Periodic solutions for a class of neutral functional differential equations with infinite delay

Nonlinear Analysis 71 (2009) 604–613

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Periodic solutions for a class of neutral functional differential equations with infinite delayI Guirong Liu ∗ , Weiping Yan, Jurang Yan School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China

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info

Article history: Received 28 April 2008 Accepted 29 October 2008

a b s t r a c t In this paper, we study the existence and uniqueness of periodic solutions of the nonlinear neutral functional differential equation with infinite delay of the form d

MSC: 34K13 34K40 Keywords: Leray–Schauder Neutral differential equation Infinite delay Periodic solution

dt



x(t ) −

Z

0

g (s, x(t + s))ds



= A(t , x(t ))x(t ) + f (t , xt ).

−∞

In the process we use the fundamental matrix solution of x0 (t ) = A(t , u(t ))x(t ) and construct appropriate mappings, where u ∈ C (R, Rn ) is an ω-periodic function. Then we employ matrix measure and the Leray–Schauder fixed point theorem to show the existence of periodic solutions of this neutral differential equation. In the special case where g (s, u) ≡ 0 and A(t , x) = A(t ), some sufficient conditions which ensure the uniform stability and global attractivity of a unique periodic solution are derived. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The theory of functional differential equations with delay has undergone a rapid development in the last fifty years. More recently researchers have given special attention to the study of neutral functional differential equations. Neutral functional differential equations are not only an extension of functional differential equations but also provide good models in many fields including Biology, Mechanics and Economics. In particular, qualitative analysis such as periodicity and stability of solutions of neutral functional differential equations has been studied extensively by many authors. We refer to [4,7,8,12, 16,20,21] for some recent work on the subject of periodicity and stability of neutral equations. In the last few years, the study of functional differential equations of various classes with infinite delay has acquired an evergrowing interest of many authors. We only mention the works of some authors [2,3,5,6,9,10,14,15,17,18]. It is well known that the development of the theory of functional differential equations with infinite delay primarily depends on the choice of a phase space. Many authors have actively devoted their research to this topic, and various phase spaces have been proposed (see [14] and references cited therein). The global existence of periodic solutions of differential equations and difference equations is a very basic and important problem, which plays a role similar to that of globally stable equilibrium in an autonomous model. Thus, it is reasonable to seek conditions under which the resulting periodic nonautonomous system would have a periodic solution. Much progress has been seen in this direction and many criteria have been established based on different approaches. But, only a few papers

I This work was supported by the National Natural Science Foundation of China (10071045).

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (G. Liu).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.096

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

605

have studied the existence of periodic solutions of neutral functional differential equations with infinite delay (see [1,4,13, 19,22,23]). In addition, to our knowledge, most of the existing research on the stability of partial functional differential equations deals with finite delay. Only a few papers have considered the stability of an equilibrium of the autonomous case of partial functional differential equations with infinite delay. See [3,10]. Using Krasnoselskii’s fixed point theorem and the contraction mapping principle, Islam and Raffoul [15] studied the existence and uniqueness of a periodic solution to the following neutral nonlinear system of differential equations d d x(t ) = A(t )x(t ) + Q (t , x(t − g (t ))) + G(t , x(t ), x(t − g (t ))). (1.1) dt dt In [19], using Krasnoselskii’s fixed point theorem, the authors studied the existence of periodic solutions to the neutral functional differential equation with infinite delay of the form d



Z

0

d



Z

0



Q (s)x(t + s)ds = A(t , x(t ))x(t ) + f (t , xt ). (1.2) x(t ) − dt −∞ In the current paper, we study the existence and uniqueness of periodic solutions of the nonlinear system of neutral differential equations with infinite delay of the form

dt

x(t ) −

g (s, x(t + s))ds



= A(t , x(t ))x(t ) + f (t , xt ),

(1.3)

−∞

where g ∈ C (R × Rn , Rn ), A ∈ C (R × Rn , Rn×n ), f ∈ C (R × BCh , Rn ), and BCh is called a phase space that will be defined ∂ g (t ,u) in the next section. In addition, for any t ∈ R, u ∈ Rn , ϕ ∈ BCh and s ∈ (−∞, 0], ∂ u ∈ C (R × Rn , Rn×n ), g (t , 0) = 0, A(t + ω, u) = A(t , u), f (t + ω, ϕ) = f (t , ϕ), xt (s) = x(t + s), where ω > 0, and f maps a bounded set into a bounded set. In the analysis, we use the fundamental matrix solution of x0 (t ) = A(t , u(t ))x(t ), where u ∈ C (R, Rn ) is an ω-periodic function. Then we employ matrix measure and the Leray–Schauder fixed point theorem, which is different from the method in [15,19], and show the existence and uniqueness of a periodic solution of Eq. (1.3) in Theorems 3.1 and 3.2. When g (s, u) ≡ 0 and A(t , x) = A(t ), we also obtain the uniform stability and global attractivity of a unique periodic solution of Eq. (1.3) in Theorems 3.3 and 3.4. Our results extend and improve the corresponding results in [19]. 2. Preliminaries Let x = (x1 , x2 , . . . , xn )T ∈ Rn , |x| and kAk = sup|x|6=0 |x| denote the norm of x and the n × n real matrix A, respectively. µ(A) = limε→0 1ε [kI + ε Ak − 1] denotes the matrix measure of the n × n real matrix A. Obviously, if A(t ) is a n × n matrix continuous function and A(t + ω) = A(t ), then µ(A(t )) is an ω-periodic function. Consider the following linear differential equation |Ax|

x0 (t ) = A(t )x(t ),

(2.1)

and x0 (t ) = A(t )x(t ) + e(t ), where A ∈ C (R, R

n×n

(2.2)

), e ∈ C (R, R ). n

Lemma 2.1 ([19]). Let X (t ) represent the fundamental matrix solution of Eq. (2.1); then

kX (t )X −1 (s)k ≤ exp

t

Z

 µ(A(τ ))dτ ,

t ≥ s.

