Volume 2013, Issue 1 325310

Research Article

Open Access

Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Corresponding Author

Qi Luo

[email protected]

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Xinjie Miao,

Xinjie Miao

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Qian Wei,

Qian Wei

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Zhengxin Zhou,

Zhengxin Zhou

School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225009, China nuist.edu.cn

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Qi Luo,

Corresponding Author

Qi Luo

[email protected]

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Xinjie Miao,

Xinjie Miao

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Qian Wei,

Qian Wei

School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China nuist.edu.cn

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Zhengxin Zhou,

Zhengxin Zhou

School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225009, China nuist.edu.cn

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First published: 12 May 2013

https://doi.org/10.1155/2013/325310

Citations: 2

Academic Editor: Xuerong Mao

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Abstract

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.

1. Introduction

Since cellular neural networks (CNNs) were proposed by Chua and Yang in 1988 [1, 2], many researchers have put great effort into this subject due to their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.

Owing to the finite switching speed of amplifiers, there is no doubt that time delays exist in the communication and response of neurons. Moreover, as neural networks usually have a spatial extent due to the presences of a multitude of parallel pathways with a variety of axon sizes and lengths, there is a distribution of conduction velocities along these pathways and a distribution of propagation designed with discrete delays. Therefore, a more appropriate and ideal way is to incorporate continuously distributed delays with a result that a more effective model of cellular neural networks with time-varying and distributed delays proposed.

In fact, beside delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. So far, there have been many results [3–11] on the study of dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs. Summing up the existing researches on the stability of complex CNNs, we see that the primary method is Lyapunov theory. However, there are also lots of difficulties in the applications of corresponding theories to specific problems. It is therefore necessary to seek some new methods to deal with the stability in order to overcome those difficulties.

Recently, it is inspiring that Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable conclusions, for example, see the monograph [12] and the work in [13–24]. In addition, more recently, there have been a few papers where the fixed point theory is employed to investigate the stability of stochastic (delayed) differential equations, for instance, see [25–31]. Precisely, in [26–28], Luo used the fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [29, 30], Sakthivel used the fixed point theory to discuss the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [31], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations. We wonder if we can obtain some new and more applicable stability criteria of complex CNNs by applying the fixed point theory.

With this motivation, in this paper, we aim to discuss the global exponential stability of impulsive CNNs with time-varying and distributed delays. It is worth noting that our research technique is based on the contraction mapping principle rather than the usual method of Lyapunov theory. We deal with, by employing the fixed point theorem, the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time, for which Lyapunov method feels helpless. The obtained stability criteria are easily checked and do not require the differentiability of delays.

2. Preliminaries

Let Rⁿ denote the n-dimensional Euclidean space and ∥·∥ represent the Euclidean norm 𝒩≜{1,2, …, n} and R₊ = [0, ∞). C[X, Y] corresponds to the space of continuous mappings from the topological space X to the topological space Y.

In this paper, we consider the following impulsive cellular neural networks with time-varying and distributed delays:

(1)

(2)

where i ∈ 𝒩 and n is the number of neurons in the neural network. x_i(t) corresponds to the state of the ith neuron at time t. f_j, g_j, and σ_j denote the activation functions, respectively. The constant a_i > 0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. The constants b_ij, c_ij, and d_ij represent the connection weights of the jth neuron to the ith neuron, respectively. τ_ij(t) and ρ(t) correspond to the transmission delays meeting 0 ≤ τ_ij(t) ≤ τ (τ = constant) and 0 ≤ ρ(t) ≤ ρ (ρ = constant). The fixed impulsive moments t_k (k = 1,2, …) satisfy 0 = t₀ < t₁ < t₂ < ⋯ and lim_k→∞t_k = ∞. x_i(t_k + 0) and x_i(t_k − 0) stand for the right-hand and left-hand limits of x_i(t) at time t_k, respectively. I_ik(x_i(t_k)) shows the impulsive perturbation of the ith neuron at the impulsive moment t_k.

Throughout this paper, we always assume that f_i(0) = g_i(0) = σ_i(0) = I_ik(0) = 0 for i ∈ 𝒩 and k = 1,2, …. Thereby, problems (1) and (2) admit a trivial equilibrium x = 0.

