Anti-periodic solutions for high-order Hopfield neural networks

Computers and Mathematics with Applications 56 (2008) 1838–1844

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Anti-periodic solutions for high-order Hopfield neural networksI Chunxia Ou Applied Mathematics Institute, Guangdong University of Technology, Guangzhou, 510290, PR China

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Article history: Received 17 January 2008 Received in revised form 10 April 2008 Accepted 17 April 2008

In this paper high-order Hopfield neural networks (HHNNs) with time-varying delays are considered. Sufficient conditions for the existence and exponential stability of anti-periodic solutions are established, which are new and complement previously known results. © 2008 Elsevier Ltd. All rights reserved.

Keywords: High-order Hopfield neural networks Anti-periodic Exponential stability Delays

1. Introduction Consider high-order Hopfield neural networks (HHNNs) with time-varying delays described by x0i (t ) = −ci (t )xi (t ) +

n X

aij (t )gj (xj (t − τij (t ))) +

j=1

n X n X

bijl (t )gj (xj (t − σijl (t )))gl (xl (t − νijl (t ))) + Ii (t ),

j=1 l=1

i = 1, 2, . . . , n,

(1.1)

in which n corresponds to the number of units in a neural network, xi (t ) corresponds to the state vector of the ith unit at time t, ci (t ) > 0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, aij (t ) and bijl (t ) are the first- and second-order connection weights of the neural network, τij (t ) ≥ 0, σijl (t ) ≥ 0 and υijl (t ) ≥ 0 correspond to the transmission delays, Ii (t ) denotes the external inputs at time t, and gj is the activation function of signal transmission. Due to the fact that high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of equilibrium points and periodic solutions of HHNNs (1.1) in the literature. We refer the reader to [1–7] and the references cited therein. However, to the best of our knowledge, there exist no results for the existence and stability of anti-periodic solutions of HHNNs (1.1). Moreover, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [8–13]). HHNNs can be analog voltage transmission, and voltage transmission process is often an anti-periodic process. Thus, it is worthwhile to continue the investigation of the existence and stability of anti-periodic solutions of HHNNs (1.1). The main purpose of this paper is to give the conditions for the existence and exponential stability of the anti-periodic solutions for system (1.1). We derive some new sufficient conditions ensuring the existence, uniqueness and exponential

I This work was supported by the NNSF of China.

E-mail address: [email protected]. 0898-1221/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2008.04.029

C. Ou / Computers and Mathematics with Applications 56 (2008) 1838–1844

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stability of the anti-periodic solution for system (1.1), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results. Let u(t ) : R −→ R be continuous in t. u(t ) is said to be T-anti-periodic on R if, u(t + T ) = −u(t ) for all t ∈ R. Throughout this paper, for i, j, l = 1, 2, . . . , n, it will be assumed that ci , Ii , aij , bijl : R → R and τij , σijl , νijl : R → [0 +∞) are continuous functions, aij and bijl are are 2T -periodic, and ci (t + T ) = ci (t ),

aij (t + T )gj (u) = −aij (t )gj (−u),

bijl (t + T )gj (u)gl (u) = −bijl (t )gj (−u)gl (−u),

∀t , u ∈ R,

(1.2)

∀t , u ∈ R,

(1.3)

τij (t + T ) = τij (t ), σijl (t + T ) = σijl (t ), νijl (t + T ) = νijl (t ), Ii (t + T ) = −Ii (t ),

∀t , u ∈ R .

(1.4)

Then, we can choose a constant τ such that

τ = max{ max sup τij (t ), max sup σijl (t ), max sup νijl (t )}. 1≤i,j≤n t ∈R

1≤i,j,l≤n t ∈R

(1.5)

1≤i,j,l≤n t ∈R

We also assume that the following conditions (H1 ) and (H2 ) hold. (H1 ) for each j ∈ {1, 2, . . . , n}, there exist nonnegative constants Lgj and Mjg such that

|gj (u) − gj (v)| ≤ Lgj |u − v|,

gj (0) = 0,

|gj (u)| ≤ Mjg for all u, v ∈ R.