(2.3)

s

Lemma 2.2 ([19]). Let A(t ), e(t ) be ω-periodic functions, and exp solution x(t ) =

Rω 0


µ(A(τ ))dτ < 1. Then (2.2) has a unique ω-periodic

t

Z

X (t )X −1 (s)e(s)ds,

(2.4)

−∞

where X (t ) is the fundamental matrix solution of Eq. (2.1).

R0

Let BC = {ϕ : ϕ ∈ C ((−∞, 0], Rn ), ϕ is bounded}, h ∈ C ((−∞, 0], (0, ∞)) and −∞ h(s)ds = 1. For ϕ ∈ BC , set

Z

0

|ϕ|h =

h(s)|ϕ|[s,0] ds,

−∞

where |ϕ|[s,0] = maxs≤θ≤0 |ϕ(θ )|. Then BCh = (BC , | · |h ) is a Banach space (see [23]). For any ϕ ∈ BCh , it is clear that

|ϕ(0)| ≤ |ϕ|h ≤ max |ϕ(s)|. s∈(−∞,0]

Consider the Banach space Cω = {u : u ∈ C (R, Rn ), u(t + ω) = u(t ), t ∈ R}

606

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

with the norm kuk0 = max0≤t ≤ω |u(t )|, u ∈ Cω , and the Banach space Cω1 = {u : u ∈ C 1 (R, Rn ), u(t + ω) = u(t ), t ∈ R} with the norm kuk1 = kuk0 + ku0 k0 , u ∈ Cω1 . Lemma 2.3. Let M > 0, S = {u ∈ Cω1 : kuk1 ≤ M }; then f (t , ut ) is uniformly continuous with respect to t ∈ [0, ω] and u ∈ S. Proof. Assume, for the sake of contradiction, that f (t , ut ) is not uniformly continuous with respect to t ∈ [0, ω] and u ∈ S. Then there exists a constant ε0 > 0 such that for any δ > 0, there always exist t , τ ∈ [0, ω] and u, v ∈ S satisfying |t − τ | < δ, ku − vk1 < δ and |f (t , ut ) − f (τ , vτ )| ≥ ε0 . Let δn = 1n . Then there exist sequences {tn }, {τn } ⊆ [0, ω] and {un }, {v n } ⊆ S such that |tn − τn | < 1n , kun − v n k1 < 1n and |f (tn , untn ) − f (τn , vτnn )| ≥ ε0 , n = 1, 2, . . .. Clearly, {tn } contains a convergent subsequence {tnk } which converges to t0 . That is, limk→∞ tnk = t0 . Also, limk→∞ τnk = t0 . Observe that kunk k0 ≤ M , k(unk )0 k0 ≤ M and so {unk (t )} is a family of uniformly bounded and equicontinuous functions on [0, ω]. By using the Arzela–Ascoli Theorem, we know that {unk } contains a subsequence which converges to u0 in Cω . Denote this subsequence {unk }. That is, limk→∞ unk = u0 . Similarly, limk→∞ v nk = v 0 . It is easy to see that n

n

ε0 ≤ |f (tnk , utnk ) − f (τnk , vτnnk )| ≤ |f (tnk , utnk ) − f (t0 , u0t0 )| + |f (τnk , vτnnk ) − f (t0 , u0t0 )| k

k

k

(2.5)

k

and n

n

n

n

n

|utnk − u0t0 |h ≤ |utnk − ut0k |h + |ut0k − u0t0 |h ≤ M |tnk − t0 | + |ut0k − u0t0 |h . k

k

n

n

From Lemma 2.3 in [19], we have |ut0k − u0t0 |h → 0 (k → ∞). Similarly, |vt0k − u0t0 |h → 0 (k → ∞). Then, in view of the continuity of f , it follows from (2.5) that n

|f (tnk , utnk ) − f (τnk , vτnnk )| → 0 (k → ∞). k

k

This contradicts (2.5) and completes the proof of Lemma 2.3.



Lemma 2.4. Let N > 0, S = {u ∈ Cω1 : kuk1 ≤ N }, g ∈ C (R × Rn , Rn ),

∂ g (s,u) ∂u

∈ C (R × Rn , Rn×n ),

R0 −∞



∂ g (s,u)

sup|u|<∞ ∂ u ds <

∂ g (s,u(t +s)) 0 u (t + s)ds. Then m(t , u) is uniformly continuous with respect to t ∈ [0, ω] and u ∈ S. −∞ ∂u

∂ g (s,u(t +s)) 0

∂ g (s,u) Proof. Obviously, ≤ N sup . It follows by the Weierstrass M-test that m(t , u) converges u ( t + s )

| u |<∞ ∂u ∂u

∞ and m(t , u) =

R0

uniformly with respect to t ∈ [0, ω] and u ∈ S. In view of the continuity of uniformly continuous with respect to t ∈ [0, ω] and u ∈ S. 