Denote by

the solution to (1) and (2) with the initial condition

(3)

where m^* = max {τ, ρ}, φ_i(s) ∈ C[[−m^*, 0], R] and

The solution x(t)≜x(t; s, φ) ∈ Rⁿ to (1)–(3) is, for the time variable t, a piecewise continuous vector-valued function with the first-kind discontinuous points t_k (k = 1,2, …), where it is left-continuous; that is, the following relations are true:

(4)

Definition 1. The trivial equilibrium x = 0 is said to be globally exponentially stable if for any initial condition φ(s) ∈ C[[−m^*, 0], Rⁿ], there exists a pair of positive constants λ and M such that

(5)

The consideration of this paper is based on the following fixed point theorem.

Theorem 2 (see [32].)Let Υ be a contraction operator on a complete metric space Θ, then there exists a unique point ζ ∈ Θ for which Υ(ζ) = ζ.

3. Main Results

In this section, we will, for (1)–(3), use the contraction mapping principle to prove the existence and uniqueness of the solution and the global exponential stability of trivial equilibrium all at once. Before proceeding, we firstly introduce some assumptions as follows.

(A1)
There exist nonnegative constants l_j such that for any η, υ ∈ R,
(6)
(A2)
There exist nonnegative constants k_j such that for any η, υ ∈ R,
(7)
(A3)
There exist nonnegative constants p_jk such that for any η, υ ∈ R,
(8)
(A4)
There exist nonnegative constants ω_j such that for any η, υ ∈ R,
(9)

Let ℋ = ℋ₁ × ⋯×ℋ_n, and let ℋ_i (i ∈ 𝒩) be the space embracing functions ϕ_i(t) : [−m^*, +∞) → R, wherein ϕ_i(t) satisfies the following:

(1)
ϕ_i(t) is continuous on t ≠ t_k (k = 1,2, …),
(2)
and exist; moreover, for k = 1,2, …,
(3)
ϕ_i(s) = φ_i(s) on s ∈ [−m^*, 0],
(4)
e^αtϕ_i(t) → 0 as t → ∞, where α = const and 0 < α < min _i∈𝒩{a_i},

where t_k and φ_i(s) are defined as shown in Section 2. Also ℋ is a complete metric space when it is equipped with a metric defined by

(10)

where

and

Theorem 3. Assume that conditions (A1)–(A4) hold provided that

(i)
there exists a constant μ such that inf _k=1,2,…{t_k − t_k−1} ≥ μ,
(ii)
there exist constants p_i such that p_ik ≤ p_iμ for i ∈ 𝒩 and k = 1,2, …,
(iii)
max _i∈𝒩{p_i(μ + (1/a_i))}≜χ < 1,

and then the trivial equilibrium x = 0 is globally exponentially stable.

Proof. Multiplying both sides of (1) with gives, for t > 0 and t ≠ t_k,

(11)

which yields after integrating from t_k−1 + ε (ε > 0) to t ∈ (t_k−1, t_k) (k = 1,2, …) that

(12)

Letting ε → 0 in (12), we have, for t ∈ (t_k−1, t_k) (k = 1,2, …),

(13)

Setting t = t_k − ε (ε > 0) in (13), we get

(14)

which generates by letting ε → 0

(15)

Noting x_i(t_k − 0) = x_i(t_k), (15) can be rearranged as

(16)

Combining (13) and (16), we derive that

(17)

is true for t ∈ (t_k−1, t_k] (k = 1,2, …). Hence, we get, for t ∈ (t_k−1, t_k] (k = 1,2, …),

(18)

which results in

(19)

We therefore conclude, for t > 0,

(20)

Note that x_i(0) = φ_i(0) in (20). We then define the following operator π acting on ℋ, for :

(21)

where π(y_i)(t) : [−m^*, +∞) → R (i ∈ 𝒩) obeys the rule as follows:

(22)

on t ≥ 0 and π(y_i)(s) = φ_i(s) on s ∈ [−m^*, 0].

In what follows, we will apply the contraction mapping principle to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time. The subsequent proof can be divided into two steps.

