This paper is concerned with impulsive cellular neural networks with time-varying delays in leakage terms. Without assuming bounded and monotone conditions on activation functions, we establish sufficient conditions on existence and exponential stability of periodic solutions by using Lyapunov functional method and differential inequality techniques. Our results are complement to some recent ones.

1. Introduction

It is well known that impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth [1–3]. Thus, many neural networks with impulses have been studied extensively, and a great deal of literature is focused on the existence and stability of an equilibrium point [4–7]. In [8–10], the authors discussed the existence and global exponential stability of periodic solution of a class of cellular neural networks (CNNs) with impulse. Recently, Wang et al. [11] considered the following CNNs with impulses and leakage delays:

()

where Δx_i(t_k) are the impulses at moments t_k and t₁ < t₂ < ⋯ is a strictly increasing sequence such that lim_k→∞t_k = +∞; a_i > 0 and τ_i > 0 are constants, and α_ij(t), I_i(t), and β_ij(t) are continuous periodic functions with period T. Suppose that the following conditions are satisfied.

(A₁) There exist constants , j = 1,2, …n, such that, for any α, β ∈ R,
()
(A₂) f_i(0) = 0 and for i = 1,2, …, n, there exists a constant 0 < M_i < +∞, such that
()

By using the continuation theorem of coincidence degree theory and a suitable degenerate Lyapunov-Krasovskii functional together with model transformation technique, some results were obtained in [11] to guarantee that all solutions of system (1) converge exponentially to a periodic function. However, to the best of our knowledge, few authors have considered the existence and stability of periodic solutions of system (1) without the assumptions (A₁) and (A₂). Thus, it is worthwhile to continue to investigate the convergence behavior of system (1) in this case. In view of the fact that the coefficients and delays in neural networks are usually time varying in the real world, motivated by the above discussions, in this paper, we will consider the problem on periodic solution of the following impulsive CNNs with time-varying delays in the leakage terms:

()

in which n corresponds to the number of units in a neural network, x_i(t) corresponds to the state vector of the ith unit at the time t, and c_i(t) 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 at the time t. a_ij(t) and b_ij(t) are the connection weights at the time t, η_i(t) and τ_ij(t) denote the transmission delays, I_i(t) denotes the external bias on the ith unit at the time t, f_j and g_j are activation functions of signal transmission, Δx_i(t_k) are the impulses at moments t_k, and 0 ≤ t₁ < t₂ < ⋯ is a strictly increasing sequence such that lim_k→∞t_k = +∞, and i, j = 1,2, …, n. It is obvious that when f = g and η_i(t) is a constant function, (1) is a special case of (4).

The main purpose of this paper is to give the conditions for the existence and exponential stability of the periodic solutions for system (4). By applying Lyapunov functional method and differential inequality techniques, without assuming (A₁) and (A₂), we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the periodic solution for system (4), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results.

Throughout this paper, we assume that the following conditions hold.

(H₁) For i, j = 1,2, …, n, I_i, a_ij, b_ij : R → R and c_i, η_i, τ_ij : R → R⁺ are continuous periodic functions with period T > 0, and t − η_i(t) ≥ 0 for all t ≥ 0. In addition, there exist constants

, τ_i,

, and

such that

()

(H₂) The sequence of times t_k (k ∈ N) satisfies t_k < t_k+1 and lim_k→+∞t_k = +∞, and d_ik satisfies −2 ≤ d_ik ≤ 0 for i ∈ {1,2, …, n} and k ∈ Z⁺, where Z⁺ denotes the set of all positive integers.

(H₃) There exists a q ∈ Z⁺ such that

()

(H₄) For each j ∈ {1,2, …, n}, there exist nonnegative constants

and

such that, for all u, v ∈ R,

()

(H₅) For all t > 0 and i ∈ {1,2, …, n}, there exist constants ξ_i > 0 and η > 0 such that

()

For convenience, let x = (x₁, x₂, …, x_n) ^T ∈ Rⁿ, in which “T” denotes the transposition. We define |x| = (|x₁|, |x₂|, …, |x_n|) ^T and ∥x∥ = max_1≤i≤n{|x_i|}. As usual in the theory of impulsive differential equations, at the points of discontinuity t_k of the solution t ↦ (x₁(t), x₂(t), …, x_n(t)) ^T, we assume that (x₁(t), x₂(t), …, x_n(t)) ^T ≡ (x₁(t − 0), x₂(t − 0), …, x_n(t − 0)) ^T. It is clearly that, in general, the derivative does not exist. On the other hand, according to system (4), there exists the limit . In view of the above convention, we assume that .

The initial conditions associated with (4) are assumed to be of the form

()

where ϕ_i(·) denotes a real-valued continuous function defined on [−τ_i, 0].

2. Preliminary Results

The following lemmas will be used to prove our main results in Section 3.

