Pseudo Almost Periodic Solutions of a Singularly Perturbed

Acta Mathematica Sinica, English Series Mar., 2007, Vol. 23, No. 3, pp. 423–438 Published online: Sept. 6, 2005 DOI: 10.1007/s10114-005-0588-3 Http://www.ActaMath.com

Pseudo Almost Periodic Solutions of a Singularly Perturbed Differential Equation with Piecewise Constant Argument Guo Jian LIN Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China E-mail: yinplum@ 163.com

Rong YUAN School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China E-mail: [email protected] Abstract Under suitable assumptions, the existence and the uniqueness of the pseudo-almost periodic solution for a singularly perturbed differential equation with piecewise constant argument are obtained. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data. Keywords Singular perturbation, Pseudo-almost periodic solutions, Pseudo-almost periodic sequences, Piecewise constant argument MR(2000) Subject Classification 34K15

1 Introduction There are a great deal of papers concerning the singularly perturbed problems. Of all these papers, Friedrichs and Wasow [1], Flatto and Levinson [2], and Anosov [3] dealt with the existence of periodic solutions to the following singularly perturbed system: x (t) = F (t, x(t), y(t), ε), (1.1) εy (t) = G(t, x(t), y(t), ε), where, ε > 0 is a small parameter, x ∈ Rp , y ∈ Rq . Anosov’s result is most general. Accordingly, the existence of almost periodic solutions to Eq. (1.1) was studied by Hale and Seifert [4], Chang [5], and Smith [6]. Recently, Yuan [7, 8] investigated the existence of almost periodic solutions and pseudo-almost periodic solutions to the singularly perturbed system with piecewise constant argument of the form: N x (t) = F (t, x(t), {x([t + i])}N −N , y(t), {y([t + i])}−N , ε), (1.2) N εy (t) = G(t, x(t), {x([t + i])}N −N , y(t), {y([t + i])}−N , ε), where, ε > 0 is a small parameter, x ∈ R, y ∈ R, and [·] denotes the greatest integer function. Meanwhile, Yuan [9] discussed the existence of almost periodic solutions to the following singularly perturbed system with piecewise constant argument: x (t) = F (t, x(t), x([t]), y(t), y([t]), ε), (1.3) εy (t) = G(t, x(t), x([t]), y(t), y([t]), ε), Received March 16, 2004, Accepted November 11, 2004 Supported by the National Natural Science Foundation of China (10371010) and SRFDP (20030027011)

Lin G. J. and Yuan R.

424

where, ε > 0 is a small parameter, x ∈ Rp , y ∈ Rq , and [·] denotes the greatest integer function. Naturally, in this paper, we want to show the existence of pseudo-almost periodic solutions to Eq. (1.3). Differential equations with piecewise constant argument, which were investigated in many papers [10–12], describe the hybrid dynamical systems (a combination of continuous and discrete systems) and, therefore, have the combined properties of both differential and difference equations. It is Yuan and Hong [13] who first considered the existence of almost periodic solutions to the differential equation with piecewise constant argument. Afterwards, the problem of the existence of almost periodic solutions and pseudo-almost periodic solutions to the differential equations with piecewise constant argument was extensively discussed (see [14–18] and the references cited therein). In what follows, we denote by |·| the Euclidean norm and by [·] the greatest integer function. We say a function (x, y) : R → Rp × Rq is a solution of Eq. (1.3) if the following conditions are satisfied: (i) (x, y) is continuous on R, (ii) The derivative (x , y ) of (x, y) exists on R except possibly at points t = n, n ∈ Z = {. . . , −1, 0, 1, . . .} where one-sided derivatives exist, (iii) (x, y) satisfies Eq. (1.3) in the intervals (n, n + 1), n ∈ Z. It is assumed that the degenerate system x (t) = F (t, x(t), x([t]), y(t), y([t]), 0), (1.4) 0 = G(t, x(t), x([t]), y(t), y([t]), 0) has a pseudo-almost periodic “outer” solution which we suppose to be the trivial solution, that is, we suppose F (t, 0, 0, 0, 0, 0) ≡ G(t, 0, 0, 0, 0, 0) ≡ 0, so that (x, y) = (0, 0) satisfies (1.4). Then expanding (1.3) about the trivial solution gives ⎧ x (t) = A(t, ε)x(t) + A1 (t, ε)x([t]) + B(t, ε)y(t) + B1 (t, ε)y([t]) ⎪ ⎪ ⎪ ⎪ ⎨ + f (t, x(t), x([t]), y(t), y([t]), ε), (1.5) ⎪ εy (t) = C(t, ε)x(t) + C1 (t, ε)x([t]) + D(t, ε)y(t) + D1 (t, ε)y([t]) ⎪ ⎪ ⎪ ⎩ + g(t, x(t), x([t]), y(t), y([t]), ε). One may think of, e.g., A(t, ε) as ∂F/∂x(t, 0, 0, 0, 0, ε), but, in fact, it is really (1.5) that we study in this paper. The following hypotheses are assumed to hold throughout the paper. (H1) A(t, ε), A1 (t, ), B(t, ε), B1 (t, ε), C(t, ε), C1 (t, ε), D(t, ε), D1 (t, ε) are almost periodic matrix functions (of size p × p, p × p, p × q, p × q, q × p, q × p, q × q, q × q, respectively) in t. Moreover, they are continuous in ε, uniformly in t ∈ R. We let M denote a common bound for the norm of each of these matrices for (t, ε) ∈ R × [0, ε0 ]. (H2) The real parts of the eigenvalues of D0 (t) = D(t, 0) ≤ −8β, for some β > 0, C(t, 0) ≡ C1 (t, 0) ≡ D1 (t, 0) ≡ 0. (H3) The system (1.6) x (t) = A(t, 0)x(t) has an exponential dichotomy on R, that is, there exist positive constants K1 , α1 and a projection P (P 2 = P ) such that |X(t)P X −1 (s)| ≤ K1 e−α1 (t−s) , t ≥ s,

|X(t)(I − P )X −1 (s)| ≤ K1 e−α1 (s−t) , s ≥ t,

where X(t) is the fundamental matrix solution of (1.6) such that X(0) = I, with I being the identity matrix. (H4) f , g are pseudo-almost periodic in t uniformly on (x0 , x1 , y0 , y1 ) such that t ∈ R, |xi |, |yi | ≤ ρ0 , (i = 0, 1), 0 ≤ ε ≤ ε0 , 0 ≤ ρ ≤ ρ0 . Furthermore, there exist nonde-