∂ g (s,u(t +s)) 0 u (t ∂u

+ s), we see that m(t , u) is

Lemma 2.5 (Leray–Schauder [11]). Let E be a Banach space. Suppose that A : E → E is a completely continuous map, and the set Q = {kxk : x = λAx, 0 < λ < 1, x ∈ E } is bounded. Then A has a fixed point in T , where T = {x : x ∈ E , kxk ≤ R} and R = sup{Q }. Lemma 2.6. If xt ∈ BCh , t > σ , then for any t > σ ,

|xt |h ≤ sup |x(s)| + |xσ |h . σ ≤s≤t

Proof. From the definition of | · |h , we have 0

Z |xt |h =

h(s)|xt |[s,0] ds =

Z

−∞

0

h(s) max |xt (θ )|ds = s≤θ≤0

−∞

0

Z

h(s) max |x(τ )|ds ≤

=

Z

t +s≤τ ≤t

−∞

−∞

σ +s≤τ ≤σ

= max |x(τ )| + σ ≤τ ≤t

Z

h(s) max |x(τ )|ds σ +s≤τ ≤t

|x(τ )| + max |x(τ )|)ds σ ≤τ ≤t

0

h(s) max |xσ (θ )|ds −∞

s≤θ≤0

= sup |x(s)| + |xσ |h . σ ≤s≤t

The proof of Lemma 2.6 is complete.

h(s) max |x(t + θ )|ds

0

−∞

h(s)( max

≤

0

−∞

0

Z

Z



3. Main results In this section, we shall give our main results.

s≤θ≤0

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

607

Theorem 3.1. In Eq. (1.3), assume that there is a continuous ω-periodic function a(t ) such that the following two conditions hold:

Rω
(H1 ) For any (t , x) ∈ R × Rn , µ(A(t , x)) ≤ a(t ) and k = exp 0 a(τ )dτ < 1;

Rω R0

(H2 ) Let ρ = lim supn→∞ 1n 0 sup|ϕ|h ≤n |f (t , ϕ)|dt , β = max0≤t ≤ω sup|x|<∞ kA(t , x)k, q = −∞ sup|u|<∞ ∂ g∂(su,u) ds < 1 and 1−k

ρ + qωβ <

M

where M = sup0≤s≤t ≤ω exp

(1 − q),

R

t s



a(τ )dτ .

Then Eq. (1.3) has at least one differentiable ω-periodic solution. Proof. For u ∈ Cω1 , consider the following two equations x0 (t ) = A(t , u(t ))x(t )

(3.1)

and x0 (t ) = A(t , u(t ))x(t ) + f (t , ut ) +

0

Z

−∞

∂ g (s, u(t + s)) 0 u (t + s)ds. ∂u

(3.2)

∂ g (s,u(t +s))

R0

∂ g (s,u)

u0 (t + s)ds. In view of the fact f (t + ω, ut +ω ) = f (t , ut ) and the continuity of ∂ u and Set r (t ) = f (t , ut ) + −∞ ∂u f , it follows from Lemma 2.4 that r (t ) is a continuous ω-periodic function. Let Xu (t ) be the fundamental matrix solution of Eq. (3.1). Then from (H1 ) and Lemma 2.2 we see that Eq. (3.2) has a unique ω-periodic solution xu (t ) =

t

Z

Xu (t )Xu (s)f (s, us )ds + −1

−∞

t

Z

Xu (t )Xu (s) −1

Z

−∞

0

−∞

 ∂ g (τ , u(s + τ )) 0 u (s + τ )dτ ds. ∂u

Consider the operator P on Cω1 defined as follows:

Z

(Pu)(t ) =

t

Xu (t )Xu (s)f (s, us )ds + −1

Z

−∞

t

Xu (t )Xu (s) −1

Z

−∞

0

−∞

 ∂ g (τ , u(s + τ )) 0 u (s + τ )dτ ds. ∂u

Obviously, if P has a fixed point x in Cω1 , then x is a ω-periodic solution of Eq. (3.1). Now, we shall use the Leray–Schauder fixed point theorem to prove that P has a fixed point in Cω1 . Clearly, P maps Cω1 into Cω1 . Observe that d ds

Xu−1 (s) = −Xu−1 (s)A(s, u(s))

and so

Z

 Z 0  Z t ∂ g (τ , u(s + τ )) 0 u (s + τ )dτ ds = Xu (t )Xu−1 (s)d g (τ , u(s + τ ))dτ ∂u −∞ −∞ −∞ Z 0 g (τ , u(t + τ ))dτ − lim Xu (t )Xu−1 (s) g (τ , u(s + τ ))dτ

t

Xu (t )Xu−1 (s) −∞ 0

Z =

Z

s→−∞

−∞

Z

0

t

Xu (t )Xu−1 (s)A(s, u(s))

+

Z

−∞

−∞ 0

g (τ , u(s + τ ))dτ



ds.

