It follows from the proof of Lemma 2 that sup |Qk (n) − hn | → 0
as k → ∞.
n∈Z
We define
ek (n) = k1∗
m n−1
n−(m+1) Qk (m) + k2∗ 1
mn
n−(m+1) Qk (m), 2
n ∈ Z,
where the constants k1∗ , k2∗ are defined by (24). A direct calculation shows that ek (n) =
nk
qk,m
m=1
where
Ek,m qk,m
k1∗
eim − 1
k1∗
eim − 1
+
+
k2∗
2 − eim
k2∗
2 − eim
eim n =
nk
Ek,m eim n ,
m=1
eim n .
It follows that ek (n) is a trigonometric polynomial and b (ek ) ⊂ eif . Setting C1 = |k1∗ |
1 1 + |k2∗ | , 1 − | 1 | | 2 | − 1
(28)
it is easy to get sup |ek (n) − cˆn | C1 sup |Qk (n) − hn | → 0,
n∈Z
n∈Z
(as k → ∞).
From this, we can conclude b (c) ˆ ⊂ b (h) = eif . For any fixed j0 ∈ f , we can assume nk > j0 . So, we can rewrite Qk (n) in (27) as Qk (n) =
nk
qk,m eim n + qk,j0 eij0 n .
m=1,m=j0
It is easy to see that a(eij0 ; h) = lim a(eij0 ; Qk ) = lim qk,j0 . k→∞
k→∞
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Thus, we obtain ˆ = lim a(eij0 ; ek ) = lim Ek,j0 a(eij0 ; c) k→∞ k→∞ 1 1 . + k2∗ = a(eij0 ; h) k1∗ i e j 0 − 1 2 − eij0 It follows that eij0 ∈ b (c). ˆ Therefore, b (c) ˆ = eif . Set u(t) v(t) , Vk,n (t) = t t P (s + n)u(s) ds P (s + n)v(s) ds 0 k 0 k
0 t 1,
and gk (n)
ek (n + 1) − (u(1) − qW (1))ek (n) + Vk,n (1) , v(1)
where Pk (t) is as in the proof of Lemma 2. We define xk (t) = ek (n)u(t − n) + gk (n)v(t − n) − qek (n)W (t − n) − Vk,n (t − n), n t < n + 1. From the definitions of ek and gk , we know that xk (t) is continuous on R, and satisfies the equation xk (t) + p(t)x(t) = qx([t]) + Pk (t) and |xk (t) − x(t)| → 0 as k → ∞ uniformly on R. For each fixed k, Pk (t) is almost periodic. From the construction, it is easy to know that xk (t) is almost periodic (e.g., see [14]). It can be calculated (see Appendix) that
T 1 = 0 if − = 2k , lim (29) xk (t)e−it dt = 0 if − = 2k . T →∞ 2T −T Since a(; x) = limk→∞ a(; xk ), it follows that if f = { }, then x = { } + {2j |j ∈ Z}. If | 1 | < 1, | 2 | < 1, we define a sequence {cn } by n−(m+1) hm + k 2 n−(m+1) hm . (30) cn = k1 1 2 m n−1
m n−1
If | 1 | > 1, | 2 | > 1, we define a sequence {cn } by n−(m+1) n−(m+1) 1 hm + k 2 2 hm . cn = k1 mn
(31)
mn
Similarly, we can uniquely determine a set of values (k1 , k2 ) such that cn in (30) or (31) is a solution of Eq. (10). The rest of proof in those cases is virtually the same as the proof in the case: | 1 | < 1, | 2 | > 1. Case 2: Eq. (12) possesses a double root. Denote it by 1 .
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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If | 1 | < 1, we can uniquely determine a set of values (k1∗ , k2∗ ) such that the sequence n−(m+1) cˆn = k1∗ n−(m+1) hm + k2∗ (n − (m + 1)) 1 hm , n ∈ Z 1 m n−1
m n−1
is a solution of the difference equation (10). If | 1 | > 1, we can obtain a solution of the difference equation (10) n−(m+1) n−(m+1) cˆn = k1∗ 1 hm + k2∗ (n − (m + 1)) 1 hm , mn
n ∈ Z.
mn
The rest of proof in this case is virtually the same as the proof in Case 1.
Proof of Theorem 2. Set M = mod(f, g) + {2k } and X = { ∈ AP|mod( ) ⊂ M}. X is a Banach space equipped with supremum norm | | = supt∈R | (t)|. For any ∈ X, the following equation x (t) + p(t)x(t) = qx([t]) + f (t) + g(t, t ) has a unique almost periodic solution T by using Theorem 1 and T ⊂ f (t)+ g(t, t ) + {2k } ⊂ M. Thus, mod(T ) ⊂ M. It can be proved that T : X → X is a contracting mapping (e.g., see [14] for details). 5. A second-order delay differential equation In this section, we consider the following differential equations of the form x (t) + p(t)x(t) = qx([t − 1]) + f (t),
(32)
where p : R → R, t → p(t), is 1-periodic and continuous, q(= 0) is a real constant, [·] is the greatest integer function, and f : R → R is a continuous function. Since the concept of almost periodic functions is in terms of functions on R, one should find a solution of Eq. (32) on R. If x(t) is a solution of Eq. (32) on R, then x(t) satisfies the following relations: x (t) + p(t)x(t) = qcn−1 + f (t),
n t < n + 1,
(33)
where cn =x(n). The general solution of Eq. (32) on the interval n t < n+1 can be written as xn (t) = An (t)u(t − n) + Bn (t)v(t − n), where u(t) and v(t) are the solutions of System (2) satisfying the conditions u(0) = 1, u (0) = 0
and v(0) = 0, v (0) = 1,
respectively. The variation of constants method leads to t t An (t) = −qcn−1 · v(s − n) ds − f (s)v(s − n) ds + cn , n
n
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Bn (t) = qcn−1 ·
t n
u(s − n) ds +
t n
f (s)u(s − n) ds + dn ,
where n t < n + 1 and dn = x (n). By using the notations in (4), it follows that x(t) could be written as x(t) = cn u(t − n) + dn v(t − n) − qcn−1 W (t − n) − Un (t − n), n t < n + 1,
(34)
x (t) = cn u (t − n) + dn v (t − n) − qcn−1 W (t − n) − Un (t − n), n t < n + 1.