Singularly Perturbed Differential Equation

425

creasing functions M (ε) and η(ρ, ε), 0 ≤ ε ≤ ε0 , 0 ≤ ρ ≤ ρ0 satisfying lim→0 M (ε) = 0, lim(ρ,ε)→(0,0) η(ρ, ε) = 0, such that |f (t, 0, 0, 0, 0, ε)| ≤ M (ε), |g(t, 0, 0, 0, 0, ε)| ≤ M (ε), t ∈ R, 0 ≤ ε ≤ ε0 , and |f (t, x0 , x1 , y0 , y1 , ε)−f (t, x0 , x1 , y 0 , y 1 , ε)| ≤ η(ρ, ε)[|x0 − x0 | + |x1 − x1 | + |y0 − y 0 | + |y1 − y 1 |], |g(t, x0 , x1 , y0 , y1 , ε)−g(t, x0 , x1 , y 0 , y 1 , ε)| ≤ η(ρ, ε)[|x0 − x0 | + |x1 − x1 | + |y0 − y 0 | + |y1 − y 1 |], hold for all t ∈ R, |xi |, |xi |, |yi |, |y i | ≤ ρ, (i = 0, 1), 0 ≤ ε ≤ ε0 . This paper is organized as follows. In Section 2, we recall some lemmas and definitions of almost periodic type functions and sequences. In Section 3, the existence of pseudo-almost periodic sequence solutions to the difference equations is obtained. In Section 4, we show the existence of a continuous family of pseudo-almost periodic solutions (x∗ (t, ε), y ∗ (t, ε)) of Eq. (1.3). In Section 5, we will show that if C = εC, C1 = εC 1 , D = εD1 , g = εg, where C, C 1 , D1 are pseudo-almost periodic matrix functions, g satisfies the same assumptions as g, then the stability of these pseudo-almost periodic solutions (x∗ (t, ε), y ∗ (t, ε)) is characterized by the construction of initial data. In Section 6, we give an example. 2

Some Definitions and Lemmas

Let C(R, RN ) (respectively C(R×Ω, RN ), where Ω ⊂ RN ), denote the Banach space of bounded continuous functions φ(t) (resp. φ(t, x)) from R (resp. R × Ω) to RN with the norm φ = supt∈R |φ(t)| (resp. φ = sup(t×x)∈R×Ω |φ(t, x)| ). Definition 2.1 [19] A function f ∈ C(R, RN ) (C(R × Ω, RN )) is called an almost periodic function (almost periodic function in t ∈ R uniformly on x ∈ Ω) (denoted by f ∈ AP(R, RN )(AP(R × Ω, RN ))) if the ε-translation set of f T (f, ε) = {τ ∈ R| |f (t + τ ) − f (t)| < ε, ∀ t ∈ R}, (T (f, ε) = {τ ∈ R| |f (t + τ, x) − f (t, x)| < ε, ∀ (t, x) ∈ R × W, ∀ compact set W ⊂ Ω}) is a relatively dense set in R for all ε > 0. τ ∈ T (f, ε) is called an ε-period for f. Definition 2.2 [20, 21] A function f ∈ C(R, RN )(C(R × Ω, RN )) is called a pseudo-almost periodic function (denoted by f ∈ (R, RN ) (PAP(R × Ω, RN ))), if f can be written as a sum f = f1 + f0 , where f1 ∈ AP(R, RN ) (AP(R × Ω, RN )) and f0 ∈ PAP (R, RN ) (PAP (R × Ω, RN )). Here T 1 PAP (R, RN ) = φ ∈ C(R, RN ) | M (|φ|) := lim |φ(t)|dt = 0 T →∞ 2T −T ⎧ ⎫ T ⎪ ⎪ ⎨ φ ∈ C(R × Ω, RN ) | M (|φ|) := lim 1 |φ(t, x)|dt = 0 ⎬ N T →∞ 2T −T PAP (R × Ω, R ) = , ⎪ ⎪ ⎩ ⎭ uniformly in x ∈ Ω f1 and f0 are, respectively, called the almost periodic component and the ergodic perturbation of the function f . Remark 2.1 [21]

Note that f1 and f0 are uniquely determined.

Definition 2.3 [19] A sequence x : Z → RN is called an almost periodic sequence (denoted by x ∈ AP(Z, RN )) if the ε-translation set of x, T (x, ε) = {τ ∈ Z||x(n + τ ) − x(n)| < ε, ∀ n ∈ Z}, is a relatively dense set in Z for all ε > 0. τ ∈ T (x, ε) is called an ε-period for x. Definition 2.4 [15, 16] A sequence x : Z → RN is called a pseudo-almost periodic sequence (denoted by x ∈ PAP(Z, RN )), if x can be written as a sum x(n) = x1 (n) + x0 (n), n ∈ Z,


426

where x1 ∈ AP(Z, RN ) and x0 ∈ PAP (Z, RN ). Here ⎧ N ⎪ ⎪ ⎨ φ : Z → R is a bounded sequence and K N PAP (Z, R ) = 1 ⎪ M (|φ|) := lim |φ(n)| = 0 ⎪ ⎩ K→∞ 2K n=−K

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

,

x1 and x0 are, respectively, called the almost periodic component and the ergodic perturbation of the sequence x. Remark 2.2 [14]

Note that x1 and x0 are uniquely determined.