(1.6)

(H2 ) there exist constants η > 0 and λ > 0, i = 1, 2, . . . , n, such that for all t > 0, there holds n X n n X X |bijl (t )|(Lgj Mlg + Mjg Lgl )eλτ < −η < 0. |aij (t )|Lgj eλτ + (λ − ci (t )) + j=1 l=1

j=1

For convenience, we introduce some notations. We will use x = {xj } = (x1 , x2 , . . . , xn )T ∈ Rn to denote a column vector, in which the symbol (T ) denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| = (|x1 |, |x2 |, . . . , |xn |)T , and define kxk = max1≤i≤n |xi |. The initial conditions associated with system (1.1) are of the form xi (s) = ϕi (s),

s ∈ [−τ , 0], i = 1, 2, . . . , n,

(1.7)

where ϕ = (ϕ1 (t ), ϕ2 (t ), . . . , ϕn (t ))T ∈ C ([−τ , 0]; Rn ). Definition 1. Let x∗ (t ) = (x∗1 (t ), x∗2 (t ), . . . , x∗n (t ))T be an anti-periodic solution of system (1.1) with initial value ϕ ∗ = (ϕ1∗ (t ), ϕ2∗ (t ), . . . , ϕn∗ (t ))T . If there exist constants λ > 0 and M > 1 such that for every solution Z (t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T of system (1.1) with any initial value ϕ ∈ C ([−τ , 0]; Rn ),

|xi (t ) − x∗i (t )| ≤ M kϕ − ϕ ∗ k∞ e−λt ,

∀t > 0, i = 1, 2, . . . , n,

(1.8)

where kϕ − ϕ k∞ = sup−τ ≤s≤0 max1≤j≤n |ϕj (s) − ϕj (s)|. Then Z (t ) is said to be globally exponentially stable. ∗

∗

∗

The remaining part of this paper is organized as follows. In Section 2, we shall derive new sufficient conditions for checking the existence of bounded solutions of (1.1). In Section 3, we present some new sufficient conditions for the existence, uniqueness and exponential stability of the anti-periodic solution of (1.1). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections. 2. Preliminary results The following lemmas will be used to prove our main results in Section 3. Lemma 2.1. Let (H1 ) and (H2 ) hold. Suppose that e x(t ) = (e x1 (t ),e x2 (t ), . . . ,e xn (t ))T is a solution of system (1.1) with initial conditions

e xi (s) = e ϕi (s),

|e ϕi (s)| < γ , s ∈ [−τ , 0],

(2.1)

where

γ > ci (t )

−1

" n X j=1

g Mj

|aij (t )|

+

n X n X

# g g Mj Ml

|bijl (t )|

+ |Ii (t )|

for all t > 0, i = 1, 2, . . . , n.

(2.2)

j=1 l=1

Then

|e xi (t )| < γ ,

for all t ≥ 0, i = 1, 2, . . . , n.

(2.3)

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Proof. Assume, by way of contradiction, that (2.3) does not hold. Then, there must exist i ∈ {1, 2, . . . , n} and ρ > 0 such that

|e xi (ρ)| = γ ,

and |e xj (t )| < γ

for all t ∈ [−τ , ρ), j = 1, 2, . . . , n.

(2.4)

Calculating the upper left derivative of |e xi (t )|, together with (2.2), (H1 ) and (H2 ), (2.4) implies that 0 ≤ D+ (|e xi (ρ)|)

n n X n X X ≤ −ci (ρ)e xi (ρ) + aij (ρ)gj (e xj (ρ − τij (ρ))) + bijl (ρ)gj (xj (ρ − σijl (ρ)))gl (xl (ρ − νijl (ρ))) + Ii (ρ) j=1 j=1 l=1 n X

≤ −ci (ρ)γ +

|aij (ρ)|Mjg +

n X n X

j=1

|bijl (ρ)|Mjg Mlg + |Ii (ρ)|

j=1 l=1

< 0, which is a contradiction and implies that (2.3) holds. The proof of Lemma 2.1 is now completed.