(35)
If set t = n + 1, we will arrive at the following difference equations:
cn+1 = cn u(1) + dn v(1) − qcn−1 W (1) − Un (1), dn+1 = cn u (1) + dn v (1) − qcn−1 W (1) − Un (1).
(36)
We can eliminate dn and dn+1 from (36) and derive the equation 1 v(s) ds cn+2 − (u(1) + v (1))cn+1 + (1 + qW (1))cn − qcn−1 0 1 f (s + n)v(s) ds − Un+1 (1)hn . =
(37)
The corresponding characteristic equation of Eq. (37) is 1 3 − (u(1) + v (1))2 + (1 + qW (1)) − q v(s) ds = 0
(38)
0
0
that has three nontrivial solutions if 1 v(s) ds = 0. 0
Similar to Theorem 1, we have the following results to Eq. (32). 1 Theorem 3. Suppose that q = 0, 0 v(s) ds = 0, and that the norm of all roots of Eq. (38) is not equal to one. Then for any almost periodic function f (t), Eq. (32) possesses a unique almost periodic solution x(t) and x = f + {2k |k ∈ Z}. Remark 3. The existence of almost periodic solution has been shown in [14], but no explanation is given to the spectrum or module of almost periodic solution in [14]. So, the present paper sharpens the corresponding results in [14]. Corollary 4. Assume that 1, 1 , . . . , r are rationally independent. Suppose that q = 1 0, 0 v(s) ds = 1, and that the norm of all roots of Eq. (38) is not equal to one. Then for any f (t) ∈ QP(), Eq. (32) possesses a unique quasi-periodic solution x(t) with frequencies (1, ).
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1 Corollary 5. Suppose that q = 0, 0 v(s) ds = 1, and that the norm of all roots of Eq. (38) is not equal to one. If f (t) is T -periodic, then the following results hold: (1) If T = n0 , n0 ∈ Z+ , then Eq. (32) possesses a T -periodic solution. n0 (2) If T = m , n0 , m0 ∈ Z+ , n0 and m0 are mutually prime, then Eq. (32) possesses a 0 m0 T -periodic solution. (3) If T is irrational and f (t) ∈ PT , then Eq. (32) possesses a unique quasi-periodic solution with frequencies (1, T1 ). Example 2. We consider the following equation: x (t) − a 2 x(t) = qx([t − 1]) + f (t)
(39)
which was discussed by Wiener and Lakshmikantham in [10] under f (t) = 0 and Yuan in [14]. The characteristic equation (38) deduces to the following form: q(cosh a − 1) q(cosh a − 1) 3 − 22 cosh a + 1 − − = 0. (40) 2 a a2 If a = 0, q = 0, and q = −a 2 ,
q = a 2
1 + 2 cosh a , cosh a − 1
(41)
the norm of all roots of Eq. (40) is not equal to one (e.g., see [14]). It follows from Theorem 3 that Eq. (39) possesses an almost periodic solution x(t) and x = f + {2k } for each almost periodic f (t), if conditions (41) hold. Example 3. We consider the following equation: x (t) + 2 x(t) = bx([t − 1]) + f (t)
(42)
which was discussed by Wiener in [9] under f (t) = 0 and Yuan in [13,14], respectively. The characteristic equation (38) of Eq. (42) deduces to (1 − cos )b (1 − cos )b 3 − 22 cos + 1 − − = 0. (43) 2 2 If = k (k ∈ Z) and b = 2 ,
b = 2
1 + 2 cos 1 − cos
(44)
the norm of all roots of Eq. (43) is not equal to one (e.g., see [14]). It follows from Theorem 3 that Eq. (42) possesses an almost periodic solution x(t) and x = f + {2k } for each almost periodic f (t), if conditions (44) hold. Note that the spectrum relations in Examples 2, 3 are our new results. Finally, we will consider the following quasilinear differential equations with piecewise constant argument x (t) + p(t)x(t) = qx([t − 1]) + f (t) + g(t, xt ),
(45)
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
where p : R → R, t → p(t), is 1-periodic and q(= 0) is a real constant, f : R1 → R1 is almost periodic, g : R1 × C → R1 is almost periodic in t uniformly on C, where C = C([−1, 0], R1 ). We define xt ∈ C as xt ( ) = x(t + ), −1 0. 1 Theorem 4. Suppose that q 0 v(s) ds = 1. Assume that the norm of all roots of Eq. (38) is not equal to one. Then there exists a ∗ > 0, such that when 0 < | | < ∗ , Eq. (45) has a unique determined almost periodic solution x(t, ) with the module containment mod(x) ⊂ mod(f, g) + {2k}. Appendix During the proof of Theorem 1, we meet with what is the exponents of xk (t) in (29). If we look at the formula of xk (t), we can find that xk (t) consists of the following term with the form s (t) = (t − n)ei n , n t < n + 1. We give the following calculation for completeness. We can calculate that m T 1 1 −it a(; s ) := lim s (t)e−it dt s (t)e dt = lim m→∞ 2m −m T →∞ 2T −T m−1 n+1 1 = lim (t − n)ei n e−it dt m→∞ 2m n=−m n m−1 1 i( −)n 1 = lim e (s)e−it dt m→∞ 2m 0 n=−m
0 if − = 2j , j ∈ Z, = 1 −i s ds = 0 if − = 2j , j ∈ Z. 0 (s)e References [1] S. Busenberg, K.L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, in: V. Lakshmikantham (Ed.), Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, pp. 179–187. [2] M.L. Cartwright, Almost periodic differential equations and almost periodic flows, J. Differ. Equations 5 (1969) 167–181. [3] K.L. Cooke, J. Wiener, A survey of differential equation with piecewise continuous argument, Lecture Notes in Mathematics, vol. 1475, Springer, Berlin, 1991, pp. 1–15. [4] C. Corduneanu, Almost Periodic Functions, Wiley, New York, London, Sydney, Toronto, 1968. [5] A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [6] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982. [7] G.H. Meisters, On almost periodic solutions of a class of differential equations, Proc. Amer. Math. Soc. 10 (1959) 113–119. [8] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. [9] J. Wiener, A second-order delay differential equation with multiple periodic solutions, J. Math. Anal. Appl. 229 (1999) 659–676.