Step 1. We need to prove that π(ℋ) ⊂ ℋ. For y_i(t) ∈ ℋ_i (i ∈ 𝒩), it is necessary to show that π(y_i)(t) ⊂ ℋ_i. As defined above, we see that π(y_i)(s) = φ_i(s) on s ∈ [−m^*, 0]. Owing to the continuity of φ_i(s) on s ∈ [−m^*, 0], we immediately know that π(y_i)(t) is continuous on t ∈ [−m^*, 0].

Choose a fixed time t > 0, and it is then derived from (22) that

(23)

where,

(24)

Since y_i(t) ∈ ℋ_i, we know that y_i(t) is continuous on t ≠ t_k (k = 1,2, …); moreover, and exist, in addition, .

Letting t ≠ t_k (k = 1,2, …) in (23), it is easy to see that Q_i → 0 as r → 0 for i = 1, …, 5. Thus, π(y_i)(t + r) − π(y_i)(t) → 0 as r → 0 holds on t > 0 and t ≠ t_k (k = 1,2, …).

Letting t = t_k (k = 1,2, …) in (23), it is not difficult to find that Q_i → 0 as r → 0 for i = 1, …, 4. Letting r < 0 be small enough, we compute

(25)

which implies

. Letting r > 0 be small enough, we have

(26)

which implies

According to the above discussion, we see that π(y_i)(t) : [−m^*, +∞) → R is continuous on ≠t_k(k = 1,2, …), while for t = t_k (k = 1,2, …), and exist; moreover, .

Next, we will prove that e^αtπ(y_i)(t) → 0 as t → ∞ for i ∈ 𝒩. To begin with, we give the expression of e^αtπ(y_i)(t) as follows:

(27)

Where

,
,
,
, and
.

First, it is obvious that lim_t→∞W₁ = 0 as a_i − α > 0. Furthermore, for y_j(t) ∈ ℋ_j (j ∈ 𝒩), we see lim_t→∞e^αty_j(t) = 0. Then, for any ε > 0, there exists a T_j > 0 such that s ≥ T_j implies |e^αs y_j(s)| < ε. Choose T^* = max _j∈𝒩 {T_j}. It is derived form (A1) that

(28)

which leads to W₂ → 0 as t → ∞.

Similarly, for the given ε > 0 above, there also exists a such that implies |e^αsy_j(s)| < ε. Select . It follows from (A2) that

(29)

which results in W₃ → 0 as t → ∞. In addition, it is derived from (A4) that

(30)

Since e^αζ|y_j(ζ)| → 0 as ζ → ∞, we know that, for any ε > 0, there exists a such that implies e^αζ|y_j(ζ)| < ε. Selecting , it follows from (30) that

(31)

which yields W₄ → 0 as t → ∞.

Furthermore, from (A3), we see that |I_ik(x_i(t_k))| ≤ p_ik|y_i(t_k)|. So,

(32)

As y_i(t) ∈ ℋ_i, we have lim _t→∞ e^αty_i(t) = 0. Then, for any ε > 0, there exists a nonimpulsive point T_i > 0 such that s ≥ T_i implies |e^αsy_i(s)| < ε. It then follows from conditions (i) and (ii) that

(33)

which means that W₅ → 0 as t → ∞.

Now, we can derive from (27) that e^αtπ(y_i)(t) → 0 as t → ∞ for i ∈ 𝒩. It is therefore concluded that π(y_i)(t) ⊂ ℋ_i which results in π(ℋ) ⊂ ℋ.

Step 2. We need to prove that π is contractive. For and , we estimate

(34)

where

(35)

Note that

(36)

It is then derived from (36) that

(37)

which means that

(38)

where

(39)

In view of condition (iii), we know that π is a contraction mapping, and hence, there exists a unique fixed point of π in ℋ which means that is the solution to (1)–(3) and as t → ∞. This completes the proof.

Lemma 4. Assume conditions (A1)–(A4) hold. Provided that

(i)
inf _k=1,2,… {t_k − t_k−1} ≥ 1,
(ii)
there exist constants p_i such that p_ik ≤ p_i for i ∈ 𝒩 and k = 1,2, …,
(iii)
max _i∈𝒩 {p_i(1 + (1/a_i))}≜χ < 1,

then the trivial equilibrium x = 0 is globally exponentially stable.

