# Almost periodic solutions of neutral functional differential equations

## Almost periodic solutions of neutral functional differential equations

Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Application...

Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

Contents lists available at ScienceDirect

Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa

Almost periodic solutions of neutral functional differential equationsI Xiaoxing Chen ∗ , Faxing Lin College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, 350002, PR China

article

abstract

info

Article history: Received 1 May 2008 Accepted 12 February 2009 Keywords: Almost periodic solution Exponential dichotomy Krasnoselskii’s fixed point theorem Neutral functional differential equation

We study a nonlinear neutral functional differential equation. Applying the properties of almost periodic function and exponential dichotomy of linear system as well as Krasnoselskii’s fixed point theorem, we establish the conditions for the existence of almost periodic solution of the equations. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Neutral differential equations have many applications. For example, these equations arise in the study of two or more simple oscillatory systems with some interconnections between them [1,2], and in modeling physical problems such as vibration of masses attached to an elastic bar [2]. Qualitative analysis such as periodicity, almost periodicity and stability of functional equations have been studied by many researchers (see [3–17,20] and the references cited therein). More recently researchers have given special attention to the study of neutral differential equations, see [3,4,8,18,12,13,15,16] and references cited therein. Recently, Islam and Raffoul [12] have studied the periodic solution of a nonlinear neutral system of the form dx(t ) dt

= A(t )x(t ) +

d dt

Q (t , x(t − g (t ))) + G(t , x(t ), x(t − g (t ))),

(1.1)

where A(t ) is a nonsingular n × n matrix with continuous-real-valued functions as its elements. The functions Q : R × Rn → Rn and G : R × Rn × Rn → Rn are continuous in their respective arguments. They give sufficient conditions which guarantee the existence of periodic system (1.1). However, there are few works dealing with the almost periodicity of neural equations. In [3], Abbas and Bahuguna have studied the almost periodicity of (1.1) when A(t ) is almost periodic and Q (t , u) and G(t , u, v) are almost periodic for u, v almost periodic. Under conditions: (A1) The family {A(t ) : t ∈ R} of operators in X generates an exponentially stable evolution system {U (t , s), t ≥ s}. (A2) {U (t , s), t ≥ s}, satisfies the condition that, for each  > 0, there exists a number l > 0 such that each interval of length l contains a number τ with the property that δ

kU (t + τ , s + τ ) − U (t , s)kB(X ) < Me− 2 (t −s) ,

I This work is supported by the National Natural Science Foundation of China (10501007), the Natural Science Foundation of Fujian Province (Z0511014), the Foundation of Developing Science and Technology of Fuzhou University (2005-QX-18). ∗ Corresponding author. E-mail address: [email protected] (X. Chen).

X. Chen, F. Lin / Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

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they give sufficient conditions for the existence of an almost periodic solution of the system. But it is clear that the first condition is too strict so that it cannot be satisfied even if for simple A = diag{1, −1} and the second condition is not easy to verify. The purpose of this paper is to deal with the existence of an almost periodic solution of system (1.1) in a complex Banach space X . We remove the above two conditions and only assume that the system y0 = A(t )y

(1.2)

admits exponential dichotomy (see Definition 2.4). We use the property of almost periodic function and exponential dichotomy and Krasnoselskii’s fixed point theorem to show the existence of an almost periodic solution of system (1.1). The theory of almost periodic function can be found in [9,19]. 2. Preliminaries Let X be a complex Banach space endowed with the norm k.kX . We denote by B(X ) the Banach space of all bounded linear operators from X to itself endowed with the operator norm given by

kLkB(X ) := sup{kLxkX : x ∈ X , kxkX ≤ 1},

L ∈ B(X ).

