DSpace at VNU: On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations

PURE AND APPLIED ANALYSISON THE EXISTENCE OF QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS Nguyen Minh Man and Nguyen Van Minh Department of M

PURE AND APPLIED ANALYSIS

ON THE EXISTENCE OF QUASI PERIODIC AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL

DIFFERENTIAL EQUATIONS

Nguyen Minh Man and Nguyen Van Minh

Department of Mathematics Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam

Abstract This paper is concerned with the existence of almost periodic

solutions of neutral functional differential equations of the form dtdDx t =

Lx t + f (t), where D, L are bounded linear operators from C := C([−r, 0], C n )

to C n , f is an almost (quasi) periodic function We prove that if the set of

imaginary solutions of the characteristic equations is bounded and the equation

has a bounded, uniformly continuous solution, then it has an almost (quasi)

periodic solution with the same set of Fourier exponents as f

1 Introduction The problem we consider in this paper is to find such conditions that linear neutral functional differential equations of the form

d

where D, L are bounded linear operators from C := C([−r, 0], Cn

) to Rn, f is an almost (quasi) periodic function in the sense of Bohr, have an almost (quasi) periodic solution with the same Fourier exponents as f

This problem has a long history and is one of the main concerns in the qualitative theory of differential equations In the simplest case of ordinary differential equa-tions, when one deals with the τ -periodicity of the integral F (t) =Rt

0f (ξ)dξ, where

f is τ -periodic and continuous, an easy computations shows that F is τ -periodic

if and only if it is bounded This simple conclusion turns out to be a motivation for numerous works In this direction, for more details on results on the existence

of periodic solutions we refer the reader to [19, 4, 16, 25, 22] When f is periodic the method of study in these works is to prove the existence of fixed points of the period maps However, this method does not work in the more general case where

f is almost periodic A new method of study was introduced in [23, 7, 13, 12, 20]

to overcome this obstacle which makes use of the so-called evolution semigroups associated with the evolutionary processes generated by equations In our equation (1.1), with general assumptions on D and L, the Cauchy problem may have no so-lutions, so one has no evolutionary processes (or in this case, solution semigroups)

In our recent paper [21], we have begun studying conditions for the existence of almost periodic solutions to Eq (1.1) We have showed that if (1.1) has a bounded, uniformly continuous solution, ∆i\Sp(f ) is closed, ∆i is bounded, and Sp(f ) is

2000 Mathematics Subject Classification 34K14, 34K06.

Key words and phrases Neutral functional differential equation, almost periodic solution, quasi periodic solution.

countable, where Sp(f ) is the Beurling spectrum of f , that is equal to the closure

of the set of the Fourier exponents of f in the case of almost periodic functions, and

∆i := {ξ ∈ R : det ∆(iλ) = 0}

∆(λ) := λDeλ·− Leλ·, λ ∈ C, then (1.1) has an almost periodic solution w such that Sp(w) ⊂ Sp(f )

In this paper, by showing that every bounded, uniformly continuous solution

of (1.1) is almost periodic, we will prove (see Theorem 3.6) that if the set ∆i is bounded, then the existence of a bounded, uniformly continuous solution of (1.1) yields readily the existence of an almost periodic solution w with the same set of Fourier exponents as f without any additional conditions This implies in particular (see Corollary 3.7) the existence of quasi periodic solutions if the forcing term f is quasi periodic

2 Spectral Theory of Functions and Almost Periodicity

2.1 Notations In the remaining part of this paper we will use the following no-tations: C, and <z stand for the set of complex numbers and the real part of a complex number z, respectively BC(R, Cn) and BU C(R, Cn) denote the space

of Cn-valued bounded continuous functions and the space of Cn-valued bounded uniformly continuous functions on R, respectively