Lemma 2.1 [20]

PAP(R, RN ) is a closed subspace of C(R, RN ).

Lemma 2.2 [14] B0 = {φ | φ ∈ PAP (R, RN ) and limK→∞ closed subset of PAP (R, RN ).

1 2K

K

n=−K |φ(n)|

= 0} is a

Lemma 2.3 [14] B = {φ | φ ∈ PAP(R, RN ) and its ergodic perturbation φ0 ∈ B0 } is a closed subset of PAP(R, RN ). Lemma 2.4 [14] Suppose that f (t, ·, ·) is a pseudo-almost periodic function for t uniformly on RN × RN and φ ∈ B. Then the function f (t, φ(t), φ([t])) ∈ PAP(R, RN ). 3

Pseudo-Almost Periodic Solutions to Differential Equations with Piecewise Constant Argument

To begin with, we consider the following differential equation x (t) = A(t, 0)x(t) + A1 (t, 0)x([t]) + f (t),

(3.1)

where f ∈ PAP(R, RN ). Let X(t) be the fundamental matrix solution of x (t) = A(t, 0)x(t), t ∈ R,

(3.2)

such that X(0) = I, with I being the identity matrix. Obviously, if x(t) is a solution of Eq. (3.1), we have the following relation t −1 −1 x(t) = X(t) X (n) + X (s)A1 (s, 0)ds x(n)

n

t

X −1 (s)f (s)ds, n ≤ t < n + 1.

+ X(t) n

(3.3)

In view of the continuity of a solution at a point, we arrive at the following difference equation: n+1 x(n + 1) = X(n + 1) X −1 (n) + X −1 (s)A1 (s, 0)ds x(n)

n+1

+ X(n + 1)

n

X −1 (s)f (s)ds.

(3.4)

n

Setting

E(n) = X(n + 1) X −1 (n) +

n+1

h(n) = X(n + 1)

n+1 n

X −1 (s)A1 (s, 0)ds ,

X −1 (s)f (s)ds,

n

then we can rewrite equation (3.4) as x(n + 1) = E(n)x(n) + h(n).

(3.5)


427

If A(t, 0), A1 (t, 0) are almost periodic, and f (t) is pseudo-almost periodic, it follows from [13, 14] that {E(n)} is an almost periodic sequence, and {h(n)} is a pseudo-almost periodic sequence. Proposition 3.1 [12] Assume that Eq. (3.2) admits an exponential dichotomy on R with projection P and constants K1 , α1 . Set α = α1 /2. If α1 e −1 M eα − 1 min , |A1 (t, 0)| ≤ , (3.6) 2K1 eM 1 + eα1 (1 + e−α )eα1 then the difference equation x(n + 1) = E(n)x(n)

(3.7)

possesses an exponential dichotomy on Z, i.e., there exists K such that −1 (m)| ≤ Ke−α(n−m) , |E(n)P E n ≥ m, −1 −α(m−n) (m)| ≤ Ke , m ≥ n, |E(n)(I − P )E where E(n) is the fundamental matrix solution of Eq. (3.7) such that E(0) = I. Remark 3.1 In fact, we can replace the condition (H2) by the assumption that the difference equation (3.7) possesses an exponential dichotomy on Z. Proposition 3.2 [9, 13, 14] Suppose that {E(n)} is an almost periodic sequence, {h(n)} is a pseudo-almost periodic sequence, and the linear difference equation (3.7) has an exponential dichotomy on Z. Then the inhomogeneous difference equation (3.5) has a unique pseudo-almost periodic sequence solution {x∗ (n)}n∈Z . Moreover, |x∗ (n)| ≤ K(1 + e−α )(1 − e−α )−1 sup|h(n)|, n∈Z

∀ n ∈ Z.

(3.8)

Proposition 3.3 [9, 13, 14] If (3.6) holds, then Eq. (3.1) has a unique pseudo-almost periodic solution x(t) ∈ B. Moreover, |x(t)| ≤ N |f |, (3.9) where N = e2M (1 + M )K(1 + e−α )(1 − e−α )−1 + eM . In the following, we sometimes use the notations D (t) = D(t, ε), D1,ε (t) = D1 (t, ε), etc. for convenience. Next, we consider the differential equation dy = D0 (t)y. dt Setting t = εs and z(s) = y(εs), then (3.10) can be written in the form ε

(3.10)

dz = D0 (εs)z. ds

(3.11)

Proposition 3.4 [5] Suppose that the real parts of the eigenvalues of D0 (t) ≤ −8β. Then when ε is sufficiently small, Eq. (3.11) has a fundamental matrix Z(s) = Zε (s) such that

|Z(s)Z −1 (s )| ≤ Le−β(s−s ) ,

s ≥ s ,

where L = L(M, β) is a constant. Now, we want to investigate the existence of pseudo-almost periodic solutions to the following equation εy (t) = D0 (t)y(t) + D1,ε (t)y0 ([t]) + g(t), (3.12) where y0 (t) ∈ B and g(t) ∈ PAP(R, RN ).