Definition 2.1 ([9]). A function f : R → X is said to be almost periodic, if for any  > 0, there is a constant l() > 0, such that in any interval of length l() there exists τ such that the inequality

kf (t + τ ) − f (t )kX <  is satisfied for all t ∈ (−∞, +∞). The number τ is called an  -translation number of f (t ). In what follows, we need the following notation. For every real sequence α = (αn ) and a continuous function f : R → X , if limn→∞ f (t + αn ) exists, we define Tα f = limn→∞ f (t + αn ). Similarly to the proof of Fink [9], we have: Lemma 2.1. f : R → X is almost periodic if and only if f is continuous and for each α = (αn ), there exists a subsequence α 0 of (αn ) such that Tα0 f = g uniformly on R. Let AP (X ) be the set of all almost periodic functions from R to X . Then (AP (X ), k.kAP (X ) ) is a Banach space with the supremum norm given by

kukAP (X ) = sup ku(t )kX . t ∈R

Definition 2.2. A function f : R × X → X is said to be almost periodic in t uniformly for x ∈ X , if for any  > 0 and for each compact subset D of X , there is a constant l() > 0, such that in any interval of length l() there exists τ such that the inequality

kf (t + τ , x) − f (t , x)kX <  is satisfied for all (t , x) ∈ R × D. Suppose that X (t ) is the fundamental matrix solution of the linear differential system dx dt

= A(t )x,

(2.1)

where the n × n matrix A(t ) is continuous on the real axis. Definition 2.3. System (2.1) is said to admit exponential dichotomy if there are a projection P and positive numbers α > 0 and β ≥ 1 such that

kX (t )PX −1 (s)kB(X ) ≤ β exp(−α(t − s)), kX (t )(I − P )X

−1

t ≥ s,

(s)kB(X ) ≤ β exp(α(t − s)),

t ≤ s.

(2.2) (2.3)

Remark 2.1. It is clear that when A(t ) = diag(1, −1), (2.1) admits exponential dichotomy. More generally, in the case A(t ) ≡ A, a constant matrix, (2.1) admits exponential dichotomy if and only if the eigenvalues of A have a nonzero real part. Definition 2.4. By an almost periodic solution u : R → X of the differential equation (1.1) we mean that u ∈ AP (X ), u(t ) is continuously differentiable and satisfies (1.1) on R. Now we state some lemmas which will be useful in the proof of our main results. Lemma 2.1. If f : R → X is almost periodic, then f (t ) is bounded and uniformly continuous on R.

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Lemma 2.2. If f : R × X → X is almost periodic in t uniformly for x ∈ X , then f (t , x) is bounded on R × D, where D is any compact subset of X . Lemma 2.3. If f : R × X → X is almost periodic in t uniformly for x ∈ X , φ(t ) is almost periodic and φ(t ) ⊂ S for all t ∈ R, with S being a compact subset of X , then f (t , φ(t )) is almost periodic. The proofs of Lemmas 2.1–2.3 are similar to those of Theorem 1.1, Theorem 2.1 and Theorem 2.8 in [19]. Hence we omit it. Lemma 2.4. If u(t ), g (t ) : R → R are almost periodic, then u(t − g (t )) is almost periodic. Proof. It is clear that u(t − g (t )) is continuous for t ∈ R. For any sequence α 0 = (αn0 ), since u(t ), g (t ) are almost periodic, there is a subsequence α = (αn ) of (αn0 ), such that Tα u(t ) = u¯ (t ),

Tα g (t ) = g¯ (t )

(2.4)

uniformly for t ∈ R. On the other hand, since u(t ) is almost periodic, it is uniformly continuous on R. For any  > 0, there exists a positive number δ(), such that |t1 − t2 | < δ implies |u(t1 ) − u(t2 )| <  . From (2.4), there exists a positive integer N, such that

|u(t + αn ) − u¯ (t )| <

 2

,

|g (t + αn ) − g¯ (t )| < min

n 2

o ,δ ,

t ∈ R,

when n > N. So we have

|u(t + αn − g (t + αn )) − u¯ (t − g¯ (t ))| ≤ |u(t + αn − g (t + αn )) − u(t + αn − g¯ (t ))| + |u(t + αn − g¯ (t )) − u¯ (t − g¯ (t ))|   ≤ + = , 2

2

when n > N. Hence, {u(t + αn − g (t + αn ))} converges to u¯ (t − g¯ (t )) uniformly on R, so u(t − g (t )) is almost periodic.