Proof. Lemma 4 is a direct conclusion by letting μ = 1 in Theorem 3.

Remark 5. In Theorem 3, we use the fixed point theorem to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium all at once, while Lyapunov method fails to do this.

Remark 6. The presented sufficient conditions in Theorem 3 and Lemma 4 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

4. Example

Consider the following two-dimensional impulsive cellular neural network with time-varying and distributed delays.

(40)

with the initial conditions x₁(s) = cos (s), x₂(s) = sin(s) on −m^* ≤ s ≤ 0, where τ_ij(t) = 0.8 + 0.4cos (t), ρ(t) = 0.5 + 0.3sin(t), m^* is defined as shown in (3), a₁ = a₂ = 7, b_ij = 0, c₁₁ = 0, c₁₂ = 1/7, c₂₁ = −1/7, c₂₂ = −1/7, d₁₁ = 3/7, d₁₂ = 2/7, d₂₁ = 0, d₂₂ = 1/7, f_j(s) = g_j(s) = σ_j(s) = (|s + 1| − |s − 1|)/2, and t_k = t_k−1 + 0.5k.

It is easily to find that μ = 0.5, l_j = k_j = ω_j = 1, and p_ik = 0.4. Let p_i = 0.8 and compute

(41)

From Theorem 3, we conclude that the trivial equilibrium x = 0 of this two-dimensional impulsive cellular neural network with time-varying and distributed delays is globally exponentially stable.

5. Conclusions

This article is a new attempt of applying the fixed point theory to the stability analysis of impulsive neural networks with time-varying and distributed delays, which is different from the existing relevant publications where Lyapunov theory is the main technique. From what have been discussed above, we see that the contraction mapping principle is effective for not only the investigation of the existence and uniqueness of solution but also for the stability analysis of trivial equilibrium. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of complex neural networks.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant no. 61174077 and Natural Science Foundation of Jiangsu of China under Grant no. BK2012682.