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[10] J. Wiener, V. Lakshmikantham, Excitability of a second-order delay differential equation, Nonlinear Anal. 38 (1999) 1–11. [11] J. Wiener, V. Lakshmikantham, Complicated dynamics in a delay Klein–Gordon equation, Nonlinear Anal. 38 (1999) 75–85. [12] R. Yuan, On the existence of almost periodic solutions of a singularly perturbed differential equations with piecewise constant argument, Z. Angew. Math. Phys. 50 (1999) 94–119. [13] R. Yuan, On the regular solution of a second order delay differential equation, Z. Angew. Math. Phys. 53 (2002) 407–437. [14] R. Yuan, On the second-order differential equation with piecewise constant argument and almost periodic coefficients, Nonlinear Anal. 52 (2003) 1411–1440. [15] R. Yuan, J. Hong, Almost periodic solutions of differential equations with piecewise constant argument, Analysis 16 (1996) 171–180. [16] R. Yuan, Z. Jing, L. Chen, Uniform asymptotic stability of hybrid dynamical systems with delay, IEEE Trans. Automat. Contr. 48 (2) (2003) 344–348.
On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument夡 Rong Yuan Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China Received 10 April 2003; accepted 21 July 2004
Abstract We study the spectrum containment of almost periodic solution to second order differential equations with piecewise constant argument. Some known (periodic, quasi-periodic) results would be expanded. As a corollary, it is shown that such equations with periodic perturbations possess a quasiperiodic solution and no periodic solution. This phenomena is due to the piecewise constant argument. The results are extended to nonlinear equations. 䉷 2004 Elsevier Ltd. All rights reserved. MSC: 34K14; 34K13 Keywords: Almost periodic solution; Spectrum; Module containment; Delay equations; Piecewise constant argument
1. Introduction Differential equations with piecewise constant argument (EPCA, for short) have received extensive investigations (see, e.g., [1,3,8–15] and references therein). In these equations, the derivatives of the unknown functions depend on not just the time t at which they are determined, but on constant values of the unknown functions in certain intervals of the time t before t. These equations combine the properties of differential equations and difference 夡 This
work was supported by NSFC(10371010) and RFDP. E-mail address: [email protected] (R. Yuan).
0362-546X/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.07.031
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equations, which are the properties of hybrid systems. The hybrid dynamical systems have widely been studied (see [16] and its references therewith). Some of these systems may be described by EPCA. The present paper deals with the following differential equation of the form x (t) + p(t)x(t) = qx([t]) + f (t),
(1)
where p : R → R, t → p(t), is 1-periodic and continuous, q(= 0) is a real constant, [·] is the greatest integer function, and f : R → R is a continuous function. Eq. (1) under f (t)=0 has been discussed by Wiener and Lakshmikantham in [11]. Let AP be the set of almost periodic functions and PT be the set of T -periodic functions. The existence of almost periodic solution and quasi-periodic solution of Eq. (1) has been shown in [14]. Furthermore, if f ∈ PT and T is rational, the existence of periodic solution for Eq. (1) has been shown in [14]. The main purpose of this paper is to investigate the spectrum containment of almost periodic solution to the second-order differential equations with piecewise constant argument (1). It should be explained that there is no discussion about the spectrum or module containment of almost periodic solutions of Eq. (1). The main question we address in the present paper is what is the spectrum or module of almost periodic solution for Eq. (1) if f ∈ AP. If g(t) is another almost periodic function then the module containment property mod(g) ⊂ mod(f ) can be characterized in several ways (see [4–6]). For periodic functions this inclusion just means that the minimal period of g(t) is a multiple of the minimal period of f (t). Therefore the discussion about module containment is very important. For doing those, we will discuss the Bohr spectrum for almost periodic sequence and compare the spectrum’s relations between almost periodic sequence (solution) and almost periodic function (solution). We also note that there was a discussion about the spectrum containment of almost periodic solution to ordinary differential equations in [2], but no paper to EPCA. Definition 1 (Wiener and Lakshmikantham [10,11]). A solution of Eq. (1) on R is a function x(t) that satisfies the conditions: (i) x(t) is continuously differentiable on R; (ii) x (t) exists at each point t ∈ R, with the possible exception of the points [t] ∈ R where it has one-sided limits; (iii) Eq. (1) is satisfied on each interval [n, n + 1) with integral endpoints. 2. Elements of almost periodic function and sequence Definition 2 (Corduneanu [4], Fink [5], Levitan and Zhikov [6]). A function f : R → R is called an almost periodic function introduced by Bohr, if it is continuous and for any > 0, the -translation set of f T (f, ) = { ∈ R; |f (t + ) − f (t)| < , t ∈ R} is a relative dense set on R.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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It is well-known that for every almost periodic function f (t), T (f, )nZ is relatively dense (see [7]) and the mean value T +a T 1 1 Mt {f } = lim f (t) dt f (t) dt = lim T →∞ 2T −T T →∞ 2T −T +a exists uniformly with respect to a. Let {j } denote the set of all real numbers such that T 1 a(j ; f ) := lim f (t) exp(−i j t) dt = 0. T →∞ 2T −T It is well-known that the set of numbers {j } in the above formula is countable. With each almost periodic function f we associate the Fourier series ak exp(i k t), f (t) ∼ k
where ak = a(k ; f ). The elements ak are called the Fourier coefficients and the numbers {k } the Fourier exponents of f . Denoted by f the set of all Fourier exponents {k } of f , which is called the spectrum of f . The set { N 1 nj j } for all integers N and integers nj is called the module of f (t), denoted by mod(f ), which is the least additive subgroup of the integral numbers containing the Fourier exponents of f (t). A finite or countable set of real numbers 1 , 2 , . . . , n , . . . is said to be rationally independent, if the equality r1 1 + r2 2 + · · · + rn n = 0 (r1 , . . . , rn are rational and n is an arbitrary natural number) implies that all of r1 , r2 , . . . , rn are zero. A finite or countable set of rationally independent real numbers 1 , 2 , . . . , n , . . . is called a rational basis of a countable set of real numbers 1 , 2 , . . . , n , . . . if every n is representable as a finite linear combination of the j with rational coefficients, that is,
n = r1(n) 1 + r2(n) 2 + · · · + rm(n)k mk
(n = 1, 2, . . .),
(n)
(n)
where the rj are rational numbers. If all the rj are integers, then the basis is called an integer basis. If a basis consists of a finite number of terms, then it is called a finite basis. When a basis in mod(f ) is both integer and finite, f is called to be quasi-periodic. If we denote by P, QP, AP the set of periodic functions, quasi-periodic functions, almost periodic functions, respectively, it is well-known that the following relation holds:
P ⊂ QP ⊂ AP, and every element of the module for QP is linear combinations with integer coefficients of only finitely many frequencies 1 , . . . , r , whereas in Bohr’s theory any denumerable set of real numbers is admitted for the frequencies. Set i2m,t ¯ QP() = f (t) = fm e |fm | < + ∞, f−m = fm . m
m
It is easy to see that every function in QP() is quasi-periodic.