2.2 Spectrum of a function We denote by F the Fourier transform, i.e

(F f )(s) :=

Z +∞

−∞

(s ∈ R, f ∈ L1

(R)) Then the Beurling spectrum of u ∈ BU C(R, Cn) is defined to

be the following set

Sp(u) := {ξ ∈ R : ∀ > 0 ∃f ∈ L1(R),

suppF f ⊂ (ξ − , ξ + ), f ∗ u 6= 0}

where

f ∗ u(s) :=

Z +∞

−∞

f (s − t)u(t)dt

Theorem 2.1 Under the notation as above, Sp(u) coincides with the set consisting

of ξ ∈ R such that the Fourier- Carleman transform of u

ˆ u(λ) =

( R∞

0 e−λtu(t)dt, (<λ > 0);

−R∞

0 eλtu(−t)dt, (<λ < 0) (2.2) has no holomorphic extension to any neighborhood of iξ

Proof For the proof we refer the reader to [24, Proposition 0.5, p.22]

We collect some main properties of the spectrum of a function, which we will need in the sequel, for the reader’s convenience

Theorem 2.2 Let f, gn∈ BU C(R, Cn

), n ∈ N such that gn→ f as n → ∞ Then

1 Sp(f ) is closed,

2 Sp(f (· + h)) = Sp(f ) ∀h ∈ R,

3 If α ∈ C\{0}, Sp(αf ) = Sp(f ),

4 If Sp(gn) ⊂ Λ for all n ∈ N then Sp(f ) ⊂ Λ,

5 Sp(ψ ∗ f ) ⊂ Sp(f ) ∩ suppF ψ, ∀ψ ∈ L1(R).

Proof For the proof we refer the reader to [15] or [24, pp 20-21]

As an immediate consequence of Theorem 2.2 we have that if Λ is a closed subset

of R, then the function space Λ(Cn) := {f ∈ BU C(R, Cn) : Sp(f ) ⊂ Λ} is a closed subspace of BU C(R, Cn)

Definition 2.3 Let g ∈ BU C(R, Cn) Then the almost periodic spectrum of g is defined to be the set

SpAP(g) := {ξ ∈ R : ∀ > 0, ∃φ ∈ L1

(R) s.t

suppF φ ⊂ (ξ − , ξ + ), φ ∗ g /∈ AP (Cn)}

Obviously, for any g ∈ BU C(R, Cn), we have SpAP(g) ⊂ Sp(g)

2.3 Almost periodic functions We recall that a subset E ⊂ R is said to be relatively dense if there exists a number l > 0 (inclusion length) such that every interval [a, a + l] contains at least one point of E Let f be a continuous function

on R taking values in Cn f is said to be almost periodic in the sense of Bohr if to every  > 0 there corresponds a relatively dense set T (, f ) (of -periods ) such that

sup

t∈R

kf (t + τ ) − f (t)k ≤ , ∀τ ∈ T (, f )

For all reals λ such that the following integrals (Fourier coefficients)

a(λ, f ) := lim

T →∞

1 2T

Z T

−T

f (t)e−iλtdt exist As is known (see e.g [6]), there are at most countably reals λ (Fourier exponents) such that a(λ, f ) 6= 0, the set of which will be denoted by σb(f ) and called Bohr spectrum of f Throughout the paper we will use the relation Sp(f ) = σb(f ) and denote by AP (X) the space of all almost periodic functions taking valued in X, here X is either Rn or Cn with sup norm We summarize several main properties

of almost periodic functions, whose proofs can be found in [6], in the following theorem:

Theorem 2.4 The following assertions hold:

1 f ∈ BC(R, X) is almost periodic if and only if for every sequence {τn}∞

n=1⊂ R the sequence of functions {fτn = f (τn+ ·)}∞

n=1contains at least a convergent subsequence;

2 If f is X-valued almost periodic, then f ∈ BU C(R, X);

3 AP (X) is a closed subspace of BU C(R, X);

4 If f ∈ AP (X) and f0 exists as an element of BU C(R, X), then f0∈ AP (X) Example 2.5 The following examples illustrate close relations between spectrum

of a function with its behavior:

1 If u(t) = aeiλt

, 0 6= a ∈ Cn

, λ ∈ R, ∀t ∈ R, then Sp(u) = {λ}

2 If u(·) is a bounded Cn-valued continuous function, then u is τ -periodic (for some τ > 0 if and only if Sp(u) ⊂ 2πZ

τ

3 For u ∈ L1

(R) we have Sp(u) = suppFu

The most important criterion for the almost periodicity of a bounded continuous function which we will use in this paper is the following:

Theorem 2.6 Let u ∈ BU C(R, Cn) Then u is almost periodic provided that

SpAP(u) is countable

Proof For the proof (of more general results) we refer the reader to [15], [14, Chap 4]

Next, we gather some further properties of the almost periodic spectrum SpAP(f )

by quoting here another approach to this kind of spectrum via the notion of quotient groups (see [1]) Consider the shift group (S(t))t∈R on BU C(R, X) with generator

B Since (S(t))t∈R leaves AP (X) invariant there is an induced C0-group ( ¯S(t))t∈R

on Y := BU C(R, X)/AP (X) given by

¯ S(t)π(u) = π(S(t)u), ∀t ∈ R, u ∈ BUC(R, X), where π : BU C(R, X) → BU C(R, X)/AP (X) denotes the quotient mapping Its generator is denoted by ˜B Let u ∈ BU C(R, X) Define

˜ u(λ) =

(R∞

0 e−λtS(t)π(u)dt,¯ (<λ > 0)

−R∞

0 eλtS(−t)π(u)dt,¯ (<λ < 0).

Then we have

Proposition 2.7 SpAP(u) coincides with the set of ξ ∈ R such that ˜u(λ) has no holomorphic extension to any neighborhood of iξ from the right half plane

Proof For the proof see [1, §2, §3]

3 Main Results Consider now the equation Eq (1.1) Using the Riesz repre-sentation of bounded linear functionals we can re-write D and L in the form

Dφ =

Z 0

−r

Lφ =

Z 0

−r

where µ(θ) and η(θ) are n × n-matrices whose entries are functions of θ of bounded variation

In the remaining part of this paper we denote ∆(λ) := λDeλ·− Leλ·, for every

λ ∈ C, where

Deλ·:=

Z 0

−r

eλθdµ(θ); Leλ·:=

Z 0

−r

eλθdη(θ)

ρ(D, L) := {λ ∈ C : ∃ ∆−1(λ)} Hence, for every λ ∈ C, ∆(λ) is a matrix depending analytically on λ Throughout this paper we make the standing assumption that

Lemma 3.1 ([21]) Under the above notations, the following assertions hold true

1 ρ(D, L) is an open subset of C and σ(D, L) := C\ρ(D, L) is discrete;

2 ∆−1(λ) is an analytic function on ρ(D, L)

Below we let ∆i:= {ξ ∈ R : ∆(iξ) = 0} Then we have

Lemma 3.2 Let x be a bounded, uniformly continuous solution of Eq (1.1) Then

Sp (x) ⊂ ∆

Proof Taking Carleman transforms of both sides of (1.1) and applying Fubini The-orem, for <λ > 0 we have

Z +∞

0

d

dtDxte

−λtdt =

Z +∞

0

Lxte−λtdt + ˆf (λ) Thus we have

Z +∞

0

d

dt[

Z 0

−r

dµ(θ)xt(θ)]e−λtdt =

Z +∞

0

[

Z 0

−r

dη(θ)xt(θ)]e−λtdt + ˆf (λ) Consequently,

[

Z 0

−r

dµ(θ)xt(θ)]e−λt|+∞

0 + λ

Z +∞

0

[

Z 0

−r

dµ(θ)xt(θ)]e−λtdt

=

Z +∞

0

[

Z 0

−r

dη(θ)xt(θ)]e−λtdt + ˆf (λ)

Therefore,

−

Z 0

−r

dµ(θ)x0(θ) + λ

Z 0

−r

dµ(θ)[

Z +∞

0

xt(θ)e−λtdt

=

Z 0

−r

dη(θ)[

Z +∞

0

xt(θ)e−λtdt + ˆf (λ)