Remark 2.1. In view of the boundedness of this solution, from the theory of functional differential equations in [14], it follows that e x(t ) can be defined on [−τ , ∞), provided that the initial conditions are bounded by γ . Lemma 2.2. Suppose that (H1 ) and (H2 ) are satisfied. Let x∗ (t ) = (x∗1 (t ), x∗2 (t ), . . . , x∗n (t ))T be the solution of system (1.1) with initial value ϕ ∗ = (ϕ1∗ (t ), ϕ2∗ (t ), . . . , ϕn∗ (t ))T being bounded by γ , and x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T be the solution of system (1.1) with initial value ϕ = (ϕ1 (t ), ϕ2 (t ), . . . , ϕn (t ))T . Then there exist constants λ > 0 and Mϕ > 1 such that

|xi (t ) − x∗i (t )| ≤ Mϕ kϕ − ϕ ∗ k∞ e−λt ,

∀t > 0, i = 1, 2, . . . , n.

Proof. Let y(t ) = {yj (t )} = {xj (t ) − x∗j (t )} = x(t ) − x∗ (t ). Then y0i (t ) = −ci yi (t ) +

n X

aij (t )(gj (xj (t − τij (t ))) − gj (x∗j (t − τij (t )))) +

n X n X

bijl (t )(gj (xj (t − σijl (t )))

j=1 l=1

j=1

× gl (xl (t − νijl (t ))) − gj (x∗j (t − σijl (t )))gl (x∗l (t − νijl (t )))),

i = 1, 2, . . . , n.

(2.5)

We consider the Lyapunov functional Vi (t ) = |yi (t )|eλt ,

i = 1, 2, . . . , n.

(2.6)

Calculating the upper left derivative of Vi (t ) along the solution y(t ) = {yj (t )} of system (2.5) with the initial value ϕ¯ = ϕ−ϕ ∗ , from (2.5) and (H1 ), we have D (Vi (t )) ≤ −ci (t )|yi (t )|e +

λt

+

" n X

|aij (t )||gj (xj (t − τij (t ))) − gj (x∗j (t − τij (t )))|

j=1

+

n X n X

|bijl (t )||gj (xj (t − σijl (t )))gl (xl (t − νijl (t ))) − gj (x∗j (t − σijl (t )))

j=1 l=1

# × gl (xl (t − νijl (t )))| eλt + λ|yi (t )|eλt ∗

λt

≤ (λ − ci (t ))|yi (t )|e +

n X

g Lj

|aij (t )| |yj (t − τij (t ))| +

" n X n X

j=1

|bijl (t )|(|gj (xj (t − σijl (t )))

j=1 l=1

× gl (xl (t − νijl (t ))) − gj (x∗j (t − σijl (t )))gl (xl (t − νijl (t )))| + |gj (x∗j (t − σijl (t ))) # × gl (xl (t − νijl (t ))) − gj (x∗j (t − σijl (t )))gl (x∗l (t − νijl (t )))|) eλt λt

≤ (λ − ci (t ))|yi (t )|e +

" n X

|aij (t )|Lgj |yj (t − τij (t ))| +

j=1

n X n X

|bijl (t )|(Lgj Mlg

j=1 l=1

# × |yj (t − σijl (t ))| + Mjg Lgl |yl (t − νijl (t ))|) eλt , where i = 1, 2, . . . , n.

(2.7)

C. Ou / Computers and Mathematics with Applications 56 (2008) 1838–1844

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Let M > 1 denote an arbitrary real number and set

kϕ − ϕ ∗ k∞ = sup max |ϕj (s) − ϕj∗ (s)| > 0. −τ ≤s≤0 1≤j≤n

It follows from (2.6) that Vi (t ) = |yi (t )|eλt < M kϕ − ϕ ∗ k∞ ,

for all t ∈ [−τ , 0], i = 1, 2, . . . , n.

(2.8)

for all t > 0, i = 1, 2, . . . , n.

(2.9)

We claim that Vi (t ) = |yi (t )|eλt < M kϕ − ϕ ∗ k∞ ,

Contrarily, there must exist i ∈ {1, 2, . . . , n} and ti > 0 such that Vi (ti ) = M kϕ − ϕ ∗ k∞

and Vj (t ) < M kϕ − ϕ ∗ k∞ ,

∀ t ∈ [−τ , ti ), j = 1, 2, . . . , n.