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Definition 3 (Corduneanu [4], Fink [5]). A sequence g := {g(n)}n∈Z is called an almost periodic sequence, if for any > 0, there exists an l = l() ∈ N with the property that for any a ∈ Z there is a ∈ Z satisfying a < a + l and |g(n + ) − g(n)| < , ∀n ∈ Z. Suppose that g is an almost periodic sequence. Then the continuous function G : R → R defined by G(n) = g(n), n ∈ Z, and G(n + s) = g(n) + s[g(n + 1) − g(n)] for s ∈ [0, 1] is an almost periodic function (see [4,5] ). By the approximation theorem (see [4–6]) for almost periodic functions, G can be approximated by some trigonometric polynomials Pk (t), k = 1, 2, . . . , of the form Pk (t) = nk x exp(i k,m t), where xk,m ∈ C and k,m ∈ R, such that m=1 k,m 1 sup |G(t) − Pk (t)| < , k t∈R
k = 1, 2, . . . .
k n − g(n)|) = 0, where z xk,m zk,m In particular, it follows that limk→∞ (supn∈Z | nm=1 k,m := exp(ik,m ), which shows that f is a uniform limit of sequences of the form: finite m=1
n xm z m ,
xm ∈ C, zm ∈ S 1 := {z ∈ C; |z| = 1}.
It follows from [4] that the limit: a(z; g) = lim
N→∞
N 1 −k 1 z g(k) = lim N→∞ 2N 2N k=−N
N−m
z−k g(k)
k=−N−m
exists uniformly for m ∈ Z when z ∈ S 1 . a(z; g) is called the Bohr transform of g. In case that g is trigonometric polynomial, that is, g is of the following form: g(n) =
p k=1
xk zkn ,
xk ∈ C, zk ∈ S 1 (zk = zl (k = l))
it is easy to check that a(z; g) = xk if z = zk for some k = 1, 2, . . . , p, and a(z; g) = 0 if z = zk for any k = 1, . . . , p. We denote by b (g) the set which consists of all numbers z ∈ S 1 satisfying a(z; g) = 0, and call b (g) the Bohr spectrum of g. It is easy to see that b (g) is at most countable. With each almost periodic sequence {g(n)} we also associate the Fourier series xk zkn , zk ∈ S 1 , g(n) ∼ k
where xk = a(zk ; g). The elements xk are called the Fourier coefficients and the numbers zk the Fourier exponents of {g(n)}. We can formulate the approximation theorem for an almost periodic sequence as follows.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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Lemma 1. Suppose that g is an almost periodic sequence. For any > 0 there is a sequence of approximating polynomials P (n) of the form P (n) =
n k=1
n xk, zk, ,
zk, ∈ b (g)
such that sup |g() − P ()| < , ∈Z
where xk, is the product of a(zk, ; g) and certain positive number. Proof. This Lemma can be established by modifying the arguments in [6, pp. 14–21]. We omit its proof. It follows from this Lemma that if f and g are almost periodic sequences and a(z; f ) ≡ a(z; g), then f ≡ g. 3. Statement of main theorems Suppose that u(t) and v(t) are the solutions of system x (t) + p(t)x(t) = 0
(2)
satisfying the conditions u(0) = 1, u (0) = 0
and v(0) = 0, v (0) = 1, u(t) v(t) = 1, for t ∈ R by using respectively. The Wronskian w(t) = w[u, v](t) = u (t) v (t) the Liouville’s formula. Since the concept of almost periodic functions is in terms of functions on R, it is suggested that one should find a solution of Eq. (1) on R. If x(t) is a solution of Eq. (1) on R, then x(t) satisfies the following relations: x (t) + p(t)x(t) = qcn + f (t),
n t < n + 1,
(3)
where cn = x(n). The general solution of the Eq. (3) on the interval n t < n + 1 can be written as xn (t) = An (t)u(t − n) + Bn (t)v(t − n). The variation of constants method leads to the system An (t)u(t − n) + Bn (t)v(t − n) = 0, An (t)u (t − n) + Bn (t)v (t − n) = qcn + f (t),
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whence
An (t) = −qcn · Bn (t) = qcn ·
t n
t
v(s − n) ds −
u(s − n) ds +
n
n t
n
t
f (s)v(s − n) ds + cn ,
f (s)u(s − n) ds + dn ,
where n t < n + 1 and dn = x (n). If we set u(t) v(t) W (t) = t t , 0 u(s) ds 0 v(s) ds u(t) v(t) t Un (t) = t 0 f (s + n)u(s) ds 0 f (s + n)v(s) ds
(4)
it follows that x(t) could be written as x(t) = cn u(t − n) + dn v(t − n) − qcn W (t − n) − Un (t − n),
n t < n + 1 (5)
and x (t) = cn u (t − n) + dn v (t − n) − qcn W (t − n) − Un (t − n), n t < n + 1.
(6)
At t = n + 1, we will arrive at cn+1 = (u(1) − qW (1))cn + dn v(1) − Un (1),
(7)
dn+1 = (u (1) − qW (1))cn + dn v (1) − Un (1).