428

The variation of constants formula implies that the solutions of Eq. (3.12) can be written in the form εt t n t y(t) = Z( )Z −1 Z y(n) + Z −1 (s)D1,ε (εs)dsy0 (n) n ε ε ε ε εt t + Z (3.13) Z −1 (s)g(εs)ds, n ≤ t ≤ n + 1. n ε ε In view of the continuity of a solution at a point, we arrive at the following difference equation y(n + 1) = G0 (n, ε)y(n) + G1 (n, ε)y0 (n) + e(n, ε), where

(3.14)

n+1 −1 n G0 (n, ε) = Z Z , ε ε n+1 ε n+1 Z G1 (n, ε) = Z −1 (s)D1,ε (εs)ds, n ε ε n+1 ε n+1 Z e(n, ε) = Z −1 (s)g(εs)ds, n ≤ t ≤ n + 1. n ε ε

0 (n, ε) of the homogeIt follows from Proposition 3.4 that the fundamental matrix solution G neous difference equation y(n + 1) = G0 (n, ε)y(n), (3.15) satisfies

β

−1 (m, ε)| ≤ Le− ε (n−m) , 0 (n, ε)G |G 0

n ≥ m.

Proposition 3.5 [9] For fixed ε > 0, the following sequences n+1 ε n+1 n+1 n (i) Z Z Z −1 ; (ii) Z −1 (s)D1,ε (εs)ds n ε ε ε ε are almost periodic. Proposition 3.6 For fixed ε > 0, the sequence n+1 ε n+1 Z Z −1 (s)g(εs)ds n ε ε is pseudo-almost periodic. Proof Since g = g1 + g0 , g1 ∈ AP(R, RN ), g0 ∈ PAP (R, RN ), then n+1 ε n+1 Z Z −1 (s)g(εs)ds n ε ε n+1 n+1 ε ε n+1 n+1 Z Z = Z −1 (s)g1 (εs)ds + Z −1 (s)g0 (εs)ds. n n ε ε ε ε and

K n+1 ε 1 n+1 −1 Z (s)g (εs)ds Z 0 n 2K ε ε n=−K K n+1 1 1 n+1 −1 s Z (s)ds Z g = 0 ε 2K ε ε n n=−K

(3.16)


429

n+1 K+1 K 1 1 1 L L|g0 (s)|ds = |g0 (s)|ds ε 2K ε 2K −K n=−K n K+1 1 1 2(K + 1)L · ≤ |g0 (s)|ds → 0, (as N → ∞). ε 2K 2(K + 1) −(K+1)

≤

n+1 ε

−1 Z( n+1 (s)g1 (εs)ds is an almost periodic sequence. ε )Z n+1 −1 (s)g(εs)ds is a pseudo-almost periodic sequence. So, for each fixed ε > 0, n ε Z( n+1 ε )Z ε This finishes the proof of Proposition 3.6. Since the fundamental matrix solution of the homogeneous difference Eq. (3.15) satisfies (3.16), then for each fixed ε > 0, the only bounded solution of Eq. (3.14) is

It is shown in [9] that

n ε

n−1

y(n, ε) =

0 (n, ε)G −1 (m + 1, ε)[G1 (m, ε)y0 (m) + e(m, ε)]. G 0

(3.17)

m=−∞

β

If we choose ε0 such that when 0 < ε ≤ ε0 , 1 ≤ (1 − e− ε )−1 ≤ 2 and ||D1,ε || ≤ 1, then | y(n, ε)| ≤ 2L sup| G1 (n, ε)y0 (n) + e(n, ε)|, n∈Z

where ||D1,ε || = supt∈R |D1,ε (t)|. Proposition 3.7

For a fixed ε > 0, the sequence n−1

0 (n, ε)G −1 (m + 1, ε)[G1 (m, ε)y0 (m) + e(m, ε)] G 0

m=−∞

is pseudo-almost periodic. Proof It follows from Proposition 3.6, that e(m, ε) can be written as e(m, ε) = e1 (m, ε) + e0 (m, ε), e1 ∈ AP(Z, RN ), e0 ∈ PAP (Z, RN ). N −1 Let φn = n−1 m=−∞ G0 (n, ε)G0 (m + 1, ε)e0 (m, ε). We will show that φn ∈ PAP (Z, R ). In fact, −K−1 K 1 1 −1 G0 (−K, ε)G0 (m + 1, ε)e0 (m, ε) |φn | = 2K 2K m=−∞

n=−K

K−1 −1 + ······ + G0 (K, ε)G0 (m + 1, ε)e0 (m, ε) m=−∞

≤

L 2K

−K−1

β

e− ε (−K−m−1) |e0 (m, ε)|

m=−∞

+ ······ +

K−1

β e− ε (K−m−1) |e0 (m, ε)|

m=−∞ β L = (|e0 (K − 1, ε)| + (1 + e− ε )|e0 (K − 2, ε)| 2K β 2Kβ + · · · · · · + (1 + e− ε + · · · · · · + e− ε )|e0 (−K − 1, ε)| β

β

2Kβ ε

−β ε

− 2Kβ ε

+ e− ε |e0 (−K − 2, ε)|(1 + e− ε + · · · · · · + e− − 2β ε

+e

|e0 (−K − 3, ε)|(1 + e

+ ······ + e

) )


430 3β

β

2Kβ

+ e− ε |e0 (−K − 4, ε)|(1 + e− ε + · · · · · · + e− ε ) + · · · · · ·) 1 L (|e0 (K − 1, ε)| + |e0 (K − 2, ε)| + · · · · · · + |e0 (−K − 1, ε)|) ≤ 2K 1 − e− βε β

e− ε L sup|e0 (n, ε)| + 2K (1 − e− βε )2 n∈Z ≤

1 L 2(K + 1) β − 2K 1 − e ε 2(K + 1)

K+1

|e0 (m, ε)|

m=−(K+1)

β

+

L e− ε sup|e0 (n, ε)| → 0, 2K (1 − e− βε )2 n∈Z

(as K → ∞).