Lemma 2.5. Suppose (2.1) admits exponential dichotomy, that is, there exist constants α > 0, β ≥ 1 such that (2.2) and (2.3) hold. If {A(t + tk )} converges to A¯ (t ) uniformly on any compact subset of R, then {X (t + tk )PX −1 (s + tk )}, {X (t + tk )(I − P )X −1 (s + tk )} converges to X¯ (t )P¯ X¯ −1 (s), X¯ (t )(I − P¯ )X¯ −1 (s) uniformly on any compact subset of R × R, respectively. Furthermore, the following inequalities hold:

kX¯ (t )P¯ X¯ −1 (s)kB(X ) ≤ β exp(−α(t − s)), t ≥ s, kX¯ (t )(I − P¯ )X¯ −1 (s)kB(X ) ≤ β exp(α(t − s)), t ≤ s, where X¯ (t ) is the fundamental matrix solution of the following equation (which is called the hull equation of (2.1)) dx dt

= A(t )x.

(2.5)

Proof. We first prove that {X (tk )PX −1 (tk )} is convergent. From (2.2), we see that

kX (tk )PX −1 (tk )kB(X ) ≤ β. Suppose {X (tk )PX −1 (tk )} is not convergent. Then we can find two subsequences {X (tkm )PX −1 (tkm )} and {X (tk0m )PX −1 (tk0m )} such that lim X (tkm )PX −1 (tkm ) = P¯ ,

m→∞

lim X (tk0m )PX −1 (tk0m ) = P

m→∞

and P¯ 6= P. Then from (2.2) we have

kX (t + tkm )PX −1 (s + tkm )kB(X ) ≤ β exp{−α(t − s)},

t ≥s

(2.6)

kX (t + tk0m )PX −1 (s + tk0m )kB(X ) ≤ β exp{−α(t − s)},

t ≥ s.

(2.7)

and

Assume that Xkm (t ), Xk0m (t ) are the fundamental matrix solutions of systems dx dt

= A(t + tkm )x

and dx dt

= A(t + tk0m )x

X. Chen, F. Lin / Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

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respectively, then X (t + tkm ) = Xkm (t )X (tkm ), X (t + tk0m ) = Xk0m (t )X (tk0m ). Since {A(t + tk )} converges to A(t ) uniformly on

any compact subset of R, then {A(t + tk )x} converges to A(t )x uniformly on any compact subset of R × X . It follows that {A(t + tkm )x} and {A(t + tk0m )x} converge to A(t )x uniformly on any compact set of R × X . So Xkm (t ), Xk0m (t ) converge to X (t ) uniformly on any compact set of R. Furthermore, it follows from (2.6) and (2.7) that

kXkm (t )X (tkm )PX −1 (tkm )Xk−m1 (s)kB(X ) ≤ β exp{−α(t − s)},

t ≥s

kXk0m (t )X (tk0m )PX −1 (tk0m )Xk−0 1 (s)kB(X ) ≤ β exp{−α(t − s)},

t ≥ s.

and m

Let m → ∞, we have

kX¯ (t )P¯ X¯ −1 (s)kB(X ) ≤ β exp(−α(t − s)),

t ≥s

(2.8)

kX¯ (t )P X¯ −1 (s)kB(X ) ≤ β exp(−α(t − s)),

t ≥ s.

(2.9)

and

Similarly, we can obtain

kX¯ (t )(I − P¯ )X¯ −1 (s)kB(X ) ≤ β exp(α(t − s)),

t ≤s

(2.10)

kX¯ (t )(I − P )X¯ −1 (s)kB(X ) ≤ β exp(α(t − s)),

t ≤ s.

(2.11)

and

From (2.8)–(2.11) we see that (2.5) admits exponential dichotomy; both P¯ and P are its projections. So P¯ = P, which is a contradiction. Hence, {X (tk )PX −1 (tk )} is convergent. Let {X (tk )PX −1 (tk )} → P as k → ∞. Now assume that Xk (t ) is the fundamental matrix solution of the system dx dt