(8)
We can eliminate dn and dn+1 from (7) and (8) and derive the equation cn+2 − (u(1) + v (1) − qW (1))cn+1 + [1 + q(v(1)W (1) − v (1)W (1))]cn = Un (1)v (1) − v(1)Un (1) − Un+1 (1).
(9)
It is easy to calculate v(1)W (1) − v (1)W (1) = −
1
v(s) ds,
0
Un (1)v
(1) − v(1)Un (1) =
1
f (s + n)v(s) ds.
0
Thus, Eq. (9) can be rewritten as
cn+2 − (u(1) + v (1) − qW (1))cn+1 + 1 − q 1 = f (s + n)v(s) ds − Un+1 (1)hn . 0
1 0
v(s) ds cn (10)
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
The corresponding homogeneous equation of Eq. (10) is 1 cn+2 − (u(1) + v (1) − qW (1))cn+1 + 1 − q v(s) ds cn = 0.
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(11)
0
Following [3], we can seek the particular solutions as cn = n for homogeneous difference equation (11). will satisfy the following equation: 1 2 − (u(1) + v (1) − qW (1)) + 1 − q v(s) ds = 0 (12) 0
that has two nontrivial solutions if 1 q v(s) ds = 1. 0
Now, we are in a position to formulate our main theorems. 1 Theorem 1. Suppose that q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. Then for any almost periodic function f (t), Eq. (1) possesses a unique almost periodic solution x(t). Furthermore, we have the following spectrum relation x = f + {2k|k ∈ Z}. Remark 1. The existence and uniqueness of almost periodic solution of Eq. (1) have been proved in [14] under the conditions of Theorem 1. It should be pointed out that no any explanation is given to spectrum and module in [14]. The study of the spectrum relation is the motivation of this paper. Corollary 1. Assume that 1, 1 , . . . , r are rationally independent. Suppose that 1 q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. Then for any f (t) ∈ QP(), Eq. (1) possesses a unique quasi-periodic solution x(t) with frequencies (1, ). Indeed, f = {2k1 1 + · · · + 2kr r |∀kj ∈ Z, 1 j r} is spanned by 21 , 22 , . . . , 2r with integer coefficients. 1 Corollary 2. Suppose that q 0 v(s) ds = 1, and that the norm of all roots of Eq. (12) is not equal to one. If f (t) is T -periodic, then the following results hold: (1) If T = n0 , n0 ∈ Z+ , then Eq. (1) possesses a unique T -periodic solution; n0 , n0 , m0 ∈ Z+ , n0 and m0 are mutually prime, then Eq. (1) possesses a unique (2) If T = m 0 m0 T -periodic solution. (3) If T is irrational and f (t) ∈ PT , then Eq. (1) possesses a unique quasi-periodic solution with frequencies (1, T1 ), where i 2m t ¯ for T > 0. PT = f (t) = fm e T |fm | < + ∞, f−m = fm m
m
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Remark 2. The conclusion of corollary 1 is given in [14]. Thus, the present paper would improve the main results in [14]. Corollary 2 shows that EPCA with periodic perturbations could possess a quasi-periodic solution and no periodic solution. The appearance of quasiperiodic rather than periodic solutions is due to the piecewise constant argument. This new phenomenon illustrates a crucial difference between ODE and EPCA. Example 1. We consider the perturbation of the Mathieu equation with piecewise constants argument x (t) + (2 + 2 cos 2t)x(t) = qx([t]) + f (t).
(13)
Let u(t) = u(t; , ) and v(t) = v(t; , ) be the fundamental solutions of Mathieu equation x (t) + (2 + 2 cos 2t)x(t) = 0
(14)
satisfying the initial values u(0) = 1, u (0) = 0 and v(0) = 0, v (0) = 1, respectively. If = 0, Eq. (14) deduces to x (t) + 2 x(t) = 0
(15)
which has fundamental solutions u0 (t) = cos t and v0 (t) = 1 sin t, satisfying the initial values u0 (0)=1, u0 (0)=0 and v0 (0)=0, v0 (0)=1, respectively. The continuous dependence on parameters implies u(t) = cos t + o(), v(t) = 1 sin t + o(), and v (t) = cos t + o() as → 0, on 0 t 1. The characteristic Eq. (12) deduces to (1 − cos )q (1 − cos )q 2 − 2 cos + + o( ) + 1 − + o( ) = 0. (16) 2 2 If = 0, (16) deduces to (1 − cos )q (1 − cos )q =0 2 − 2 cos + + 1 − 2 2 which has nontrivial solutions if q = 0,
= k ,
(1−cos )q
2
(17)
= 1. It can be shown that if
q = 2
(18)
then the algebraic equation (17) does not possess roots of unity (see [14] for details). The roots of Eq. (16) are continuous on . So, there exists 0 > 0, such that if 0 || 0 , the norm of all roots of Eq. (16) is not equal to one. Therefore, if conditions (18) hold, there exists 0 > 0, such that if 0 || 0 , Eq. (13) possesses a unique almost periodic solution x(t) for any almost periodic function f (t) and x = f + {2k |k ∈ Z}. Note that Example 1 has been discussed in [14] except the spectrum that is a new result. In the present paper, we will also consider the following quasilinear differential equations with piecewise constant argument x (t) + p(t)x(t) = qx([t]) + f (t) + g(t, xt ),
(19)
where f : R1 → R1 is almost periodic, g : R1 × C → R1 is almost periodic in t uniformly on C, where C = C([−1, 0], R1 ). We define xt ∈ C as xt ( ) = x(t + ), −1 0.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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1 Theorem 2. Suppose that q 0 v(s) ds = 1. Assume that the norm of all roots of Eq. (12) is not equal to one. Then there exists a ∗ > 0, such that when 0 < | | < ∗ , Eq. (19) has a unique determined almost periodic solution x(t, ) with the module containment x ⊂ mod(f, g) + {2k}. The existence of almost periodic solution in Theorem 2 has been shown in [14]. But, the module containment is not discussed in [14]. Thus, Theorem 2 is an improvement of the results in [14].