(2.10)

It follows from (2.10) that Vi (ti ) − M kϕ − ϕ ∗ k∞ = 0

and Vj (t ) − M kϕ − ϕ ∗ k∞ < 0,

∀ t ∈ [−τ , ti ), j = 1, 2, . . . , n.

(2.11)

Together with (2.7) and (2.11), we obtain 0 ≤ D+ (Vi (ti ) − M kϕ − ϕ ∗ k∞ )

= D+ (Vi (ti )) ≤ (λ − ci (ti ))|yi (ti )|e

λti

+

" n X

|aij (ti )|Lgj |yj (ti − τij (ti ))| +

n X n X

|bijl (ti )|

j=1 l=1

j=1

# g g Lj Ml

×(

|yj (ti − σijl (ti ))| +

= (λ − ci (ti ))|yi (ti )|e

λti

+

g g Mj Ll

" n X

|yl (ti − νijl (ti ))|) eλti

|aij (ti )|Lgj |yj (ti − τij (ti ))|eλ(ti −τij (ti )) eλτij (ti )

j=1

+

n X n X

# g g Lj Ml

|bijl (ti )|(

λ(ti −σijl (ti ))

|yj (ti − σijl (ti ))|e

eλσijl (ti )

j=1 l=1

+ Mjg Lgl |yl (ti − νijl (ti ))|eλ(ti −νijl (ti )) eλνijl (ti ) ) n X n n X X |bijl (ti )| |aij (ti )|Lgj eλτ M kϕ − ϕ ∗ k∞ + ≤ (λ − ci (ti ))M kϕ − ϕ ∗ k∞ + j=1 l=1

j=1 g g Lj Ml

×( " ≤

+

g g Mj Ll

(λ − ci (ti )) +

λτ

)M kϕ − ϕ k∞ e ∗

n X

|aij (ti )|Lgj eλτ +

j=1

n X n X

|bijl (ti )|

j=1 l=1

# g g Lj Ml

×(

+

g g Mj Ll

λτ

)e

M kϕ − ϕ ∗ k∞ .

(2.12)

Thus, 0 ≤ (λ − ci (ti )) +

n X

|aij (ti )|Lgj eλτ +

n X n X

j=1

|bijl (ti )|(Lgj Mlg + Mjg Lgl )eλτ ,

j=1 l=1

which contradicts (H2 ). Hence, (2.9) holds. It follows that

|yi (t )| < M kϕ − ϕ ∗ k∞ e−λt ,

t > 0, i = 1, 2, . . . , n.

This completes the proof of Lemma 2.2.

(2.13)



Remark 2.2. If x∗ (t ) = (x∗1 (t ), x∗2 (t ), . . . , x∗n (t ))T is the T-anti-periodic solution of system (1.1) with initial value ϕ ∗ being bounded by γ , it follows from Lemma 2.2 and Definition 1 that x∗ (t ) is globally exponentially stable.

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C. Ou / Computers and Mathematics with Applications 56 (2008) 1838–1844

3. Main results The following is our main result. Theorem 3.1. Suppose that (H1 ) and (H2 ) are satisfied. Then system (1.1) has exactly one T-anti-periodic solution x∗ (t ). Moreover, x∗ (t ) is globally exponentially stable. Proof. Let v(t ) = (v1 (t ), v2 (t ), . . . , vn (t ))T be a solution of system (1.1) with initial conditions

|ϕiv (s)| < γ , s ∈ [−τ , 0], i = 1, 2, . . . , n.

vi (s) = ϕiv (s),

(3.1)

By Lemma 2.1, the solution v(t ) is bounded and

|vi (t )| < γ ,

for all t ∈ [−τ , ∞), i = 1, 2, . . . , n.