= A(t + tk )x,

then {Xk (t )} converges to X¯ (t ) uniformly on any compact set of R. It is easy to see that {Xk−1 (s)} converges to X¯ −1 (s) uniformly on any compact subset of R. So {X (t + tk )PX −1 (s + tk )} and {X (t + tk )(I − P )X −1 (s + tk )} converge to {X¯ (t )P¯ X¯ −1 (s)} and {X¯ (t )(I − P¯ )X¯ −1 (s)} uniformly on any compact set of X × R respectively. Furthermore, from (2.6) and (2.7), we have

kX (t + tk )PX −1 (s + tk )kB(X ) ≤ β exp{−α(t − s)} t ≥ s and

kX (t + tk )(I − P )X −1 (s + tk )kB(X ) ≤ β exp{−α(t − s)} t ≤ s. That is,

kXk (t )X (tk )PX −1 (tk )Xk−1 (s)kB(X ) ≤ β exp{−α(t − s)} t ≥ s and

kXk (t )X (tk )(I − P )X −1 (tk )Xk−1 (s)kB(X ) ≤ β exp{−α(t − s)} t ≤ s. Let k → ∞, we obtain

kX¯ (t )P¯ X¯ −1 (s)kB(X ) ≤ β exp(−α(t − s)), t ≥ s, kX¯ (t )(I − P¯ )X¯ −1 (s)kB(X ) ≤ β exp(α(t − s)), t ≤ s. The proof is completed.



Lemma 2.6 ([21], Krasnoselskii). Let M be a closed convex nonempty subset of X . Suppose that B and C map M into X such that: (1) x, y ∈ M, implies Bx + Cy ∈ M, (2) C is continuous and C M is contained in a compact set, (3) B is a contraction mapping. Then there exists z ∈ M with z = Bz + Cz. 3. Main results We make the following assumptions: (H1) There exist positive numbers LF , LG such that

kF (t , φ) − F (t , ψ)kX ≤ LF kφ − ψkAP (X )

(3.1)

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X. Chen, F. Lin / Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

for all t ∈ R and each φ, ψ ∈ AP (X ) and

kG(t , u, φ) − G(t , v, ψ)kX ≤ LG (ku − vkX + kφ − ψkAP (X ) )

(3.2)

for all t ∈ R and each (u, φ), (v, ψ) ∈ X × AP (X ); (H2) A(t ) is almost periodic, F (t , u) is almost periodic in t uniformly with respect to u ∈ X . G(t , u, v) is almost periodic in t uniformly with respect to u, v ∈ X × X ; (H3) System (2.1) admits exponential dichotomy, that is, there exist constants α > 0, β ≥ 1, such that (2.2) and (2.3) hold. We define a mapping Φ by

(Φ u)(t ) = F (t , u(t − g (t ))) +

Z

t

X (t )PX −1 (s)G(s, u(s), u(s − g (s)))ds −∞

Z

X (t )(I − P )X −1 (s)G(s, u(s), u(s − g (s)))ds.

(3.3)

t

Lemma 3.1. The operator Φ u is almost periodic for u almost periodic. Proof. For u(t ) being almost periodic, from (H2), Lemmas 2.1–2.4, we see that F (t , u(t − g (t ))), G(t , u(t ), u(t − g (t ))) are all almost periodic, so they are uniformly bounded on R. Let M1 , M2 be positive numbers such that

kF kAP (X ) ≤ M1 ,

kGkAP (X ) ≤ M2 .

Now we prove that (Φ u)(t ) is almost periodic. First, it is clear that (Φ u)(t ) is continuous on R. For any sequence α = (αn ), since F (t , u(t − g (t ))), G(t , u(t ), u(t − g (t ))) are almost periodic, combining with Lemma 2.5, we can find a common subsequence of (αn ). We still denote it as α = (αn ), such that Tα F (t , u(t − g (t ))) = F1 (t ),

Tα G(t , u(t ), u(t − g (t ))) = G1 (t )

(3.4)

uniformly for t ∈ R and lim X (t + αk )PX −1 (s + αk ) = X¯ (t )P¯ X¯ −1 (s),

k→∞

t ≥ s,

lim X (t + αk )(I − P )X −1 (s + αk ) = X¯ (t )(I − P¯ )X¯ −1 (s),

k→∞

(3.5) t ≤ s.