Lemma 1. Let (H₁)–(H₅) hold. Suppose that x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T is a solution of system (1) with the initial conditions

()

where

, i = 1,2, …, n. Then

()

Proof. Assume that (11) does not hold. From (H₂), we have

()

So, if

, then |x_i(t_k)| > γ. Thus, we may assume that there exist i ∈ {1,2, …, n} and t_* ∈ (t_k, t_k+1) such that

()

According to (4), we get

()

Calculating the upper left derivative of |x_i(t)|, together with (13), (14), (H₅), and

()

we obtain

()

It is a contradiction and shows that (11) holds. The proof is now completed.

Remark 2. After the conditions (H₁)–(H₅), the solution of system (4) always exists (see [1, 2]). In view of the boundedness of this solution, from the theory of impulsive differential equations in [1], it follows that the solution of system (4) with initial conditions (10) can be defined on [0, +∞).

Lemma 3. Suppose that (H₁)–(H₅) are true. Let be the solution of system (4) with initial value , and let x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T be the solution of system (4) with initial value φ = (φ₁(t), φ₂(t), …, φ_n(t)) ^T. Then, there exists a positive constant λ such that

()

Proof. Let y(t) = x(t) − x^*(t). Then, for i ∈ {1,2, …, n}, it is followed by

()

Define continuous functions Γ_i(ω) by setting

()

Then

()

which, together with the continuity of Γ_i(ω), implies that we can choose two positive constants λ and

such that

()

Let

()

Then

()

We define a positive constant M as follows:

()

Let K be a positive number such that

()

We claim that

()

Obviously, (27) holds for t = 0. We first prove that (27) is true for 0 < t ≤ t₁. Otherwise, there exist i ∈ {1,2, …, n} and ρ ∈ (0, t₁] such that one of the following two cases must occur;

()

Now, we distinguish two cases to finish the proof.

Case (i). If (28) holds. Then, from (21), (23), and (H₁)–(H₅), we have

()

Case (ii). If (29) holds. Then, from (21), (23), and (H₁)–(H₅), we get

()

Therefore, (27) holds for t ∈ [0, t₁]. From (24) and (27), we know that

()

Thus, for t ∈ [t₁, t₂], we may repeat the above procedure and obtain

()

Further, we have

()

That is,

()

Remark 4. If is the T-periodic solution of system (4), it follows from Lemma 3 that x^*(t) is globally exponentially stable.

3. Main Results

In this section, we will study existence and exponential stability for periodic solutions of system (4).

Theorem 5. Suppose that all conditions in Lemma 3 are satisfied. Then system (4) has exactly one T-periodic solution x^*(t). Moreover, x^*(t) is globally exponentially stable.

Proof. Let x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T be a solution of system (4) with initial conditions (10). By Remark 2, the solution x(t) can be defined for all t ∈ [0, +∞). By hypothesis (H₁), we have, for any natural number h,

()

Further, by hypothesis of (H₃), we obtain

()

Thus, for any natural number h, we obtain that x(t + (h + 1)T) is a solution of system (4) for all t + (h + 1)T ≥ 0. Hence, x(t + T) is also a solution of (4) with initial values

()

Then, by the proof of Lemma 3, there exists a constant K > 0 such that for any natural number h,

()

Moreover, for any natural number m, we can obtain

()

Combining (39) with (40), we know that x(t + mT) will converge uniformly to a piecewise continuous function on any compact set of R.

Now we are in the position of proving that x^*(t) is a T-periodic solution of system (4). It is easily known that x^*(t) is T-periodic since

()

where i = 1,2, …, n. Noting that the right side of (4) is piecewise continuous, together with (36) and (37), we know that

converges uniformly to a piecewise continuous function on any compact set of R∖{t₁, t₂, …}. Therefore, letting m → +∞ on both sides of (36) and (37), we get

()

Thus,

is a T-periodic solution of system (4).

Finally, by Lemma 3, 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 the previous sections.

Example 6. Consider the following impulsive cellar neural network consisting of two neurons with time-varying delays in the leakage terms, which is described by

()

Here, it is assumed that the activation functions are

()

Noting that

()

then we obtain

()

This yields that system (43) satisfies (H₁)–(H₅). Hence, from Theorem 5, system (43) has exactly one 2-periodic solution. Moreover, the 2-periodic solution is globally exponentially stable.

Remark 7. Since g₁(x) = g₂(x) = x + 2sinx, f₁(x) = f₂(x) = x + 3sinx and CNNs (43) is a very simple form of CNNs with time-varying delays in the leakage terms, it is clear that the conditions (A₁) and (A₂) are not satisfied. Therefore, all the results in [11–19] and the references therein cannot be applicable to system (43) to obtain the existence and exponential stability of the 2-periodic solutions.

Acknowledgments

The authors would like to express their sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Hunan Provincial of China (Grant no. 11JJ6006), the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grants nos. 11C0916 and 11C0915), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).

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Periodic Solution for Impulsive Cellar Neural Networks with Time-Varying Delays in the Leakage Terms

Abstract

1. Introduction

2. Preliminary Results

3. Main Results

4. An Example

Acknowledgments

References

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Periodic Solution for Impulsive Cellar Neural Networks with Time-Varying Delays in the Leakage Terms

Abstract

1. Introduction

2. Preliminary Results

3. Main Results

4. An Example

Acknowledgments

References

Citing Literature

References

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