Re-writing the above expression we have

−Dx0+ λ

Z 0

−r

dµ(θ)[

Z +∞

0

x(t + θ)e−λ(t+θ).eλθdt]

=

Z 0

−r

dη(θ)[

Z +∞

0

x(t + θ)e−λ(t+θ)eλθdt] + ˆf (λ)

=

Z 0

−r

eλθdη(θ)[

Z +∞

θ

x(ξ)e−λξdξ] + ˆf (λ)

=

Z 0

−r

eλθdη(θ)[

Z +∞

0

x(ξ)e−λξdξ −

Z θ 0

x(ξ)eλξdξ] + ˆf (λ)

So, we have

−Dx0+ λ

Z 0

−r

eλθdµ(θ).ˆx(λ) − λ

Z 0

−r

eλθdµ(θ).[

Z θ 0

x(ξ)e−λξdξ]

= Leλ·x(λ) − Leˆ λ·[

Z θ 0

x(ξ)e−λξdξ] + ˆf (λ)

Therefore,

[λDeλ·− Leλ·]ˆx(λ) = φ(λ) + ˆf (λ), where

φ(λ) = Dx0+ λ

Z 0

−r

dµ(θ)eλθ[

Z θ 0

x(ξ)e−λξdξ] −

Z 0

−r

dη(θ)eλθ[

Z θ 0

x(ξ)e−λξdξ] Finally,

∆(λ)ˆx(λ) = φ(λ) + ˆf (λ) (3.4)

Now for s ∈ R let us denote S(s)u by us By a similar computation we have

∆(λ)xbs(λ) = cfs(λ) + ϕs(λ), (3.5) where

ϕs(λ) := Dxs0+ λ

Z 0

−r

dµ(θ)eλθ[

Z θ 0

xs(ξ)e−λξdξ] −

Z 0

−r

dη(θ)eλθ[

Z θ 0

xs(ξ)e−λξdξ] which is analytic in λ ∈ C

Let λ06∈ ∆i Then det ∆(iλ0) 6= 0 As shown in [21], ∆−1(λ) is analytic in λ in

a neighborhood of iλ0 of the complex plane Hence,

b

xs(λ) = ∆−1(λ)cfs(λ) + ∆−1(λ)φs(λ), (3.6) for λ in a neighborhood of iλ0 Obviously, for <λ > 0

b

xs(λ) =

Z ∞

0

e−λtxs(ξ)dξ =

Z ∞ 0

e−λtx(ξ + s)dξ

=

Z ∞

0

e−λt[S(ξ)x](s)dξ =

Z ∞ 0

e−λtS(ξ)xdξ

 (s) = (R(λ, B)x)(s) Similarly,

c

fs(λ) = (R(λ, B)f )(s), ∀<λ > 0

Substituting these expresions into (3.6) we have

R(λ, B)x = ∆−1(λ)R(λ, B)f + ∆−1(λ) ˜ϕ(λ), (3.7) where ˜ϕ(λ) : R 3 s 7→ ϕs(λ) Hence, in a neighborhood of iλ0, we have

π(R(λ, B)x) = π(∆−1(λ)R(λ, B)f ) + π(∆−1(λ)ϕ(λ)) (3.8) Since f ∈ AP (X) we have

π(∆−1(λ)R(λ, B)f ) = 0 ∈ Y

Therefore, using the indentity ˜x(λ) = R(λ, ˜B)π(x) for <λ > 0, we have

˜

x(λ) = R(λ, ˜B)π(x) = π(R(λ, B)x) = ∆−1(λ)π(ϕ(λ))

= ∆−1(λ) ˜ϕ(λ) <λ > 0 (3.9)

On the other hand, ˜ϕ(λ) is analytic in λ ∈ C Finally, around iλ0the function ˜x(λ) has a holomorphic extension from the right half plane That is λ06∈ SpAP(x) This proves the lemma

Corollary 3.3 Under the standing assumption, any bounded and uniformly con-tinuous solution of Eq (1.1) is almost periodic