(3.6)

Then

Z t +αk X (t + αk )PX −1 (s)G(s, u(s), u(s − g (s)))ds (Φ u)(t + αk ) = F (t + αk , u(t + αk − g (t + αk ))) + −∞ Z ∞ − X (t + αk )(I − P )X −1 (s)G(s, u(s), u(s − g (s)))ds t +αk

= F (t + αk , u(t + αk − g (t + αk ))) Z t + X (t + αk )PX −1 (s + αk )G(s + αk , u(s + αk ), u(s + αk − g (s + αk )))ds −∞ Z ∞ − X (t + αk )(I − P )X −1 (s + αk )G(s + αk , u(s + αk ), u(s + αk − g (s + αk )))ds. t

From (3.4)–(3.6) and Lebesgue’s control convergence theorem, we see that (Φ u)(t + αk ) converges to F1 (t ) +

Z

t

X¯ (t )P¯ X¯ −1 (s)G1 (s)ds − −∞

Z

X¯ (t )(I − P¯ )X¯ −1 (s)G1 (s)ds t

uniformly for t ∈ R. Hence (Φ u)(t ) is almost periodic, i.e., Φ acts on AP (X ). We note that to apply Krasnoselskii’s theorem we need to construct two mappings; one is a contraction and the other is compact. Let

(Φ u)(t ) = (Bu)(t ) + (Cu)(t ). where B, C : AP (X ) → AP (X ) are given by

(Bu)(t ) = F (t , u(t − g (t )))

(3.7)

X. Chen, F. Lin / Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

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and

(Cu)(t ) =

t

Z

X (t )PX −1 (s)G(s, u(s), u(s − g (s)))ds −∞ ∞

Z

X (t )(I − P )X −1 (s)G(s, u(s), u(s − g (s)))ds. 

(3.8)

t

Lemma 3.2 ([3]). The operator B is a contraction provided LF < 1. Lemma 3.3. The operator C is continuous and the image CD is contained in a compact set, where D = {u ∈ AP (X ), kukAP (X ) ≤ R}, R is a fixed constant. Proof. First, we have

k(Cu)(t )kX ≤

t

Z

kX (t )PX −1 (s)kkG(s, u(s), u(s − g (s)))kX ds −∞ ∞

Z

kX (t )(I − P )X −1 (s)kkG(s, u(s), u(s − g (s)))kX ds.

+ t

Thus we obtain

k(Cu)(.)kAP (X ) ≤ kG(., u(.), u(. − g (.)))kAP (X )

Z

≤ kG(., u(.), u(. − g (.)))kAP (X )

Z

t

kX (t )PX

−1

(s)kds +

−∞

=

α

kX (t )(I − P )X −1 (s)kds



t

t

β exp{−α(t − s)}ds + −∞

Z

Z

β exp{α(t − s)}ds



t

kG(., u(.), u(. − g (.)))kAP (X ) .

(3.9)

To see that C is continuous, we let u, v ∈ AP (X ). Given  > 0, take δ = 4βαL We have G

k(Cu)(t ) − (C v)(t )kX ≤

Z

t

kX (t )PX −1 (s)kkG(s, u(s), u(s − g (s))) − G(s, v(s)), v(s − g (s))kX ds −∞

Z ∞ + kX (t )(I − P )X −1 (s)kkG(s, u(s), u(s − g (s))) − G(s, v(s)), v(s − g (s))kX ds t Z t ≤ β e−α(t −s) LG ku(s) − v(s)kX + ku(s − g (s)) − v(s − g (s))kX ds −∞ Z ∞ + β eα(t −s) LG ku(s) − v(s)kX + ku(s − g (s)) − v(s − g (s))kX ds t Z ∞ Z t  ≤ 2LG ku − vkAP (X ) β e−α(t −s) ds + β eα(t −s) ds −∞

=

4β LG

α < ,

t

ku − vkAP (X )

whenever ku − vkAP (X ) < δ . This proves that C is continuous. For D = {u ∈ AP (X ) : kukAP (X ) ≤ R}, to show that the image of C (D) is contained in a compact set, let {un } be a sequence in D. In view of (3.2), we have

kG(t , u(t ), φ(t ))kX ≤ kG(t , u(t ), φ(t )) − G(t , 0, 0)kX + kG(t , 0, 0)kX ≤ LG (kukAP (X ) + kφkAP (X ) ) + a ≤ LG (2R + a),

(3.10)

where a = kG(t , 0, 0)kX . From (3.9) and (3.10), we have 2β LG

(2R + a) := L. α Next, we calculate (Cun )0 (t ) and show that it is uniformly bounded. We can calculate that kCun kAP (X ) ≤

(3.11)

(Cun )0 (t ) = A(t )(Cun )(t ) + X (t )PX −1 (t )G(t , un (t ), un (t − g (t ))) − X (t )(I − P )X −1 (t )G(t , un (t ), un (t − g (t ))).