−∞

Let t − s = nω + ω1 , 0 ≤ ω1 < ω, where n is a nonnegative integer. In view of the mean value inequality, it follows from Lemma 2.1 and (H1 ) that

Z Xu (t )X −1 (s) u

0

−∞





0

(g (τ , u(s + τ )) − g (τ , 0))dτ −∞

Z 0

∂ g (τ , θ u(s + τ ))

|u(s + τ )|dτ

≤ kXu (t )Xu−1 (s)k

∂u −∞ Z t  ≤ qkuk0 exp a(τ )dτ s Z nω+s+ω1  Z s+ω1  = qkuk0 exp a(τ )dτ exp a(τ )dτ

g (τ , u(s + τ ))dτ = Xu (t )Xu−1 (s)

Z

s+ω1

≤ qkuk0 kn M ,

s

608

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

R0

where 0 < θ < 1. Hence lims→−∞ Xu (t )Xu−1 (s) −∞ g (τ , u(s + τ ))dτ = 0. Furthermore,

(Pu)(t ) =

t

Z

Xu (t )Xu−1 (s)f (s, us )ds +

g (τ , u(t + τ ))dτ −∞

−∞ t

Z

0

Z

Xu (t )Xu−1 (s)A(s, u(s))

+

0

Z

−∞

g (τ , u(s + τ ))dτ



ds.

(3.3)

−∞

Now, we prove that P is a completely continuous operator on Cω1 . Let S ⊂ Cω1 be a bounded set, and kuk1 ≤ δ for any u ∈ S. From (H2 ), for any t ∈ [0, ω], we see that 0

Z

g (τ , u(t + τ ))dτ =

0

Z

(g (τ , u(t + τ )) − g (τ , 0)) dτ ≤ qkuk0 −∞

−∞

and

Z

t

−∞



t

Z

Xu (t )Xu−1 (s)f (s, us )ds ≤

Z exp

=

s −∞ ∞ Z t −n ω X n =0

t −(n+1)ω

∞ X

≤M

kn

ω

Z

M 1−k

a(τ )dτ



|f (s, us )|ds

t

Z exp

t −n ω

a(τ )dτ



t −n ω

Z exp

a(τ )dτ



|f (s, us )|ds

s

|f (s, us )|ds

0

n=0

=

ω

Z

t

|f (s, us )|ds.

0

Similarly, for any t ∈ [0, ω], we have

Z

t

Xu (t )Xu (s)A(s, u(s)) −1

Z

−∞

0

−∞

M g (τ , u(s + τ ))dτ ds ≤ qkuk0 βω . 1−k 

Therefore, for any t ∈ [0, ω],

|(Pu)(t )| ≤

M 1−k

Z

ω

|f (s, us )|ds + qkuk0 + qkuk0 βω

0

M 1−k

.

(3.4)

In view of the fact that f maps a bounded set into a bounded set, let b0 = max0≤t ≤ω sup|ϕ|h ≤δ |f (t , ϕ)|. Hence, for any u ∈ S , t ∈ [0, ω],

|(Pu)(t )| ≤

M 1−k

b0 ω + qδ +

M 1−k

qδβω,

and



Z 0 d(Pu)(t )

∂ g (s, u(t + s)) 0 ≤ kA(t , u(t ))k|(Pu)(t )| + |f (t , ut )| +

|u (t + s)|ds dt

∂u −∞   M M ≤β b0 ω + qδ + qδβω + b0 + qδ. 1−k 1−k

(3.5)

Then, P (S ) is precompact in Cω . Furthermore, for any {Pun }, un ∈ S , n = 1, 2, . . ., there exists a convergent subsequence {Punk } ⊆ {Pun } in Cω . It follows from (3.5) that {(Pu)0 : u ∈ S } is a family of uniformly bounded functions on [0, ω]. For any u ∈ S, we have

(Pu)0 (t1 ) − (Pu)0 (t2 ) = A(t1 , u(t1 ))[(Pu)(t1 ) − (Pu)(t2 )] + [A(t1 , u(t1 )) − A(t2 , u(t2 ))](Pu)(t2 )  Z 0  ∂ g (s, u(t1 + s)) 0 ∂ g (s, u(t2 + s)) 0 + [f (t1 , ut1 ) − f (t2 , ut2 )] + u (t1 + s) − u (t2 + s) ds. ∂u ∂u −∞ In view of the uniform continuity of A(t , x) on {(t , x) : 0 ≤ t ≤ ω, |x| ≤ δ}, it follows from Lemmas 2.3 and 2.4 that {(Pu)0 : u ∈ S } is a family of equicontinuous functions on [0, ω]. Therefore, there exists a convergent subsequence {(Punkl )0 } ⊆ {(Punk )0 } in Cω . Further, there exists a convergent subsequence {Punkl } ⊆ {Punk } in Cω1 . That is, P is compact on Cω1 . Next, we shall prove the continuity of P. Let un , u0 ∈ Cω1 , kun − u0 k1 → 0 (n → ∞) and vn = Pun − Pu0 . Clearly,

vn 0 (t ) = A(t , un (t ))vn (t ) + fn (t ),

(3.6)

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

609

where fn (t ) = [A(t , un (t )) − A(t , u0 (t ))](Pu0 )(t ) + [f (t , unt ) − f (t , u0t )] 0

Z



+ −∞

 ∂ g (s, un (t + s)) n ∂ g (s, u0 (t + s)) 0 (u (t + s))0 − (u (t + s))0 ds. ∂u ∂u

Let c = sup{ku0 k1 , ku1 k1 , ku2 k1 , . . .}. In view of the uniform continuity of A(t , x) on {(t , x) : 0 ≤ t ≤ ω, |x| ≤ c }, it follows from Lemmas 2.3 and 2.4 that kfn k0 → 0 (n → ∞). Let Xn (t ) be the fundamental matrix solution of x0 (t ) = A(t , un (t ))x(t ). In view of the fact that vn (t ) is an ω-periodic solution of Eq. (3.6), it follows from Lemma 2.2 that

vn (t ) =

Z

t

Xn (t )Xn−1 (s)fn (s)ds, −∞

which yields

kvn k0 ≤

ω

Z

M 1−k

|fn (s)|ds.