4. Proofs of Theorems We begin with a lemma. Lemma 2. If u(t), v(t) are fundamental solutions of Eq. (2) and f : R → R is an almost (1) (2) periodic function with spectrum f , then the sequences {hn } and {hn } are almost periodic sequences with the Bohr spectrums b (h(1) ) = eif and b (h(1) ) = eif , where
1 } = f (n + s)u(s) ds , {h(1) n n∈Z n∈Z
0
{h(2) n }n∈Z =
1
f (n + s)v(s) ds
0
n∈Z
.
(20) (1)
(1)
Proof. Without loss of generality, it suffices to consider hn . The almost periodicity of {hn } has been proved in [14]. Since f (t) is an almost periodic function, it follows from the approxk pk,m eim t , m ∈ f imation theorem (see [4–6]) that for any k > 0 there exists Pk (t)= nm=1 such that 1 |f (t) − Pk (t)| < , k
k = 1, 2, . . . ,
where we can assume that pk,m eim t and p¯ k,m e−im t appear together in the trigonometric polynomial Pk (t). Define 1 1 (1) (2) Pk (n + s)u(s) ds, Qk (n) := Pk (n + s)v(s) ds, Qk (n) := 0
∀n ∈ Z, k = 1, . . . . (1)
A direct calculation shows Qk (n) =
0
(21) 1 (1) im n (1) , where qk,m = pk,m 0 eim s u(s) ds, m=1 qk,m e
nk
(1)
which is the Fourier coefficient corresponding to the Fourier exponent eim . Qk (n) is a (1) trigonometric polynomial, b (Qk ) = eiPk ⊂ eif := {eim t , m ∈ f }. It is easy to see that 1 (1) 0 |u(s)| ds sup |Qk (n) − h(1) | < , k = 1, 2, . . . . n k n∈Z
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
This implies that b (h(1) ) ⊂ eif . For any fixed j0 ∈ f , we can assume nk > j0 . So, we can rewrite nk
Pk (t) =
pk,m eim t + pk,j0 eij0 t .
m=1,m=j0
It is easy to see that a(j0 ; f ) = limk→∞ a(j0 ; Pk ) = limk→∞ pk,j0 . Thus, we can obtain 1 (1) (1) eij0 s u(s) ds. a(eij0 ; h(1) ) = lim a(eij0 ; Qk ) = lim qk,j0 = a(j0 ; f ) k→∞
It follows that e
ij0
∈ b
(h(1) ).
k→∞
Thus, b
(h(1) ) = eif .
0
Corollary 3. If f (t) is almost periodic and u, v are fundamental solutions of Eq. (2), then the sequence {Un (1)}n∈Z defined in (4) is almost periodic sequence with the Bohr spectrum b (Un (1)) ⊂ eif . Proof of Theorem 1. The existence and uniqueness of almost periodic solution of Eq. (1) have been proved in [14]. It is only needed to study the spectrum containment. Let 1 hn f (s + n)v(s) ds − Un+1 (1). (22) 0
We would like to distinguish several cases. Case 1: All roots of Eq. (12) are simple (Denoted by 1 , 2 , respectively). Without loss of generality, we assume that | 1 | < 1, | 2 | > 1. We define a sequence {cˆn } by n−(m+1) n−(m+1) 1 hm + k2∗ 2 hm , n ∈ Z , (23) cˆn = k1∗ m n−1
mn
where the constants k1∗ , k2∗ are defined by
k1 − k2 = 0, k1 1 − k2 2 = 1.
(24)
It is easy to know that the sequence {cˆn } satisfies the difference equation (10) and is an almost periodic sequence (e.g., see [14]). We define a sequence cˆn+1 − (u(1) − qW (1))cˆn + Un (1) dˆn v(1)
(25)
which is an almost periodic sequence. The above mentioned almost periodic sequences {cˆn } and {dˆn } satisfy Eqs. (7) and (8). For such almost periodic sequences {cˆn } and {dˆn } defined by (23) and (25), respectively, we can construct a solution x(t) of Eq. (1) by using (5) as x(t) = cˆn u(t − n) + dˆn v(t − n) − q cˆn W (t − n) − Un (t − n), n t < n + 1.
(26)
Since {cˆn } is a solution of difference equation (10), we know that the x(t) is continuous on R. It is easily shown that the solution x(t) defined by (26) is an almost periodic solution of Eq. (1) (e.g., see [14]).
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Set
(2) Qk (n)Qk (n) −
u(1) v(1) (1) (2) Qk (n + 1) Qk (n + 1)
(2)
(2)
(1)
= Qk (n) − u(1)Qk (n + 1) + v(1)Qk (n + 1) := (1)
nk
1199
qk,m eim n ,
(27)
m=1
(2)
where Qk (n) and Qk (n) are defined as in (21) and (1)
(2)
(2)
qk,m v(1)qk,m eim + qk,m − u(1)qk,m eim . It follows from the proof of Lemma 2 that sup |Qk (n) − hn | → 0
as k → ∞.
n∈Z
We define
ek (n) = k1∗
m n−1
n−(m+1) Qk (m) + k2∗ 1
mn
n−(m+1) Qk (m), 2
n ∈ Z,
where the constants k1∗ , k2∗ are defined by (24). A direct calculation shows that ek (n) =
nk
qk,m
m=1
where
Ek,m qk,m
k1∗
eim − 1
k1∗
eim − 1
+
+
k2∗
2 − eim
k2∗
2 − eim
eim n =
nk
Ek,m eim n ,
m=1
eim n .
It follows that ek (n) is a trigonometric polynomial and b (ek ) ⊂ eif . Setting C1 = |k1∗ |
1 1 + |k2∗ | , 1 − | 1 | | 2 | − 1
(28)
it is easy to get sup |ek (n) − cˆn | C1 sup |Qk (n) − hn | → 0,
n∈Z
n∈Z
(as k → ∞).