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Since A(t ) is almost periodic, A(t ) is bounded. There is a positive constant K such that kAkAP (X ) ≤ K . This, combined with (3.10) and (3.11) implies

k(Cun )0 kX ≤ KL + 2β LG (2R + a). Thus the sequence {(Cun )} is uniformly bounded and equi-continuous. Hence by the Arzela–Ascoli theorem, C (D) is compact.  Theorem 3.4. Suppose (H1) and (H2) hold. If LF +

4LG β

< 1,

α

(3.12)

then Eq. (1.1) has an almost periodic solution. Proof. From (3.12), we can take R0 satisfying LF R0 + b +

2LG β(2R0 + a)

α

≤ R0 ,

(3.13)

where a = kG(., 0, 0)kAP (X ) , b = kF (., 0)kAP (X ) . Let M = {u ∈ AP (X ) : kukAP (X ) ≤ R0 }. Noting that condition (3.12) implies LF < 1, then the mapping B defined by (3.7) is a contraction by Lemma 3.2. The mapping C defined by (3.8) is continuous by Lemma 3.3 and C M is contained in a compact set. For u, v ∈ M, we have

k(Bu)(t ) + (C v)(t )kX ≤ kF (t , u(t − g (t ))) − F (t , 0)kX + kF (t , 0)kB(X ) Z t + kX (t )PX −1 (s)kB(X ) kG(s, v(s), v(s − g (s)))kX ds −∞ Z ∞ + kX (t )(I − P )X −1 (s)kB(X ) kG(s, v(s), v(s − g (s)))kX ds t

≤ LF kukAP (X ) + b + ≤ R0 .

α

(2R0 + a)

Thus Bu + c v ∈ M. Hence all the conditions of Krasnoselskii’s theorem are satisfied. Hence there exists a fixed point z ∈ M, such that z = bz + cz. By Lemma 2.1, this fixed point is a solution of (1.1). Hence (1.1) has an almost periodic solution.  Remark 3.1. From the proof of the main result, we see that in condition (H2), we only need (3.2) and (3.3) to hold when (u, v) ∈ D = {(u, v) : k(u, v)k ≤ M } with M being a fixed positive number. Remark 3.2. If the conditions of the main result of [3] hold, then (2.1) admits exponential dichotomy with projection P = I, hence system (1.1) has an almost periodic solution. So our main result greatly improves the main result of [3]. 4. Example As in [12], we consider the following perturbed Van Der Pol equation for small 1 and 2 : x00 + (2 x2 − 1)x0 + x − 1

d



sin(t )) + sin( t )x2 (t − g (t ) − 2 (cos(t ) + cos( 2t )) = 0,

(4.1) dt where g (t ) is a nonnegative, continuous and almost periodic function. Using the transformations x0 = x1 and x01 = x2 , we can transfer the above equation X 0 = AX +

d dt

F ((t , x(t − g (t )))) + G(t , x(t ), x(t − g (t ))),

(4.2)

where

  X =

x1 x2

,

 A=

0

−1

1 0



,

and F (t , x(t − g (t ))) =



0√ 1 (sin(t )) + sin( t )x2 (t − g (t ))

G(t , x(t ), x(t − g (t ))) =



0√

2 (cos(t ) + cos( 2t )) − 

 

2 2 x1 x2

.

Since the real part of the eigenvalues of A is nonzero, we see that X 0 (t ) = A(t )X (t ) admits exponential dichotomy, and as in [12], the other conditions of Theorem 3.4 are satisfied. So (4.1) has an almost periodic solution.

X. Chen, F. Lin / Nonlinear Analysis: Real World Applications 11 (2010) 1182–1189

1189

Remark 4.1. Since the real part of the eigenvalues of A is 12 > 0, the condition that ‘‘the family {A(t ) : t ∈ R} of operators in R generates an exponentially stable evolution system {U (t , s), t ≥ s}’’, cannot be satisfied. So the main result of [3] cannot ensure the existence of almost periodic solution of (4.1).

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