That is, {φn }n∈Z ∈ PAP (Z, RN ). n−1 −1 Meanwhile, it is shown in [9] that the sequence m=−∞ G 0 (n, ε)G0 (m + 1, ε)e1 (m, ε) is almost periodic. −1 Therefore, the sequence n−1 m=−∞ G0 (n, ε)G0 (m + 1, ε)e(m, ε) is pseudo-almost periodic. −1 Similarly, it can be shown that the sequence n−1 m=−∞ G0 (n, ε)G0 (m + 1, ε)G1 (m, ε)y0 (m) is pseudo-almost periodic. n−1 −1 So, the sequence m=−∞ G 0 (n, ε)G0 (m + 1, ε)[G1 (m, ε)y0 (m) + e(m, ε)] is pseudo-almost periodic. This ends the proof of Proposition 3.7. Similarly to [8], it can be shown that the solution y(t) defined by Eq. (3.13), with y(n) = y(n, ε), n ∈ Z, is the unique pseudo-almost solution of Eq. (3.12), which is denoted by T (y0 , g, ε)(t) ∈ B . ||T (y0 , g, ε)|| ≤ where U := 4

L2 β (2L

L2 (2L + 1)(||D1,ε || · ||y0 || + ||g||) ≤ U (||y0 || + ||g||), β

(3.18)

+ 1), 0 ≤ ε ≤ ε0 (for details, see [9, 13]).

Existence of Pseudo-Almost Periodic Solution

First of all, we consider the inhomogeneous equations x (t) = A(t, ε)x(t) + A1 (t, ε)x([t]) + B(t, ε)y(t) + B1 (t, ε)y([t]) + f (t), εy (t) = C(t, ε)x(t) + C1 (t, ε)x([t]) + D(t, ε)y(t) + D1 (t, ε)y([t]) + g(t),

(4.1)

where f, g ∈ PAP(R, RN ). The Banach space PAP(R, RN ) is equipped with supremum norm || · ||. Theorem 4.1 If (3.6) holds, then we have ε1 , 0 < ε1 ≤ ε0 , and the positive functions a(ε), b(ε), c(ε), d(ε) defined for 0 < ε ≤ ε1 , satisfying lim a(ε) = N,

ε→0+

lim b(ε) = 2N M U,

ε→0+

lim c(ε) = 0,

ε→0+

lim d(ε) = U,

ε→0+

a(ε), b(ε), c(ε), d(ε) ≤ 2N M U + N + U, such that for each (f, g) ∈ PAP, 0 < ε ≤ ε1 ,there is a unique solution (x(f, g, ε), y(f, g, ε)) ∈ B ⊂ PAP of Eq. (4.1). The solution satisfies ||x|| ≤ a(ε)||f || + b(ε)||g||, ||y|| ≤ c(ε)||f || + d(ε)||g||.

(4.2)

The map (f, g) → (x(f, g, ε), y(f, g, ε)) defines a bounded linear operator L (ε) satisfying ||L (ε)|| ≤ 4N M U + 2N + 2U and ε → L (ε) is continuous for 0 < ε ≤ ε1 .


431

Proof Given (f, g) ∈ PAP, (x0 , y0 ) ∈ B, define (x(t), y(t)) as the solution of ⎧ x (t) = A0 (t)x(t) + A1,0 (t)x([t]) + (Aε (t) − A0 (t))x0 (t) ⎪ ⎪ ⎪ ⎪ ⎨ + (A (t) − A (t))x ([t]) + B (t)y(t) + B (t)y([t]) + f (t), 1,ε

1,0

0

ε

1,ε

⎪ εy (t) = D0 (t)y(t) + D1,ε (t)y0 ([t]) + (Dε (t) − D0 (t))y0 (t) ⎪ ⎪ ⎪ ⎩ + Cε (t)x0 (t) + C1, (t)x0 ([t]) + g(t). From the variation of constants formula, it follows that t ⎧ ⎪ −1 −1 ⎪ x(t) = X(t)X (n) + X(t)X (s)A (s)ds x(n) ⎪ 1,0 ⎪ ⎪ n ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ + X(t) X −1 (s){(Aε (s) − A0 (s))x0 (s) + Bε (s)y(s) + f (s)}ds ⎪ ⎪ ⎪ n ⎪ ⎪ t ⎪ ⎪ ⎪ ⎪ ⎪ X −1 (s)(A1,ε (s) − A1,0 (s))dsx0 (n) + X(t) ⎪ ⎪ ⎪ n ⎪ t ⎪ ⎪ ⎨ + X(t) X −1 (s)B1,ε (s)dsy(n), n ⎪ ⎪ εt ⎪ ⎪ t t ⎪ −1 n ⎪ y(t) = Z Z Z y(n) + Z −1 (s)D1,ε (εs)dsy0 (n) ⎪ ⎪ ⎪ n ε ε ε ⎪ ε ⎪ ⎪ εt ⎪ ⎪ ⎪ t ⎪ ⎪ + Z Z −1 (s){(Dε (εs) − D0 (εs))y0 (εs) + Cε (εs)x0 (εs) + g(εs)}ds ⎪ ⎪ n ε ⎪ ⎪ ε ⎪ ⎪ εt ⎪ ⎪ t ⎪ ⎪ + Z n ≤ t < n + 1. Z −1 (s)C1,ε (εs)dsx0 (n), ⎩ n ε ε

(4.3)

(4.4)