0

In view of the fact that kfn k0 → 0 (n → ∞), it follows from the Lebesgue dominated convergence theorem that kvn k0 → 0 (n → ∞). From (3.6) we have

|vn 0 (t )| ≤ kA(t , un (t ))kkvn k0 + kfn k0 ≤ βkvn k0 + kfn k0 . Hence, kvn 0 k0 → 0. Furthermore, kvn k1 = kvn k0 + kvn 0 k0 → 0 (n → ∞). That is, P is continuous on Cω1 . Consequently, P is a completely continuous operator on Cω1 . Let W = {u : u = λPu, 0 < λ < 1, u ∈ Cω1 } and Q = {kuk1 : u ∈ W }. If u ∈ W , then from (3.4), for any t ∈ [0, ω],

|u(t )| = λ|(Pu)(t )| ≤

ω

Z

M 1−k

|f (s, us )|ds + qkuk0 +

0

M 1−k

qωβkuk0 .

From (H2 ), we know that there exists constants ρ1 > ρ and N > 0 such that ρ1 + qωβ <

Rω

1 −k M

(1 − q), and

sup|ϕ|h ≤n |f (t , ϕ)|dt < ρ1 for any n ≥ N. We now claim that kuk0 ≤ N for any u ∈ W . Otherwise, there exists 0 u0 ∈ W such that ku0 k0 > N. Then, for any t ∈ [0, ω], 1 n

|u0 (t )| M 1 ≤ k u0 k 0 1 − k k u0 k 0

ω

Z

|f (s, ϕ)|ds + q + qβω

sup 0

|ϕ|h ≤ku0 k0

M 1−k

|u (t )|

<

M 1−k

ρ1 + q +

M 1−k

qβω < 1.

|u (t )|

Hence, max0≤t ≤ω ku0 k ≤ 1M (ρ + qβω) + q < 1, which contradicts max0≤t ≤ω ku00 k0 = 1. So kuk0 ≤ N for any u ∈ W . −k 1 0 0  M  1 Let b1 = max0≤t ≤ω sup|ϕ|h ≤N |f (t , ϕ)| and η > 1− ωβ(b1 + Nqβ) + Nqβ + b1 . Assume u ∈ W and ku0 k0 > η. q 1−k Then for any t ∈ [0, ω],

Z 0

∂ g (s, u(t + s)) |u0 (t + s)| |u0 (t )| 1 1

≤ kA(t , u(t ))k |(Pu)(t )| + |f (t , ut )| +

ku0 k ds k u0 k 0 η η ∂u 0 −∞   1 M M 1 ≤ β b1 ω + qN + qN ωβ + b1 + q η 1−k 1−k η   1 M = ωβ(b1 + Nqβ) + Nqβ + b1 + q η 1−k < 1, |u0 (t )|

|u0 (t )|

which yields max0≤t ≤ω ku0 k < 1. This contradicts max0≤t ≤ω ku0 k = 1. Hence, ku0 k0 ≤ η for any u ∈ W . Furthermore, 0 0 kuk1 = kuk0 + ku0 k0 ≤ N + η for any u ∈ W . Hence, Q is bounded. It follows from Lemma 2.5 that P has a fixed point u∗ in Cω1 satisfying ku∗ k1 ≤ N + η. That is, u∗ is a differentiable ω-periodic solution of Eq. (1.3). The proof of Theorem 3.1 is complete.  Remark 3.1. When g (s, u) = Q (s)u, Theorem 3.1 reduces to Theorem 3.1 in [19]. Thus, Theorem 3.1 generalizes Theorem 3.1 in [19]. In addition, in this paper, we use the Leray–Schauder fixed point theorem which is different from the method in [19], and the proof of Theorem 3.1 is simpler than that in [19]. Consider the special case of Eq. (1.3), namely, d dt



x(t ) −

Z

0

g (s, x(t + s))ds



= A(t )x(t ) + f (t , x(t )).

−∞

We give sufficient conditions to guarantee the existence of a unique periodic solution of Eq. (3.7).

(3.7)

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G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

Theorem 3.2. In Eq. (3.7), assume that:

Rω
(H3 ) k = exp 0 µ(A(τ ))dτ < 1; (H4 ) There exists an ω-periodic continuous function L(t ) such that for any t ∈ R, ϕ1 , ϕ2 ∈ BCh , |f (t , ϕ1 ) − f (t , ϕ2 )| ≤ L(t )|ϕ1 − ϕ2 |h and Z ω 1−k L(t )dt + qωβ < (1 − q), M

0

 R0

∂ g (s,u) µ( A (τ )) d τ , q = ds < 1, β = sup0≤t ≤ω kA(t )k. sup

| u |<∞ s −∞ ∂u Then Eq. (3.7) has a unique differentiable ω-periodic solution.

where M = sup0≤s≤t ≤ω exp

R t

Proof. From (H4 ), for any ϕ ∈ BCh , we see that |f (t , ϕ) − f (t , 0)| ≤ L(t )|ϕ|h . Furthermore, |f (t , ϕ)| ≤ L(t )|ϕ|h + |f (t , 0)|, which yields 1

lim sup

n

n→∞

ω

Z

sup |f (t , ϕ)|dt ≤ |ϕ|h ≤n

0

ω

Z

L(t )dt .