Proof Since the zero set of ∆(λ) in this case consists of isolated points, it is count-able So is ∆i By the above lemma, the almost periodic spectrum of any bounded and uniformly continuous solution should be countable, so the solution is almost periodic

Remark 3.4 The almost periodicity of bounded solutions of ordinary differential equations has been proved in [6] For abstract ordinary differential equations in Banach spaces this result has been considered in [14, Chap 6] with additional conditions (see also [1] for a more complete proof)

Lemma 3.5 Let x be a bounded and uniformly continuous solution of Eq (1.1) Then

σb(f ) ⊂ σb(x) ⊂ ∆i∪ σb(f ) (3.10) Proof By Corollary 3.3, this solution x is almost periodic Thus, the Bohr trans-forms of both sides of Eq (1.1) exist Therefore,

lim

T →∞

1 2T

Z T

−T

e−iλt d dt

Z 0

−r

dµ(ξ)x(t + ξ)

 dt

= lim

T →∞

1 2T

Z T

−T

e−iλt

Z 0

−r

dη(ξ)x(t + ξ)



dt + a(λ, f )

Therefore, integrating the above left side by parts we have

lim

T →∞

1 2Te

−iλtZ 0

−r

dµ(ξ)x(t + ξ)|T−T

− lim

T →∞

1 2T

Z T

−T

(−iλ)e−iλt

Z 0

−r

dµ(ξ)x(t + ξ)dt

= lim

T →∞

1 2T

Z T

−T

Z 0

−r

dη(ξ)x(t + ξ)e−iλt



dt + a(λ, f )

The first term of the left side vanishes as x and e−iλt are bounded Hence,

iλ

Z 0

−r

dµ(ξ) lim

T →∞

1 2T

Z T

−T

e−iλtx(t + ξ)dt

=

Z 0

−r

dη(ξ) lim

T →∞

1 2T

Z T

−T

e−iλtx(t + ξ)dt + a(λ, f )

Thus,

iλ

Z 0

−r

dµ(ξ)[ lim

T →∞

1 2T

Z T

−T

e−iλθx(θ)dθ

+ lim

T →∞

1 2T

Z T +ξ T

x(θ)e−iλθ− lim

T →∞

1 2T

Z −T +ξ

−T

x(θ)e−iλθ]eiλξ

=

Z 0

−r

dη(ξ)[ lim

T →∞

1 2T

Z T

−T

x(θ)eiλθdθ

+ lim

T →∞

1 2T

Z T +ξ T

e−iλθdθ − lim

T →∞

1 2T

Z −T +ξ

−T

x(θ)e−iλθ]eiλξ Consequently,

iλa(λ, x)

Z 0

−r

dµ(ξ)eiλξ= a(λ, x)

Z 0

−r

dη(ξ)eiλξ+ a(λ, f )

Hence,

iλDeiλ·− Leiλ·
a(λ, x) = a(λ, f )

If λ ∈ R, λ 6∈ ∆i and λ 6∈ σb(f ), then a(λ, x) = 0, i.e., λ 6∈ σb(x) This shows that

σb(x) ⊂ ∆i∪ σb(f ) Next, if λ ∈ R and λ 6∈ σb(x), then, by definition, a(λ, x) = 0 Thus, a(λ, f ) = 0 By definition, λ 6∈ σb(f ), i.e., σb(f ) ⊂ σb(x)

We are now in a position to prove the main result of this note

Theorem 3.6 Let ∆i be bounded Moreover, assume that Eq (1.1) has a bounded, uniformly continuous solution Then, there exists an almost periodic solution w to

Eq (1.1) such that

Proof If ∆iis bounded, since it is part of the zero set of det ∆(λ) = 0 it consists of finitely many points By the Corollary 3.3 the bounded solution, denoted by x, of