In view of the continuity of a solution at a point, we arrive at the following difference equation ⎧ n+1 ⎪ ⎪ X −1 (s)B1,ε (s)dsy(n) ⎪ x(n + 1) = E(n)x(n) + X(n + 1) ⎪ ⎪ n ⎪ ⎪ n+1 ⎪ ⎪ ⎪ ⎪ ⎪ X −1 (s)(A1,ε (s) − A1,0 (s))dsx0 (n) + X(n + 1) ⎪ ⎪ ⎪ n ⎪ n+1 ⎪ ⎪ ⎪ ⎪ ⎪ X −1 (s){(Aε (s) − A0 (s))x0 (s) + Bε (s)y(s) + f (s)}ds, + X(n + 1) ⎪ ⎪ ⎨ n (4.5) (n, ε)y(n) + G1 (n, ε)y0 (n) y(n + 1) = G 0 ⎪ ⎪ n+1 ⎪ ⎪ ε ⎪ n+1 ⎪ ⎪ + Z Z −1 (s)C1,ε (εs)dsx0 (n) ⎪ ⎪ n ε ⎪ ⎪ ε ⎪ ⎪ n+1 ⎪ ⎪ ε n+1 ⎪ ⎪ ⎪ Z + Z −1 (s){(Dε (εs) − D0 (εs))y0 (εs) ⎪ ⎪ n ε ⎪ ⎪ ε ⎪ ⎩ + Cε (εs)x0 (εs) + g(εs)}ds, where

n+1 E(n) = X(n + 1) X −1 (n) + X −1 (s)A1,0 (s)ds , n n+1 n G0 (n, ε) = Z Z −1 , ε ε n+1 ε n+1 Z G1 (n, ε) = Z −1 (s)D1,ε (εs)ds. n ε ε


432

Note that the second equation in (4.5) is solved first. It has a unique pseudo-almost periodic sequence solution {y(n, ε)}n∈Z . Similarly to [8], it can be proved that the y(t) defined by (4.4), with values y(n, ε) at t = n, is pseudo-almost periodic, besides, it can be seen that y(t) ∈ B. Then this y(t) is put into the first equation in (4.5) which is then solved for {x(n, )}n∈Z . Therefore, the inhomogeneous difference equation (4.5) has a pseudo-almost periodic sequence solution (x(n, ε), y(n, ε)). At this time, it can be proved that the solution (x(t, ε), y(t, ε)) defined by (4.4) is the unique pseudo-almost periodic solution of Eq. (4.3). Moreover, it can be seen that (x(t, ε), y(t, ε)) ∈ B. Writing (x, y) = T (x0 , y0 , f, g, ε), then solving (4.1) is equivalent to finding a fixed point of T (·, ·, f, g, ε). If (x, y) = T (x0 , y0 ; f, g, ε), (x, y) = T (x0 , y 0 ; f, g, ε), letting u = x − x, v = y − y, we find that u and v satisfy ⎧ u (t) = A0 (t)u(t) + A1,0 (t)u([t]) + (Aε (t) − A0 (t))(x0 (t) − x0 (t)) ⎪ ⎪ ⎪ ⎪ ⎨ + (A1,ε (t) − A1,0 (t))(x0 ([t]) − x0 ([t])) + Bε (t)v(t) + B1,ε (t)v([t]), (4.6) ⎪ εv (t) = D0 (t)v(t) + D1,ε (t)(y0 ([t]) − y 0 ([t])) + (Dε (t) − D0 (t))(y0 (t) − y 0 (t)) ⎪ ⎪ ⎪ ⎩ + Cε (t)(x0 (t) − x0 (t)) + C1, (t)(x0 ([t]) − x0 ([t])). From (3.18), it follows that |v(t)| ≤ U (||Cε || + ||C1,ε ||)||x0 − x0 || + U (||D1,ε || + ||Dε − D0 ||)||y0 − y 0 ||.

(4.7)

From (3.9) and (4.7), we arrive at |u(t)| ≤ N [||Aε − A0 || + ||A1,ε − A1,0 ||]||x0 − x0 || + N [||Bε || + ||B1,ε ||]||v|| ≤ N [||Aε − A0 || + ||A1,ε − A1,0 || + U (||Bε || + ||B1,ε ||)(||Cε || + ||C1,ε ||)]||x0 − x0 || + N U (||Bε || + ||B1,ε ||)(||D1,ε || + ||Dε − D0 ||)||y0 − y 0 ||. (4.8) Choose ε1 ≤ ε0 so small that N [||Aε − A0 || + ||A1,ε − A1,0 || + U (||Bε || + ||B1,ε ||)(||Cε || + ||C1,ε ||)] < N U (||Bε || + ||B1,ε ||)(||D1,ε || + ||Dε − D0 ||)
K, and Σ(ε) is a projection such that rankΣ(ε)=rankΣ1 (ε). Denote Σ(σ, ε) = T(σ, ε)Σ(ε)T−1 (σ, ε),

σ ∈ Z.

Then Σ(σ, ε)Rp+q is the subspace of initial data z0 such that the solution of Eq. (5.6) z(n) = T(n, ε)T−1 (σ, ε)z0 tends to zero as n → +∞ at an exponential rate α and (I − Σ(σ, ε))Rp+q is the subspace of initial data for which that solution approaches zero as n → −∞ at an exponential rate α. Given δ > 0, σ ∈ Z, define δM S(σ, ε) = z ∈ Rp+q ; |Σ(σ, ε)z| < 2KeM (M + γ) ∗ ∗ and for z0 = z + z (σ, ε), |z(t, σ, z0 , ε) − z (t, ε)| ≤ δ, t ≥ σ , and U (σ, ε) =

z ∈ Rp+q ; |(I − Σ(σ, ε))z|
0 and ε4 ∈ (0, ε0 ) such that the mapping ) and for each Σ(σ, ε), σ ∈ Z, is a homeomorphism of S(σ, ε) onto Σ(σ, ε)Rp+q ∩ B( 2KeMδM (m+γ) ∗ z0 = z + z (σ, ε) with z ∈ S(σ, ε), 0 < ε ≤ ε4 , |z(t, σ, z0 , ε) − z ∗ (t, ε)| ≤ 2K

e2M (M + γ) |Σ(σ, ε)z|e−α(t−σ)/2 , t ≥ σ. M

The map gs (·, σ, ε) : Σ(σ, ε)R

p+q

δM ∩B M 2Ke (m + γ)