0

Clearly, the conditions in Theorem 3.1 are satisfied. Therefore, Eq. (3.7) has at least one differentiable ω-periodic solution x(t ). Now, we shall prove the uniqueness of the ω-periodic solution of Eq. (3.7). Otherwise, let y(t ) be another ω-periodic solution of Eq. (3.7) with kx − yk0 > 0. It follows from the proof of Theorem 3.1 that x(t ) =

t

Z

X (t )X −1 (s)A(s)

0

Z

−∞

g (τ , x(s + τ ))dτ



t

Z

X (t )X −1 (s)f (s, xs )ds +

ds +

−∞

−∞

0

Z

g (τ , x(t + τ ))dτ , −∞

and y(t ) =

t

Z

X (t )X −1 (s)A(s)

0

Z

−∞

g (τ , y(s + τ ))dτ



Z

t

X (t )X −1 (s)f (s, ys )ds +

ds +

−∞

−∞

Z

0

g (τ , y(t + τ ))dτ , −∞

where X (t ) is the fundamental matrix solution of Eq. (2.1). Hence, x(t ) − y(t ) =

t

Z

X (t )X −1 (s)[f (s, xs ) − f (s, ys )]ds + −∞

Z

Z

0

[g (τ , x(t + τ )) − g (τ , y(t + τ ))]dτ −∞

t

X (t )X −1 (s)A(s)

+ −∞

0

Z

 (g (τ , x(s + τ )) − g (τ , y(s + τ )))dτ ds.

−∞

Furthermore,

|x(t ) − y(t )| ≤ ≤ ≤

M 1−k M

ω

Z

Z0 ω

1−k

0



Z

M 1−k

|f (s, xs ) − f (s, ys )|ds + qkx − yk0 + L(s)|xs − ys |h ds + qkx − yk0 + ω

L(s)ds + q +

0

M 1−k

M 1−k

M 1−k

qβωkx − yk0

qβωkx − yk0



qβω kx − yk0 .

It follows from (H4 ) that



M 1−k

ω

Z 0

L(s)ds + q +

M 1−k

qβω



< 1.

Hence, kx − yk0 < kx − yk0 . This is a contradiction. That is, kx − yk0 = 0. The proof of Theorem 3.2 is complete.



Consider the special case of Eq. (3.7) x0 (t ) = A(t )x(t ) + f (t , xt ).

(3.8)

We give sufficient conditions to guarantee the uniform stability and global attractivity of the unique ω-periodic solution of Eq. (3.8). In order to state our results, we need the following definitions. Definition 3.1. Let u(t ) be a periodic solution of Eq. (3.8). u(t ) is said to be uniformly stable if, for any ε > 0, t0 ≥ 0, there is a δ = δ(ε) > 0 such that ϕ ∈ BCh and |ϕ − ut0 |h < δ imply |x(t , t0 , ϕ) − u(t )| < ε, t ≥ t0 , where x(t , t0 , ϕ) is a solution of Eq. (3.8) and satisfies the initial condition xt0 = ϕ . Definition 3.2. Let u(t ) be a periodic solution of Eq. (3.8). u(t ) is said to be globally attractive if, for any ϕ ∈ BCh , t0 ≥ 0, limt →+∞ |x(t , t0 , ϕ) − u(t )| = 0, where x(t , t0 , ϕ) is a solution of Eq. (3.8) and satisfies the initial condition xt0 = ϕ .

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

611

Theorem 3.3. In Eq. (3.8), assume that:

Rω
(H5 ) For any t ∈ R, µ(A(t )) ≤ 0 and k = exp 0 µ(A(t ))dt < 1; (H6 ) There exists a constant a > kln−k1 such that for any t ∈ R, ϕ1 , ϕ2 ∈ BCh , 1

|f (t , ϕ1 ) − f (t , ϕ2 )| ≤ − µ(A(t )) |ϕ1 − ϕ2 |h . a

Then Eq. (3.8) has a unique ω-periodic solution which is uniformly stable. Proof. From (H5 ) and (H6 ), we see that the conditions of Theorem 3.2 are satisfied. It follows from Theorem 3.2 that Eq. (3.8) has a unique ω-periodic solution u(t ). In view of  the fact that 0 < k < 1 implies ln k < k − 1, we have a > 1. For any ε > 0, t0 ≥ 0 and ϕ ∈ BCh , choose δ = min (a − 1)ε, a−1 1 ε . Let |ϕ − ut0 |h < δ , and x(t ) = x(t , t0 , ϕ) be a solution satisfying the initial condition xt0 = ϕ for Eq. (3.8). Let X (t ) represent the fundamental matrix solution of Eq. (2.1). Then t

Z

x(t ) = X (t )X −1 (t0 )ϕ(0) +

X (t )X −1 (s)f (s, xs )ds t0

and u(t ) = X (t )X −1 (t0 )u(t0 ) +

t

Z

X (t )X −1 (s)f (s, us )ds. t0

Now, we shall prove that for any t ≥ t0 , |x(t )− u(t )| < ε . Otherwise, there exists a constant t1 > t0 such that |x(t )− u(t )| < ε for any t0 ≤ t < t1 , but |x(t1 ) − u(t1 )| = ε . From Lemma 2.1, we have