From this, we can conclude b (c) ˆ ⊂ b (h) = eif . For any fixed j0 ∈ f , we can assume nk > j0 . So, we can rewrite Qk (n) in (27) as Qk (n) =
nk
qk,m eim n + qk,j0 eij0 n .
m=1,m=j0
It is easy to see that a(eij0 ; h) = lim a(eij0 ; Qk ) = lim qk,j0 . k→∞
k→∞
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Thus, we obtain ˆ = lim a(eij0 ; ek ) = lim Ek,j0 a(eij0 ; c) k→∞ k→∞ 1 1 . + k2∗ = a(eij0 ; h) k1∗ i e j 0 − 1 2 − eij0 It follows that eij0 ∈ b (c). ˆ Therefore, b (c) ˆ = eif . Set u(t) v(t) , Vk,n (t) = t t P (s + n)u(s) ds P (s + n)v(s) ds 0 k 0 k
0 t 1,
and gk (n)
ek (n + 1) − (u(1) − qW (1))ek (n) + Vk,n (1) , v(1)
where Pk (t) is as in the proof of Lemma 2. We define xk (t) = ek (n)u(t − n) + gk (n)v(t − n) − qek (n)W (t − n) − Vk,n (t − n), n t < n + 1. From the definitions of ek and gk , we know that xk (t) is continuous on R, and satisfies the equation xk (t) + p(t)x(t) = qx([t]) + Pk (t) and |xk (t) − x(t)| → 0 as k → ∞ uniformly on R. For each fixed k, Pk (t) is almost periodic. From the construction, it is easy to know that xk (t) is almost periodic (e.g., see [14]). It can be calculated (see Appendix) that
T 1 = 0 if − = 2k , lim (29) xk (t)e−it dt = 0 if − = 2k . T →∞ 2T −T Since a(; x) = limk→∞ a(; xk ), it follows that if f = { }, then x = { } + {2j |j ∈ Z}. If | 1 | < 1, | 2 | < 1, we define a sequence {cn } by n−(m+1) hm + k 2 n−(m+1) hm . (30) cn = k1 1 2 m n−1
m n−1
If | 1 | > 1, | 2 | > 1, we define a sequence {cn } by n−(m+1) n−(m+1) 1 hm + k 2 2 hm . cn = k1 mn
(31)
mn
Similarly, we can uniquely determine a set of values (k1 , k2 ) such that cn in (30) or (31) is a solution of Eq. (10). The rest of proof in those cases is virtually the same as the proof in the case: | 1 | < 1, | 2 | > 1. Case 2: Eq. (12) possesses a double root. Denote it by 1 .
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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If | 1 | < 1, we can uniquely determine a set of values (k1∗ , k2∗ ) such that the sequence n−(m+1) cˆn = k1∗ n−(m+1) hm + k2∗ (n − (m + 1)) 1 hm , n ∈ Z 1 m n−1
m n−1
is a solution of the difference equation (10). If | 1 | > 1, we can obtain a solution of the difference equation (10) n−(m+1) n−(m+1) cˆn = k1∗ 1 hm + k2∗ (n − (m + 1)) 1 hm , mn
n ∈ Z.
mn
The rest of proof in this case is virtually the same as the proof in Case 1.
Proof of Theorem 2. Set M = mod(f, g) + {2k } and X = { ∈ AP|mod( ) ⊂ M}. X is a Banach space equipped with supremum norm | | = supt∈R | (t)|. For any ∈ X, the following equation x (t) + p(t)x(t) = qx([t]) + f (t) + g(t, t ) has a unique almost periodic solution T by using Theorem 1 and T ⊂ f (t)+ g(t, t ) + {2k } ⊂ M. Thus, mod(T ) ⊂ M. It can be proved that T : X → X is a contracting mapping (e.g., see [14] for details). 5. A second-order delay differential equation In this section, we consider the following differential equations of the form x (t) + p(t)x(t) = qx([t − 1]) + f (t),
(32)
where p : R → R, t → p(t), is 1-periodic and continuous, q(= 0) is a real constant, [·] is the greatest integer function, and f : R → R is a continuous function. Since the concept of almost periodic functions is in terms of functions on R, one should find a solution of Eq. (32) on R. If x(t) is a solution of Eq. (32) on R, then x(t) satisfies the following relations: x (t) + p(t)x(t) = qcn−1 + f (t),
n t < n + 1,
(33)
where cn =x(n). The general solution of Eq. (32) on the interval n t < n+1 can be written as xn (t) = An (t)u(t − n) + Bn (t)v(t − n), where u(t) and v(t) are the solutions of System (2) satisfying the conditions u(0) = 1, u (0) = 0
and v(0) = 0, v (0) = 1,
respectively. The variation of constants method leads to t t An (t) = −qcn−1 · v(s − n) ds − f (s)v(s − n) ds + cn , n
n
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
Bn (t) = qcn−1 ·
t n
u(s − n) ds +
t n
f (s)u(s − n) ds + dn ,
where n t < n + 1 and dn = x (n). By using the notations in (4), it follows that x(t) could be written as x(t) = cn u(t − n) + dn v(t − n) − qcn−1 W (t − n) − Un (t − n), n t < n + 1,
(34)
x (t) = cn u (t − n) + dn v (t − n) − qcn−1 W (t − n) − Un (t − n), n t < n + 1.
(35)
If set t = n + 1, we will arrive at the following difference equations:
cn+1 = cn u(1) + dn v(1) − qcn−1 W (1) − Un (1), dn+1 = cn u (1) + dn v (1) − qcn−1 W (1) − Un (1).
(36)
We can eliminate dn and dn+1 from (36) and derive the equation 1 v(s) ds cn+2 − (u(1) + v (1))cn+1 + (1 + qW (1))cn − qcn−1 0 1 f (s + n)v(s) ds − Un+1 (1)hn . =
(37)
The corresponding characteristic equation of Eq. (37) is 1 3 − (u(1) + v (1))2 + (1 + qW (1)) − q v(s) ds = 0
(38)
0
0
that has three nontrivial solutions if 1 v(s) ds = 0. 0
Similar to Theorem 1, we have the following results to Eq. (32). 1 Theorem 3. Suppose that q = 0, 0 v(s) ds = 0, and that the norm of all roots of Eq. (38) is not equal to one. Then for any almost periodic function f (t), Eq. (32) possesses a unique almost periodic solution x(t) and x = f + {2k |k ∈ Z}. Remark 3. The existence of almost periodic solution has been shown in [14], but no explanation is given to the spectrum or module of almost periodic solution in [14]. So, the present paper sharpens the corresponding results in [14]. Corollary 4. Assume that 1, 1 , . . . , r are rationally independent. Suppose that q = 1 0, 0 v(s) ds = 1, and that the norm of all roots of Eq. (38) is not equal to one. Then for any f (t) ∈ QP(), Eq. (32) possesses a unique quasi-periodic solution x(t) with frequencies (1, ).