Eq (1.1) is almost periodic Suppose that x ∼P∞

k=0eiλktxk is the Fourier series

of x Set

P (t) = X

λk∈∆ i ∩σ b (x)

eiλk txk

Obviously, P (t) is a trigonometric polynomial Moreover, it is a solution to the homogeneous equation In fact, it suffices to show that every term Pk(t) := eiλ k txk

is a solution of the corresponding homogeneous equation By definition, ∆(iλk) = 0 Hence,

d

dtDP

k(t) − LPk(t) = ∆(iλk)xk = 0

Define

w(t) = x(t) − P (t)

Obviously, w is almost periodic Moreover,

σb(w) = σb(x)\∆i Hence,

σb(w) ⊂ σb(f )

It remains to show that w is an almost periodic solution to Eq (1.1) But this is obvious as P (t) is a solution to the corresponding homogeneous equation Combined with the inclusion σb(f ) ⊂ σb(w) of the Lemma 3.5, this proves the theorem

A set of reals S is said to have an integer and finite basis if there is a finite subset

T ⊂ S such that any element s ∈ S can be represented in the form s = n1b1+

· · · + nmbm, where nj∈ Z, j = 1, · · · , m, bj∈ T, j = 1, · · · , m An almost periodic function f is said to be quasi periodic if it is of the form f (t) = F (t, t, , t), t ∈ R, where F (t1, t2, , tp) is a Cn-valued continuous function of p variables which is periodic in each variable The function f is quasi periodic if and only if the set of its Fourier-Bohr exponents has an integer and finite basis (see [14, p.48])

Corollary 3.7 Let all assumptions of the above theorem be satisfied Moreover, let f be quasi-periodic Then if Eq (1.1) has a bounded, uniformly continuous solution, then it has a quasi-periodic solution w such that σb(w) ⊂ σb(f )

Proof This corollary is an immediate consequence of Theorem 3.6

Example 3.8 Consider the equation

˙ x(t) −1

3x(t − 2) = Bx(t − 1) + f (t),˙ x(t) ∈ Cn, (3.12) where B is a n × n-matrix and f is an almost periodic function ∆i is the set of

λ ∈ R such that the matrix

∆(iλ) = (iλ − 1

3iλe

−2iλ)I − e−iλB is not invertible

It is easy to see that if |λ| > 2kBk, then

|iλ −1

3iλe

−2iλ| = |λ||1 −1

3e

−2iλ| > 2kBk ×1

2 = kBk.

Therefore, the operator ∆(iλ) is invertible for large λ ∈ R Hence, ∆i is bounded Applying the above theorem we can conclude that if (3.12) has a bounded uniformly continuous solution, then it has an almost periodic solution w such that σb(w) =

σb(f )

Example 3.9 Consider scalar equations of the form

˙

x(t) +

N

X

k=1

Akx(t − τ˙ k) =

M

X

j=1

Bjx(t − µj) + f (t), x(t) ∈ R (3.13)

where N, M ∈ N, Ak, Bj are reals, τk, µj are positive reals Then, the characteristic operator is of the form

∆(λ) = λ + λ

N

X

k=1

e−τk λAk−

M

X

j=1

e−µj λBj

We now show that the function ∆(λ) 6≡ 0 This is obvious since

lim

λ∈R, λ→+∞∆(λ) = +∞

That is, the standing assumption holds for this class of equations If we impose the condition

N

X

k=1

|Ak| < 1, then by a simple computation we can show that for sufficiently large λ ∈ R, ∆(iλ)

is invertible That is, ∆i is bounded and our above results are applicable to this class of equations

Acknowledgements The authors are grateful to the referee for carefully reading and pointing out several inaccuracies in the previous version of the paper

REFERENCES [1] W Arendt and C.J.K Batty, Almost periodic solutions of first and second oder Cauchy problems, J Differential Equations, (2) 137 (1997), 363–383.

[2] R Benkhalti and K Ezzinbi, A Massera type criterion for some partial functional differential equations, Dynam Systems Appl., (2) 9 (2000), 221–228.

[3] Y.Q Chen, On Massera’s theorem for anti-periodic solution, Adv Math Sci Appl., 9 (1999), 125–128.