ε = |x(t1 ) − u(t1 )| ≤ kX (t1 )X −1 (t0 )k|ϕ(0) − ut0 (0)| +

t1

Z

kX (t1 )X −1 (s)k|f (s, xs ) − f (s, us )|ds

t0

< δ exp

t1

Z



µ(A(τ ))dτ

−

t0

1

Z

t1



t1

Z

  µ(A(τ ))dτ µ(A(s))|xs − us |h ds.

exp

a

t0

s

For t0 ≤ s ≤ t1 , it follows from Lemma 2.6 that

|xs − us |h ≤ sup |x(t ) − u(t )| + |xt0 − ut0 |h < ε + δ, t0 ≤ t ≤ s

which yields

ε < δ exp

t1

Z

−

ε+δ

+

ε+δ



µ(A(τ ))dτ

t0

= δ exp

t1

Z



µ(A(τ ))dτ

t0

 = ≤

δ−

ε+δ a



Z

t1

exp

t1

Z exp

a

t0

a



µ(A(τ ))dτ µ(A(s))ds

s



µ(A(τ ))dτ

t0

t1

Z

Z

t1

1 − exp

µ(A(τ ))dτ



t0

 +

ε+δ a

ε+δ

≤ ε.

a

This is a contradiction. Hence, u(t ) is uniformly stable. The proof of Theorem 3.3 is complete.



Remark 3.2. Theorem 3.3 in [19] required that a > max 3, k−1 . But, in Theorem 3.3, we only require that a > Consequently, Theorem 3.3 improves Theorem 3.3 in [19].



ln k

Theorem 3.4. In Eq. (3.8), in addition to the assumption (H5 ), assume further that

(H7 ) There exists a constant a >

ln k k−1

such that for any t ∈ R, ϕ1 , ϕ2 ∈ BCh , 1

|f (t , ϕ1 ) − f (t , ϕ2 )| ≤ − µ(A(t )) |ϕ1 (0) − ϕ2 (0)|. a

Then Eq. (3.8) has a unique ω-periodic solution which is not only uniformly stable but also globally attractive.

ln k . k−1

612

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

Proof. In view of the fact that |ϕ1 (0) − ϕ2 (0)| ≤ |ϕ1 − ϕ2 |h , it follows from (H5 ) and (H7 ) that the conditions in Theorem 3.3 are satisfied. Hence, Eq. (3.8) has a unique and uniformly stable ω-periodic solution u(t ). For any ϕ ∈ BCh , t0 ≥ 0, let x(t ) = x(t , t0 , ϕ) be a solution satisfying the initial condition xt0 = ϕ for Eq. (3.8). Let X (t ) represent the fundamental matrix solution of Eq. (2.1). Then x(t ) = X (t )X −1 (t0 )ϕ(0) +

t

Z

X (t )X −1 (s)f (s, xs )ds t0

and u(t ) = X (t )X

−1

(t0 )u(t0 ) +

t

Z

X (t )X −1 (s)f (s, us )ds. t0

From Lemma 2.1, we have t

Z

|x(t ) − u(t )| ≤ kX (t )X −1 (t0 )k|ϕ(0) − u(t0 )| +

kX (t )X −1 (s)k|f (s, xs ) − f (s, us )|ds

t0

≤ |ϕ(0) − u(t0 )| exp

Z

t

µ(A(τ ))dτ



1

−

t

Z exp

a

t0

t

Z

t0

 µ(A(τ ))dτ µ(A(s))|x(s) − u(s)|ds.

s

Hence,

|x(t ) − u(t )| 1 R  ≤ |ϕ(0) − u(t0 )| − t a exp t µ(A(τ ))dτ 0

Z

|x(s) − u(s)| R  ds. s exp t µ(A(τ ))dτ 0

t

µ(A(s)) t0

By using Gronwall’s inequality, we see that

|x(t ) − u(t )| R  ≤ |ϕ(0) − u(t0 )| exp t exp t µ(A(τ ))dτ 0

Z t 

  1 − µ(A(s)) ds . a

t0

That is, for t ≥ t0 ,

|x(t ) − u(t )| ≤ |ϕ(0) − u(t0 )| exp



1

1−

t

Z

a





µ(A(s))ds = |ϕ(0) − u(t0 )| exp t0

Z

t

µ(A(s))ds

1− 1a

.

t0

Let t0 + nω ≤ t < t0 + (n + 1)ω, where n is a nonnegative integer; then 0 ≤ |x(t ) − u(t )|



≤ |ϕ(0) − u(t0 )| exp

Z

t0 + n ω

µ(A(s))ds

1− 1a 

t

Z

exp t0 +nω

t0

µ(A(s))ds

1− 1a

1

≤ |ϕ(0) − u(t0 )|(kn )1− a . Thus, limt →+∞ |x(t ) − u(t )| = 0. Hence, u(t ) is globally attractive. The proof of Theorem 3.4 is complete.



In order to verify the applicability of our results, we shall take a special case of Eq. (1.3) as an example. Example 3.1. Consider the equation d



x( t ) −

dt

Z

0

e

20s



sin(x(t + s))ds

= −| cos t |x(t ) +

−∞

Z

t

e8(s−t ) | cos t |x(s)ds.