References

1 Chua L. O. and Yang L., Cellular neural networks: theory, IEEE Transactions on Circuits and Systems. (1988) 35, no. 10, 1257–1272, https://doi.org/10.1109/31.7600, MR960777, ZBL0663.94022.
10.1109/31.7600
Web of Science® Google Scholar
2 Chua L. O. and Yang L., Cellular neural networks: applications, IEEE Transactions on Circuits and Systems. (1988) 35, no. 10, 1273–1290, https://doi.org/10.1109/31.7601, MR960778.
10.1109/31.7601
Web of Science® Google Scholar
3 Stamov G. T. and Stamova I. M., Almost periodic solutions for impulsive neural networks with delay, Applied Mathematical Modelling. (2007) 31, no. 7, 1263–1270, 2-s2.0-33846641685, https://doi.org/10.1016/j.apm.2006.04.008.
10.1016/j.apm.2006.04.008
Web of Science® Google Scholar
4 Ahmad S. and Stamova I. M., Global exponential stability for impulsive cellular neural networks with time-varying delays, Nonlinear Analysis. Theory, Methods & Applications. (2008) 69, no. 3, 786–795, https://doi.org/10.1016/j.na.2008.02.067, MR2428750, ZBL1151.34061.
10.1016/j.na.2008.02.067
Web of Science® Google Scholar
5 Li K., Zhang X., and Li Z., Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay, Chaos, Solitons and Fractals. (2009) 41, no. 3, 1427–1434, https://doi.org/10.1016/j.chaos.2008.06.003, MR2537611, ZBL1198.34154.
10.1016/j.chaos.2008.06.003
Web of Science® Google Scholar
6 Qiu J., Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing. (2007) 70, no. 4-6, 1102–1108, 2-s2.0-33846004186, https://doi.org/10.1016/j.neucom.2006.08.003.
10.1016/j.neucom.2006.08.003
Web of Science® Google Scholar
7 Wang X. and Xu D., Global exponential stability of impulsive fuzzy cellular neural networks with mixed delays and reaction-diffusion terms, Chaos, Solitons & Fractals. (2009) 42, no. 5, 2713–2721, https://doi.org/10.1016/j.chaos.2009.03.177, MR2559991, ZBL1198.34094.
10.1016/j.chaos.2009.03.177
Web of Science® Google Scholar
8 Zhang Y. and Luo Q., Global exponential stability of impulsive delayed reaction-diffusion neural networks via Hardy-Poincare Inequality, Neurocomputing. (2012) 83, 198–204, https://doi.org/10.1016/j.neucom.2011.12.024.
10.1016/j.neucom.2011.12.024
Web of Science® Google Scholar
9 Zhang Y. and Luo Q., Novel stability criteria for impulsive delayed reaction-diffusion Cohen-Grossberg neural networks via Hardy-Poincarè inequality, Chaos, Solitons & Fractals. (2012) 45, no. 8, 1033–1040, https://doi.org/10.1016/j.chaos.2012.05.001, MR2949090.
10.1016/j.chaos.2012.05.001
Web of Science® Google Scholar
10 Zhang W., Tang Y., Fang J.-a., and Wu X., Stability of delayed neural networks with time-varying impulses, Neural Networks. (2012) 36, 59–63, https://doi.org/10.1016/j.neunet.2012.08.014.
10.1016/j.neunet.2012.08.014
CAS PubMed Web of Science® Google Scholar
11 Zhu S. and Shen Y., Robustness analysis for connection weight matrices of global exponential stability of stochastic recurrent neural networks, Neural Networks. (2013) 38, 17–22, https://doi.org/10.1016/j.neunet.2012.10.004.
10.1016/j.neunet.2012.10.004
PubMed Web of Science® Google Scholar
12 Burton T. A., Stability by Fixed Point Theory for Functional Differential Equations, 2006, Dover, Mineola, NY, USA, MR2281958.
Google Scholar
13 Becker L. C. and Burton T. A., Stability, fixed points and inverses of delays, Proceedings of the Royal Society of Edinburgh A. (2006) 136, no. 2, 245–275, https://doi.org/10.1017/S0308210500004546, MR2218152, ZBL1112.34054.
10.1017/S0308210500004546
Web of Science® Google Scholar
14 Burton T. A., Fixed points, stability, and exact linearization, Nonlinear Analysis. Theory, Methods & Applications. (2005) 61, no. 5, 857–870, https://doi.org/10.1016/j.na.2005.01.079, MR2130068, ZBL1067.34077.
10.1016/j.na.2005.01.079
Web of Science® Google Scholar
15 Burton T. A., Fixed points, Volterra equations, and Becker′s resolvent, Acta Mathematica Hungarica. (2005) 108, no. 3, 261–281, https://doi.org/10.1007/s10474-005-0224-9, MR2162564, ZBL1091.34040.
10.1007/s10474-005-0224-9
Web of Science® Google Scholar
16 Burton T. A., Fixed points and stability of a nonconvolution equation, Proceedings of the American Mathematical Society. (2004) 132, no. 12, 3679–3687, https://doi.org/10.1090/S0002-9939-04-07497-0, MR2084091, ZBL1050.34110.
10.1090/S0002-9939-04-07497-0
Web of Science® Google Scholar
17 Burton T. A., Perron-type stability theorems for neutral equations, Nonlinear Analysis. Theory, Methods & Applications. (2003) 55, no. 3, 285–297, https://doi.org/10.1016/S0362-546X(03)00240-2, MR2007475, ZBL1044.34028.
10.1016/S0362-546X(03)00240-2
Web of Science® Google Scholar