[4] S.N Chow and J.K Hale, Strongly limit-compact maps, Funkc Ekvac., 17(1974), 31–38 [5] M Fan and K Wang, Periodic solutions of linear neutral functional-differential equations with infinite delay, Acta Math Sinica, (4) 43 (2000), 695–702.

[6] A.M Fink, “Almost Periodic Differential Equations”, Lecture Notes in Math., 377, Springer, Berlin-New York, 1974.

[7] T Furumochi, T Naito and Nguyen Van Minh, Boundedness and almost periodicity of solu-tions of partial functional differential equasolu-tions, J Differential Equasolu-tions, 180 (2002), 125– 152.

[8] J.K Hale, “Theory of Functional Differential Equations”, Springer-Verlag, New York - Berlin 1977.

[9] J.K Hale and S.M Verduyn-Lunel, “Introduction to Functional Differential Equations”, Springer-Verlag, Berlin-New york, 1993.

[10] Y Hino and S Murakami, Periodic solutions of a linear Volterra system, Differential equa-tions (Xanthi, 1987), 319–326, Lecture Notes in Pure and Appl Math., 118, Dekker, New York, 1987.

[11] Y Hino, S Murakami and T Naito, “Functional Differential Equations with Infinite Delay”, Lect Notes Math 1473, Springer-Verlag, 1991.

[12] Y Hino, S Murakami and Nguyen Van Minh, Decomposition of variation of constants formula for abstract functional differential equations, Funkc Ekva To appear.

[13] Y Hino, T Naito, N.V Minh and J.S Shin, “Almost periodic solutions of differential equa-tions in Banach spaces”, Taylor and Francis, London-New York, 2002.

[14] B.M Levitan and V.V Zhikov, “Almost Periodic Functions and Differential Equations”, Moscow Univ Publ House 1978 English translation by Cambridge University Press 1982 [15] Y Katznelson, “An Introduction to Harmonic Analysis”, Dover Publications, New York , 1968.

[16] Y Li, Z Lin and Z Li, A Massera type criterion for linear functional-differential equations with advance and delay, J Math Anal Appl., (3) 200 (1996), 717–725.

[17] Y Li, F Cong, Z Lin and W Liu, Periodic solutions for evolution equations, Nonlinear Anal 36 (1999), 275–293.

[18] G Makay, On some possible extensions of Massera’s theorem, in ”Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999)” , No 16, 8

pp (electronic), Proc Colloq Qual Theory Differ Equ., Electron J Qual Theory Differ Equ., Szeged, 2000

[19] J.L Massera, The existence of periodic solutions of systems of differential equations, Duke Math J., 17, (1950), 457–475.

[20] S Murakami, T Naito and Nguyen Van Minh, Massera Theorem for Almost Periodic Solu-tions of Functional Differential EquaSolu-tions, J Math Soc Japan, To appear.

[21] Nguyen Minh Man and Nguyen Van Minh, Massera criterion for almost periodic solutions

of neutral functional differential equations, Intern J of Diff Eq and Dyn Sys., To appear [22] T Naito, Nguyen Van Minh, R Miyazaki and J.S Shin, A decomposition theorem for bounded solutions and periodic solutions of periodic equations, J Differential Equations, 160 (2000), 263-282.

[23] T Naito, Nguyen Van Minh and J.S Shin, New Spectral Criteria for Almost Periodic Solu-tions of Evolution EquaSolu-tions, Studia Mathematica, 145 (2001), 97–111.

[24] J Pr¨ uss, “Evolutionary Integral Equations and Applications”, Birkh¨ auser, Basel, 1993 [25] J.S Shin and T Naito, Semi-Fredholm operators and periodic solutions for linear functional-differential equations, J Differential Equations, 153 (1999), 407–441.

[26] E Shustin, E Fridman and L Fridman, Oscillation in second-order discontinuous system with delay, Discrete Contin Dyn Syst., (2) 9 (2003), 239–358.

Received January 2003; revised August 2003; final version January 2004 E-mail address: nvminh@netnam.vn