(3.2)

From (1.1)–(1.4), we have

((−1)k+1 vi (t + (k + 1)T ))0 = (−1)k+1 vi0 (t + (k + 1)T ) ( n X k+1 = (−1) aij (t + (k + 1)T ) −ci (t + (k + 1)T )vi (t + (k + 1)T ) + j=1

× gj (vj (t + (k + 1)T − τij (t + (k + 1)T ))) +

n X n X

bijl (t + (k + 1)T )gj (vj (t + (k + 1)T

j=1 l=1

) − σijl (t + (k + 1)T )))gl (vl (t + (k + 1)T − νijl (t + (k + 1)T ))) + Ii (t + (k + 1)T ) ( = (−1)

k+1

−ci (t + (k + 1)T )vi (t + (k + 1)T ) +

n X

aij (t + (k + 1)T )

j=1

× gj (vj (t + (k + 1)T − τij (t ))) +

n X n X

bijl (t + (k + 1)T )gj (vj (t + (k + 1)T

j=1 l=1

) − σijl (t )))gl (vl (t + (k + 1)T − νijl (t ))) + Ii (t + (k + 1)T ) = −ci (t )(−1)k+1 vi (t + (k + 1)T ) +

n X

aij (t )gj ((−1)k+1 vj (t + (k + 1)T − τij (t )))

j=1

+

n n X X

bijl (t )gj ((−1)k+1 vj (t + (k + 1)T − σijl (t )))

j=1 l=1

× gl ((−1)k+1 vl (t + (k + 1)T − νijl (t ))) + Ii (t ),

i = 1, 2, . . . , n,

(3.3)

Thus, for any natural number k, (−1)k+1 v(t + (k + 1)T ) are the solutions of system (1.1). Then, by Lemma 2.2, there exists a constant M > 0 such that

|(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )| ≤ Me−λ(t +kT ) sup max |vi (s + T ) + vi (s)| −τ ≤s≤0 1≤i≤n

≤ e−λ(t +kT ) M2γ ,

∀t + kT > 0, i = 1, 2, . . . , n.

(3.4)

Thus, for any natural number m, we obtain

(−1)m+1 vi (t + (m + 1)T ) = vi (t ) +

m X [(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )].

(3.5)

k=0

Then,

|(−1)m+1 vi (t + (m + 1)T )| ≤ |vi (t )| +

m X k=0

where i = 1, 2, . . . , n.

|(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )|,

(3.6)

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Fig. 1. Numerical solution x1 (t ) of system (4.1) for (φ1 (s), φ2 (s)) = (4, −2).

In view of (3.4), we can choose a sufficiently large constant N > 0 and a positive constant α such that

|(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )| ≤ α(e−λT )k ,

∀k > N , i = 1, 2, . . . , n,

(3.7)

on any compact set of R. It follows from (3.5)–(3.7) that {(−1) v(t + mT )} uniformly converges to a continuous function x∗ (t ) = (x∗1 (t ), x∗2 (t ), . . . , x∗n (t ))T on any compact set of R. Now we will show that x∗ (t ) is the T-anti-periodic solution of system (1). First, x∗ (t ) is T-anti-periodic, since m

x∗ (t + T ) = lim (−1)m v(t + T + mT ) = − m→∞

lim

(m+1)→∞

(−1)m+1 v(t + (m + 1)T ) = −x∗ (t ).

Next, we prove that x∗ (t ) is a solution of (1.1). In fact, together with the continuity of the right side of (1.1), (3.3) implies that {((−1)m+1 v(t + (m + 1)T ))0 } uniformly converges to a continuous function on any compact set of R. Thus, letting m −→ ∞, we obtain d dt

{x∗i (t )} = −ci (t )x∗i (t ) +

n X

aij (t )gj (x∗j (t − τij (t ))) +

n X n X

bijl (t )gj (x∗j (t − σijl (t )))gl (x∗l (t − νijl (t ))) + Ii (t ).

j=1 l=1

j=1

(3.8) Therefore, x (t ) is a solution of (1.1). Finally, by Lemma 2.2, we can prove that x∗ (t ) is globally exponentially stable. This completes the proof. ∗



4. An example In this section, we give an example to demonstrate the results obtained in previous sections. Example 4.1. Consider the following HHNNs with delays:

 1 1  x01 (t ) = −x1 (t ) + g1 (x1 (t − sin2 t )) + g2 (x2 (t − 7 sin2 t ))    16 16   1   + (sin t )g (x (t − 5 sin2 t ))g (x (t − 2 sin2 t )) + 4 sin t , 1

8

1

2

2

(4.1)

1

1   x02 (t ) = −x2 (t ) + g1 (x1 (t − cos2 t )) + g2 (x2 (t − 5 sin2 t ))   16 16    1  + (sin t )g1 (x1 (t − sin2 t ))g2 (x2 (t − 4 sin2 t )) + sin t , 8

(|x + 1| − |x − 1|). Observe that c1 = c2 = Lg1 = Lg2 = M1g = M2g = 1, aij = = (sin t ), bijl = 0, i, j, l = 1, 2, ijl 6= 112, ijl 6= 212. Then ( !) 2 2 X 2 X X 3 g g g g g −1 |bijl |(Lj Ml + Mj Ll ) = < 1, max ci |aij |Lj +

where g1 (x) = g2 (x) = b212

1 8

1≤i≤2

j=1

1 2

j=1 l=1

1 16

, i, j = 1, 2, b112 =

8

which implies that system (4.1) satisfies all the conditions in Theorem 3.1. Hence, system (4.1) has exactly one π -antiperiodic solution. Moreover, the π -anti-periodic solution is globally exponentially stable. This fact is verified by the numerical simulation in Figs. 1 and 2.

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C. Ou / Computers and Mathematics with Applications 56 (2008) 1838–1844

Fig. 2. Numerical solution x2 (t ) of system (4.1) for (φ1 (s), φ2 (s)) = (4, −2).

Remark 4.1. RCNNs (4.1) are a very simple form of recurrent neural networks with mixed delays. One can observe that all the results in [8–13] and the references therein cannot be applicable to system (4.1) to obtain the existence and exponential stability of the anti-periodic solutions. This implies that the results of this paper are essentially new. Acknowledgement The author would like to express his sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the paper. References [1] Z. Wang, J. Fang, X. Liu, Global stability of stochastic high-order neural networks with discrete and distributed delays, Chaos Solitons Fractals 36 (2) (2008) 388–396. [2] S. Mohamad, Exponential stability in Hopfield-type neural networks with impulses, Chaos Solitons Fractals 32 (2) (2007) 456–467; 81 (1984) 3088–3092. [3] Y. Liu, Z. You, Multi-stability and almost periodic solutions of a class of recurrent neural networks, Chaos Solitons Fractals 33 (2) (2007) 554–563. [4] Y. Jiang, B. Yang, J. Wang, C. Shao, Delay-dependent stability criterion for delayed Hopfield neural networks, Chaos Solitons Fractals, doi:10.1016/j.chaos.2007.06.039. [5] B. Xiao, H. Meng, Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks, Appl. Math. Modell. (2007), doi:10.1016/j.apm.2007.11.027. [6] B. Liu, L. Huang, Existence and exponential stability of periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays, Chaos Solitons Fractals 32 (2) (2007) 617–627. [7] F. Zhang, Y. Li, Almost periodic solutions for higher-order Hopfield neural networks without bounded activation functions, Electron. J. Diff. Eqns. 2007 (97) (2007) 1–10. [8] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 40 (3) (1988) 541–553. [9] A.R. Aftabizadeh, S. Aizicovici, N.H. Pavel, On a class of second-order anti-periodic boundary value problems, J. Math. Anal. Appl. 171 (1992) 301–320. [10] S. Aizicovici, M. McKibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear Anal. 43 (2001) 233–251. [11] Y. Chen, J.J. Nieto, D. ORegan, Anti-periodic solutions for fully nonlinear first-order differential equations, Math. Comput. Modelling 46 (2007) 1183–1190. [12] R. Wu, An anti-periodic LaSalle oscillation theorem, Appl. Math. Lett. (2007), doi:10.1016/j.aml.2007.10.004. [13] F.J. Delvos, L. Knoche, Lacunary interpolation by antiperiodic trigonometric polynomials, BIT 39 (1999) 439–450. [14] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.