→ S(σ, ε),

inverse to Σ(σ, ε)|S(σ,ε) , is Lipschitz with constant 2Ke2M (M + γ)/M. The mapping I − ) and for Σ(σ, ε), σ ∈ Z, is a homeomorphism of U (σ, ε) onto (I − Σ(σ, ε))Rp+q ∩ B( 2KeMδM (m+γ) ∗ each z0 = z + z (σ, ε) with z ∈ U (σ, ε), 0 < ε ≤ ε4 , |z(t, σ, z0 , ε) − z ∗ (t, ε)| ≤ 2K

e2M (M + γ) |(I − Σ(σ, ε))z|e−α(t−σ)/2 , t ≤ σ. M

The map gu (·, σ, ε) : (I − Σ(σ, ε))Rp+q ∩ B

δM 2KeM (m + γ)

→ U (σ, ε),

inverse to I − Σ(σ, ε)|U(σ,ε) , is Lipschitz with constant 2Ke2M (M + γ)/M. Proof In fact, the proof of this theorem is similar to [9], so we omit it. Remark 5.1 For σ ∈ / Z, the continuity and uniqueness of a solution with initial values imply that z(σ, [σ], ·, ε) is a homeomorphism of S([σ], ε) onto z(σ, [σ], ·, ε), which is a stable manifold of z ∗ (t, ε) at σ. 6

An Example

In this section, we consider the following singularly perturbed system with piecewise constant argument εDu (t) + u (t) − f (t, u(t), u([t]), ε) = 0, (6.1) where ε is a small positive parameter, D is an n × n matrix described below and f is continuous ∂f ∂f ∂f ∂f in all variables, as are ∂u , ∂f , ∂f ∂ε . In addition, f, ∂u0 , ∂u1 , and ∂ε are continuous in (u0 , u1 , ε) 0 ∂u1 uniformly in t ∈ R and bounded on the bounded (u0 , u1 , ε) sets uniformly in t ∈ R. The equation (6.1) comes from the paper [9]. We assume that D has no positive real part eigenvalues, and has purely imaginary eigenvalues except possibly zero. Thus, D is similar to D 0 , 0 0r where 0r is an r × r 0 matrix, r < n, and D is an (n − r) × (n − r) matrix with negative real part eigenvalues. With respect to appropriate coordinates u = (u1 , u2 ) ∈ Rn−r × Rr , Eq. (6.1) is of the form εDu1 (t) + u1 (t) − f1 (t, u1 (t), u2 (t), u1 ([t]), u2 ([t]), ε) = 0, (6.2) u2 (t) − f2 (t, u1 (t), u2 (t), u1 ([t]), u2 ([t]), ε) = 0. We assume that the reduced equation u (t) = f (t, u(t), u([t]), 0)

(6.3)


437

has a pseudo-almost periodic solution u = ( u1 (t), u 2 (t)) defined on R such that u and u 1 are bounded on R. A change of variables x1 = εDu 1 + u1 ,

u=u + u,

x2 = u2 ,

y = εDu

1

leads to the system ⎧ x (t) = A(t, ε)x(t) + A1 (t, ε)x([t]) + B(t, ε)y(t) + B1 (t, ε)y([t]) ⎪ ⎪ ⎪ ⎪ ⎨ + r1 (t, x(t), x([t]), y(t), y([t]), ε),

(6.4)

−1 ⎪ εy (t) = εC(t, ε)x(t) + εC1 (t, ε)x([t]) − [D + εE(t, ε)]y(t) + εE1 (t, ε)y([t]) ⎪ ⎪ ⎪ ⎩ + εr2 (t, x(t), x([t]), y(t), y([t]), ε),

where

A(t, ε) =

A11 (t, ε)

A12 (t, ε)

A21 (t, ε)

A22 (t, ε)

with Aij (t, ε) =

,

∂fi (t, u (t), u ([t]), ε), ∂u0j

B(t, ε) = −

A11 (t, ε) A21 (t, ε)

A1 (t, ε) =

A1ij (t, ε) =

,

B1 (t, ε) = −

A111 (t, ε)

A112 (t, ε)

A121 (t, ε)

A122 (t, ε)

,

∂fi (t, u (t), u ([t]), ε), ∂u1j

A111 (t, ε) A121 (t, ε)

,

C(t, ε) = [A11 (t, ε), A12 (t, ε)], C1 (t, ε) = [A111 (t, ε), A112 (t, ε)], E(t, ε) = A11 (t, ε), E1 (t, ε) = A111 (t, ε), r1 (t, x(t), x([t]), y(t), y([t]), ε) = (s1 , s2 ), r2 (t, x(t), x([t]), y(t), y([t]), ε) = s1 s1 = f1 (t, ( u1 + x1 − y)(t), ( u2 + x2 )(t), ( u1 + x1 − y)([t]), ( u2 + x2 )([t]), ε) −f1 (t, u 1 (t), u 2 (t), u 1 ([t]), u 2 ([t]), 0) − A11 (t, ε)(x1 (t) − y(t)) 1 −A12 (t, ε)x2 (t) − A11 (t, ε)(x1 ([t]) − y([t])) − A112 (t, ε)x2 ([t]) − εD u 1 (t), s2 = f2 (t, ( u1 + x1 − y)(t), ( u2 + x2 )(t), ( u1 + x1 − y)([t]), ( u2 + x2 )([t]), ε) −f2 (t, u 1 (t), u 2 (t), u 1 ([t]), u 2 ([t]), 0) − A21 (t, ε)(x1 (t) − y(t)) −A22 (t, ε)x2 (t) − A121 (t, ε)(x1 ([t]) − y([t])) − A122 (t, ε)x2 ([t]). We assume that the equation z (t) = A(t, 0)z(t)