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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1 Corollary 5. Suppose that q = 0, 0 v(s) ds = 1, and that the norm of all roots of Eq. (38) is not equal to one. If f (t) is T -periodic, then the following results hold: (1) If T = n0 , n0 ∈ Z+ , then Eq. (32) possesses a T -periodic solution. n0 (2) If T = m , n0 , m0 ∈ Z+ , n0 and m0 are mutually prime, then Eq. (32) possesses a 0 m0 T -periodic solution. (3) If T is irrational and f (t) ∈ PT , then Eq. (32) possesses a unique quasi-periodic solution with frequencies (1, T1 ). Example 2. We consider the following equation: x (t) − a 2 x(t) = qx([t − 1]) + f (t)
(39)
which was discussed by Wiener and Lakshmikantham in [10] under f (t) = 0 and Yuan in [14]. The characteristic equation (38) deduces to the following form: q(cosh a − 1) q(cosh a − 1) 3 − 22 cosh a + 1 − − = 0. (40) 2 a a2 If a = 0, q = 0, and q = −a 2 ,
q = a 2
1 + 2 cosh a , cosh a − 1
(41)
the norm of all roots of Eq. (40) is not equal to one (e.g., see [14]). It follows from Theorem 3 that Eq. (39) possesses an almost periodic solution x(t) and x = f + {2k } for each almost periodic f (t), if conditions (41) hold. Example 3. We consider the following equation: x (t) + 2 x(t) = bx([t − 1]) + f (t)
(42)
which was discussed by Wiener in [9] under f (t) = 0 and Yuan in [13,14], respectively. The characteristic equation (38) of Eq. (42) deduces to (1 − cos )b (1 − cos )b 3 − 22 cos + 1 − − = 0. (43) 2 2 If = k (k ∈ Z) and b = 2 ,
b = 2
1 + 2 cos 1 − cos
(44)
the norm of all roots of Eq. (43) is not equal to one (e.g., see [14]). It follows from Theorem 3 that Eq. (42) possesses an almost periodic solution x(t) and x = f + {2k } for each almost periodic f (t), if conditions (44) hold. Note that the spectrum relations in Examples 2, 3 are our new results. Finally, we will consider the following quasilinear differential equations with piecewise constant argument x (t) + p(t)x(t) = qx([t − 1]) + f (t) + g(t, xt ),
(45)
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R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
where p : R → R, t → p(t), is 1-periodic and q(= 0) is a real constant, f : R1 → R1 is almost periodic, g : R1 × C → R1 is almost periodic in t uniformly on C, where C = C([−1, 0], R1 ). We define xt ∈ C as xt ( ) = x(t + ), −1 0. 1 Theorem 4. Suppose that q 0 v(s) ds = 1. Assume that the norm of all roots of Eq. (38) is not equal to one. Then there exists a ∗ > 0, such that when 0 < | | < ∗ , Eq. (45) has a unique determined almost periodic solution x(t, ) with the module containment mod(x) ⊂ mod(f, g) + {2k}. Appendix During the proof of Theorem 1, we meet with what is the exponents of xk (t) in (29). If we look at the formula of xk (t), we can find that xk (t) consists of the following term with the form s (t) = (t − n)ei n , n t < n + 1. We give the following calculation for completeness. We can calculate that m T 1 1 −it a(; s ) := lim s (t)e−it dt s (t)e dt = lim m→∞ 2m −m T →∞ 2T −T m−1 n+1 1 = lim (t − n)ei n e−it dt m→∞ 2m n=−m n m−1 1 i( −)n 1 = lim e (s)e−it dt m→∞ 2m 0 n=−m
0 if − = 2j , j ∈ Z, = 1 −i s ds = 0 if − = 2j , j ∈ Z. 0 (s)e References [1] S. Busenberg, K.L. Cooke, Models of vertically transmitted diseases with sequential-continuous dynamics, in: V. Lakshmikantham (Ed.), Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 1982, pp. 179–187. [2] M.L. Cartwright, Almost periodic differential equations and almost periodic flows, J. Differ. Equations 5 (1969) 167–181. [3] K.L. Cooke, J. Wiener, A survey of differential equation with piecewise continuous argument, Lecture Notes in Mathematics, vol. 1475, Springer, Berlin, 1991, pp. 1–15. [4] C. Corduneanu, Almost Periodic Functions, Wiley, New York, London, Sydney, Toronto, 1968. [5] A.M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [6] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge, 1982. [7] G.H. Meisters, On almost periodic solutions of a class of differential equations, Proc. Amer. Math. Soc. 10 (1959) 113–119. [8] J. Wiener, Generalized Solutions of Functional Differential Equations, World Scientific, Singapore, 1993. [9] J. Wiener, A second-order delay differential equation with multiple periodic solutions, J. Math. Anal. Appl. 229 (1999) 659–676.
R. Yuan / Nonlinear Analysis 59 (2004) 1189 – 1205
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[10] J. Wiener, V. Lakshmikantham, Excitability of a second-order delay differential equation, Nonlinear Anal. 38 (1999) 1–11. [11] J. Wiener, V. Lakshmikantham, Complicated dynamics in a delay Klein–Gordon equation, Nonlinear Anal. 38 (1999) 75–85. [12] R. Yuan, On the existence of almost periodic solutions of a singularly perturbed differential equations with piecewise constant argument, Z. Angew. Math. Phys. 50 (1999) 94–119. [13] R. Yuan, On the regular solution of a second order delay differential equation, Z. Angew. Math. Phys. 53 (2002) 407–437. [14] R. Yuan, On the second-order differential equation with piecewise constant argument and almost periodic coefficients, Nonlinear Anal. 52 (2003) 1411–1440. [15] R. Yuan, J. Hong, Almost periodic solutions of differential equations with piecewise constant argument, Analysis 16 (1996) 171–180. [16] R. Yuan, Z. Jing, L. Chen, Uniform asymptotic stability of hybrid dynamical systems with delay, IEEE Trans. Automat. Contr. 48 (2) (2003) 344–348.