(3.9)

−∞

R0

Clearly, ω = 2π , g (s, u) = e20s sin(u), µ(A(t )) = A(t ) = −| cos t |, f (t , ϕ) = −∞ e8u | cos t |ϕ(u)du. Let h(u) = 8e8u , u ∈ (−∞, 0]; then

|f (t , ϕ1 ) − f (t , ϕ2 )| ≤ Furthermore, k = e−4 , q = 2π

Z 0

L(t )dt + qωβ =

1 8

1 20

1 2

| cos t ||ϕ1 − ϕ2 |h .

, L(t ) = 18 | cos t |, β = 1, +

π 10

,

1−k M

R 2π 0

(1 − q) =

L(t )dt = 19 20

1 2

, M = 1 and

(1 − e−4 ).

G. Liu et al. / Nonlinear Analysis 71 (2009) 604–613

613

Hence, 2π

Z

L(t )dt + qωβ <

0

1−k M

(1 − q).

It follows from Theorem 3.2 that Eq. (3.9) has a unique 2π -periodic solution. But Theorem 3.2 in [19] cannot be used to obtain the existence and uniqueness of 2π -periodic solution of Eq. (3.9). Example 3.2. Consider the following equation 0 1 e8(s−t ) | cos t |x(s)ds + p(t ), x0 (t ) = − | cos t |x(t ) + 4 −∞

Z

where p(t ) = cos t +

(33 sin t + 4 cos t ). 260 Clearly, µ(A(t )) = A(t ) = − 41 | cos t |, k = e−1 ,

(3.10)

| cos t |

h(u) = 8e8u , u ∈ (−∞, 0]; then

ln k k −1

1

1

2

8

|f (t , ϕ1 ) − f (t , ϕ2 )| ≤ − µ(A(t ))|ϕ1 − ϕ2 |h =

=

e e− 1

< 2 = a, f (t , ϕ) =

R0 −∞

e8τ | cos t |ϕ(τ )dτ + p(t ). Let

| cos t ||ϕ1 − ϕ2 |h .

It follows from Theorem 3.3 that Eq. (3.10) has a unique 2π -periodic solution which is uniformly stable. Remark 3.3. In Eq. (3.10), there is no constant a > 3 such that 1

|f (t , ϕ1 ) − f (t , ϕ2 )| ≤ − µ(A(t ))|ϕ1 − ϕ2 |h . a

Consequently, Theorem 3.3 in [19] cannot be used to obtain the uniqueness and uniformly stability of the 2π -periodic solution of Eq. (3.10). In fact, x(t ) = sin t is a unique and uniformly stable 2π -periodic solution of Eq. (3.10). Acknowledgment The authors are very grateful to the referee for his valuable comments and suggestions, which improved the presentation of this paper. References [1] M. Adimy, H. Bouzahir, K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl. 294 (2004) 438–461. [2] M. Adimy, H. Bouzahir, K. Ezzinbi, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal. 46 (2001) 91–112. [3] M. Adimy, K. Ezzinbi, A. Ouhinou, Behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay, Nonlinear Anal. 68 (2008) 2280–2302. [4] M.A. Babram, K. Ezzinbi, Periodic solutions of functional differential equations of neutral type, J. Math. Anal. Appl. 204 (1996) 898–909. [5] R. Benkhalti, H. Bouzahir, K. Ezzinbi, Existence of a periodic solution for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 256 (2001) 257–280. [6] F.D. Chen, Periodic solutions of nonlinear integrodifferential equations with infinite delay, Acta Math. Appl. Sinica 26 (2003) 141–148. [7] C. Corduneanu, Existence of solutions for neutral functional differential equations with causal operators, J. Differential Equation 168 (2000) 93–101. [8] J.G. Dix, C.G. Philos, I.K. Purnaras, Asymptotic properties of solutions to linear non-autonomous neutral differential equations, J. Math. Anal. Appl. 318 (2006) 296–304. [9] A. Elazzouzi, K. Ezzinbi, Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 329 (2007) 498–514. [10] K. Ezzinbi, A. Ouhinou, Necessary and sufficient conditions for the regularity and stability for some partial functional differential equations with infinite delay, Nonlinear Anal. 64 (2006) 1690–1709. [11] D. Guo, Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, 2001 (in Chinese). [12] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 2003. [13] W. Han, J. Ren, Some results on second-order neutral functional differential equations with infinite distributed delay, Nonlinear Anal., in press (doi:10.1016/j.na.2008.02.018). [14] Y. Hino, S. Murakami, T. Naito, Functional-Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, vol. 1473, Springer-Verlag, Berlin, 1991. [15] M.N. Islam, Y.N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl. 331 (2007) 1175–1186. [16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New Work, 1993. [17] J. Liu, T. Naito, N.V. Minh, Bounded and periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl. 286 (2003) 705–712. [18] A. Ouahab, Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay, Nonlinear Anal. 67 (2007) 1027–1041. [19] S. Peng, S. Zhu, Periodic solutions of functional differential equations with infinite delay, Chinese Ann. Math. 23A (2002) 371–380. [20] R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2005) 391–428. [21] J. Shen, Y. Liu, J. Li, Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses, J. Math. Anal. Appl. 332 (2007) 179–189. [22] L. Shi, Boundedness and periodicity of solutions neutral functional differential equations with infinite delay, Chinese Sec. Bull. 35 (1990) 409–411. [23] K. Wang, Q. Huang, Ch space, boundedness and periodic solution of functional differential equations with infinite delay, Sci. Sinica 3A (1987) 242–252.