(6.5)

admits an exponential dichotomy on R with projection P and constants K1 , α1 . It is apparent that Eq. (6.4) has the same form as Eq. (4.9) considered in Section 4. As an immediate consequence of Theorem 4.2 we have the following Theorem 6.1

Suppose that |A1 (t, 0)| ≤

α1 e −1 M eα − 1 min , , 2K1 eM 1 + eα1 (1 + e−α )eα1

(6.6)

where α = α1 /2. Then there exist positive constants ε2 and ρ1 such that for each ε satisfying || ≤ 0 < ε ≤ ε2 , Eq. (6.2) has a unique pseudo-almost periodic solution u∗ (t, ε) satisfying ||u∗ − u ρ1 . The solution is continuous in ε uniformly with respect to t ∈ R and satisfies the estimates ||u∗ (ε) − u || + ||u∗1 (ε) − u 1 || = O(ε) as ε → 0+ . From Theorem 5.1, we obtain the following result, which describes the stability properties of the solution u∗ (t, ε) of Eq. (6.2).

438


Theorem 6.2 Suppose that (6.6) holds. Then for all sufficiently small ε > 0 and for all σ ∈ R, there is a Lipschitz manifold, S(σ, ε)(U (σ, ε)), of initial data at t = σ, having dimension dim R(P ) + n − r(n − dim R(P )) in R2n−r , corresponding to the solutions of Eq. (6.2) which are asymptotic to the solution (u∗1 (t, ε), u∗1 (t, ε), u∗2 (t, ε)) at an exponential rate of attraction as t → +∞(t → −∞).In particular, the latter solution is uniformly asymptotically stable if and only if the trivial solution of (6.5) is uniformly asymptotically stable. References [1] Friedrichs, K. O., Wasow, W. R.: Singular perturbations of nonlinear oscillations. Duke Math. J., 13, 367–381 (1946) [2] Flatto, L., Levinson, N.: Periodic solutions of singularly perturbed systems. J. Rat. Mech. Anal., 4, 943–950 (1955) [3] Anosov, D. V.: On limit cycles in systems of differential equations with a small parameter in the highest derivative. Amer. Math. Soc. Trans., 33(2), 233–275 (1963) [4] Hale, J. K., Seifert, G.: Bounded and almost periodic solutions of singularly perturbed equations. J. Math. Anal. Appl., 3, 18–24 (1961) [5] Chang, K. W.: Almost periodic solutions of singularly perturbed systems of differential equations. J. Differential Equations, 4, 300–307 (1968) [6] Smith, H. L.: On the existence and stability of bounded, almost periodic and periodic solutions of a singularly perturbed nonautonomous system. Diff. Integ. Equs., 8(8), 2125–2144 (1995) [7] Yuan, R.: Almost periodic solutions of a class of singularly perturbed differential equations with piecewise constant argument. Nonlinear Anal., TMA, 37, 841–859 (1999) [8] Yuan, R.: On a new almost periodic type solution of a class of singularly perturbed differential equations with piecewise constant argument. Science in China (Series A), 45(4), 484–502 (2002) [9] Yuan, R.: On the existence of almost periodic solutions of a singularly perturbed differential equation with piecewise constant argument. Z. Angew. Math. Phys., 50, 94–119 (1999) [10] Cooke, K. L., Wiener J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl., 99, 265–297 (1984) [11] Cooke, K. L., Wiener, J.: A survey of differential equations with piecewise continuous arguments, Lecture Note in Mathematics, Springer-Verlag, Berlin, 1475, 1–15, 1991 [12] Papaschinopoulos, G.: On asymptotic behavior of the solutions of a class perturbed differential equations with piecewise constant argument. J. Math. Anal. Appl., 185, 490–500 (1994) [13] Yuan, R.: Hong, J., The existence of almost periodic solutions for a class of differential equations with piecewise constant argument. Nonlinear Anal., TMA, 28(8), 1439–1450 (1997) [14] Hong, J., Obaya, R., Sanz, A.: Almost periodic type solutions of some differential equations with piecewise constant argument. Nonlinear Anal., TMA, 45, 661–688 (2001) [15] Yuan, R.: Pseudo almost periodic solutions of second order neutral delay differential equations with piecewise constant argument. Nonlinear Anal., TMA, 41, 871–890 (2000) [16] Yuan, R., Piao, D.: Pseudo almost periodic solutions of differential equations with piecewise constant argument. Applicable Analysis, 73(3–4), 345–357 (1999) [17] Piao, D.: Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t]. Science in China, (Series A), 44, 1156–1161 (2001) [18] Piao, D.: Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [t + 1/2]. Science in China, (Series A), 47, 31–38 (2004) [19] Fink, A. M.: Almost periodic differential equations. Lecture Notes in Mathematics, 377, Springer, Berlin, 1974 [20] Zhang, C.: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl., 181, 62–76 (1994) [21] Ait Dads, E., Arino, O.: Exponential dichotomy and existence of pseudo-almost solution of some differential equations. Nonlinear Anal., TMA, 27(4), 369–386 (1996)