18 Burton T. A., Integral equations, implicit functions, and fixed points, Proceedings of the American Mathematical Society. (1996) 124, no. 8, 2383–2390, https://doi.org/10.1090/S0002-9939-96-03533-2, MR1346965, ZBL0873.45003.
10.1090/S0002-9939-96-03533-2
Web of Science® Google Scholar
19 Burton T. A. and Furumochi T., Krasnoselskii′s fixed point theorem and stability, Nonlinear Analysis. Theory, Methods & Applications. (2002) 49, no. 4, 445–454, https://doi.org/10.1016/S0362-546X(01)00111-0, MR1886230, ZBL1015.34046.
10.1016/S0362-546X(01)00111-0
Web of Science® Google Scholar
20 Burton T. A. and Zhang B., Fixed points and stability of an integral equation: nonuniqueness, Applied Mathematics Letters. (2004) 17, no. 7, 839–846, https://doi.org/10.1016/j.aml.2004.06.015, MR2072844, ZBL1066.45002.
10.1016/j.aml.2004.06.015
Web of Science® Google Scholar
21 Furumochi T., Stabilities in FDEs by Schauder′s theorem, Nonlinear Analysis, Theory, Methods and Applications. (2005) 63, no. 5–7, e217–e224, 2-s2.0-28044472264, https://doi.org/10.1016/j.na.2005.02.057.
10.1016/j.na.2005.02.057
Web of Science® Google Scholar
22 Jin C. and Luo J., Fixed points and stability in neutral differential equations with variable delays, Proceedings of the American Mathematical Society. (2008) 136, no. 3, 909–918, https://doi.org/10.1090/S0002-9939-07-09089-2, MR2361863, ZBL1136.34059.
10.1090/S0002-9939-07-09089-2
Web of Science® Google Scholar
23 Raffoul Y. N., Stability in neutral nonlinear differential equations with functional delays using fixed-point theory, Mathematical and Computer Modelling. (2004) 40, no. 7-8, 691–700, https://doi.org/10.1016/j.mcm.2004.10.001, MR2106161, ZBL1083.34536.
10.1016/j.mcm.2004.10.001
Google Scholar
24 Zhang B., Fixed points and stability in differential equations with variable delays, Nonlinear Analysis, Theory, Methods and Applications. (2005) 63, no. 5–7, e233–e242, 2-s2.0-28044433663, https://doi.org/10.1016/j.na.2005.02.081.
10.1016/j.na.2005.02.081
Web of Science® Google Scholar
25 Luo J., Fixed points and stability of neutral stochastic delay differential equations, Journal of Mathematical Analysis and Applications. (2007) 334, no. 1, 431–440, https://doi.org/10.1016/j.jmaa.2006.12.058, MR2332567, ZBL1160.60020.
10.1016/j.jmaa.2006.12.058
Web of Science® Google Scholar
26 Luo J., Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, Journal of Mathematical Analysis and Applications. (2008) 342, no. 2, 753–760, https://doi.org/10.1016/j.jmaa.2007.11.019, MR2433617, ZBL1157.60065.
10.1016/j.jmaa.2007.11.019
Web of Science® Google Scholar
27 Luo J., Stability of stochastic partial differential equations with infinite delays, Journal of Computational and Applied Mathematics. (2008) 222, no. 2, 364–371, https://doi.org/10.1016/j.cam.2007.11.002, MR2474632, ZBL1151.60336.
10.1016/j.cam.2007.11.002
Web of Science® Google Scholar
28 Luo J. and Taniguchi T., Fixed points and stability of stochastic neutral partial differential equations with infinite delays, Stochastic Analysis and Applications. (2009) 27, no. 6, 1163–1173, https://doi.org/10.1080/07362990903259371, MR2573454, ZBL1177.93094.
10.1080/07362990903259371
Web of Science® Google Scholar
29 Sakthivel R. and Luo J., Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, Journal of Mathematical Analysis and Applications. (2009) 356, no. 1, 1–6, https://doi.org/10.1016/j.jmaa.2009.02.002, MR2524209, ZBL1166.60037.
10.1016/j.jmaa.2009.02.002
Web of Science® Google Scholar
30 Sakthivel R. and Luo J., Asymptotic stability of nonlinear impulsive stochastic differential equations, Statistics & Probability Letters. (2009) 79, no. 9, 1219–1223, https://doi.org/10.1016/j.spl.2009.01.011, MR2519005, ZBL1166.60316.
10.1016/j.spl.2009.01.011
Web of Science® Google Scholar
31 Luo J., Fixed points and exponential stability for stochastic Volterra-Levin equations, Journal of Computational and Applied Mathematics. (2010) 234, no. 3, 934–940, https://doi.org/10.1016/j.cam.2010.02.013, MR2606678, ZBL1197.60053.
10.1016/j.cam.2010.02.013
Web of Science® Google Scholar
32 Smart D. R., Fixed Point Theorems, 1974, Cambridge University Press, London, UK, MR0467717.
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Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

5. Conclusions

Acknowledgment

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Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Example

5. Conclusions

Acknowledgment

References

Citing Literature

References

Related

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