Existence and Exponential Stability of Equilibrium Point for Fuzzy BAM ...

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 721586, 17 pages http://dx.doi.org/10.1155/2014/721586

Research Article Existence and Exponential Stability of Equilibrium Point for Fuzzy BAM Neural Networks with Infinitely Distributed Delays and Impulses on Time Scales Yongkun Li, Lijie Sun, and Li Yang Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China Correspondence should be addressed to Yongkun Li; [email protected] Received 28 November 2013; Revised 6 February 2014; Accepted 10 February 2014; Published 1 April 2014 Academic Editor: Ray K. L. Su Copyright © 2014 Yongkun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the fixed point theorem and constructing a Lyapunov functional, we establish some sufficient conditions on the existence, uniqueness, and exponential stability of equilibrium point for a class of fuzzy BAM neural networks with infinitely distributed delays and impulses on time scales. We also present a numerical example to show the feasibility of obtained results. Our example also shows that the described time and continuous neural time networks have the same dynamic behaviours for the stability.

1. Introduction The bidirectional associative memory (BAM) neural network models were first introduced by Kosko (see [1]). BAM neural network is a special class of recurrent neural networks that can store bipolar vector pairs. It is composed of neurons arranged in two layers, the 𝑋-layer and 𝑌-layer. This class of networks possesses good applications prospects in areas of pattern recognition, signal and image process, and automatic control. Such applications heavily depend on the dynamical behaviors of neural networks. Thus, the analysis of the dynamical behaviors is a necessary step for practical design neural networks. In particular, many researchers have studied the dynamics of BAM neural networks with or without delays including stability and existence of periodic solutions or almost periodic solutions. For the results on BAM neural networks, the reader may see [2–11] and reference therein. In mathematical modeling of real world problems, we will encounter some inconveniences, for example, the complexity and the uncertainty or vagueness. For the sake of taking vagueness into consideration, fuzzy theory is considered as a suitable method. Yang et al. proposed fuzzy cellular neural network, which integrates fuzzy logic into the structure of traditional cellular neural networks and maintains local connectedness among cells [12]. Fuzzy neural network has

fuzzy logic between its template input and/or output besides the sum of product operation. Studies have been revealed that fuzzy neural network has its potential in imagine processing and pattern recognition and some results have reported on the stability and periodicity of fuzzy neural networks. Besides, in reality, time delays often occur due to finite switching speeds of the amplifiers and communication time and can destroy a stable network or cause sustained oscillations, bifurcation, or chaos. Hence, it is important to consider both the fuzzy logic and delay effect on dynamical behaviors of neural networks. There have been many results on the fuzzy neural networks with time delays [13–21]. For example, in [22], under the assumption that the activation function 𝑓𝑗 (𝑢) is a second differentiable bounded function, 𝑗 = 1, 2, . . . , 𝑛, the authors proved the exponential stability and obtained the domain of robust attraction of the equilibrium point of the following interval fuzzy neural network: 𝑛

𝑥𝑖󸀠 (𝑡) = − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑ 𝑏𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡)) 𝑗=1

𝑛

𝑛

𝑗=1

𝑗=1

+ ⋀𝛼𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡)) + ⋀𝑇𝑖𝑗 𝑢𝑗

2

Journal of Applied Mathematics 𝑛

𝑛

𝑗=1

𝑗=1

𝑚

+ ⋁𝛽𝑖𝑗 𝑓𝑗 (𝑥𝑗 (𝑡)) + ⋁𝑆𝑖𝑗 𝑢𝑗 𝑛

+ ∑𝑐𝑖𝑗 𝜇𝑗 + 𝐼𝑖 ,

(1) In fact, both continuous and discrete systems are very important in implementation and applications. To avoid the troublesomeness of studying the dynamical properties for continuous and discrete systems, respectively, it is meaningful to study that on time scales, which was initiated by Stefan Hilger in his Ph.D. thesis in order to unify continuous and discrete analysis. Lots of scholars have studied neural networks on time scales and obtained many good results [23–29]. For example, in [30], the authors considered the existence and global exponential stability of an equilibrium point for a class of fuzzy BAM neural networks with time-varying delays in leakage terms on time scales. Moreover, many systems also undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. Therefore, it is significant to study the dynamics of impulsive systems. For example, in [31], the authors studied the exponential stability of the following impulsive system on time scales: 𝑚

𝑥𝑖Δ (𝑡) = − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑𝑏𝑗𝑖 𝑓𝑗 (𝑦𝑗 (𝑡)) 𝑗=1

𝑗=1

𝑗=1

𝑚

𝑚

𝑗=1

𝑗=1

+ ∑𝑏𝑗𝑖 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜎𝑗 )) + ⋀𝛼𝑗𝑖 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜎𝑗 )) + ⋁𝛽𝑗𝑖 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝜎𝑗 )) + ⋀𝑇𝑗𝑖 𝜇𝑗 𝑚

+ ⋁𝐻𝑗𝑖 𝜇𝑗 + 𝐼𝑖 , 𝑗=1

Δ𝑥𝑖 (𝑡𝑘 ) = 𝐼𝑘 (𝑥𝑖 (𝑡𝑘 )) ,

𝑡 ≠ 𝑡𝑘 , 𝑡 ∈ T + , 𝑖 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . . , 𝑛, 𝑘 = 1, 2, . . . , (2)

where T + = T ∩ (0, +∞) and T is a time scale, and in [32], authors studied a class of fuzzy BAM neural networks with finite distributed delays and impulses, by using fixed point theorem and differential inequality techniques; they established the existence and global exponential stability of unique equilibrium point to the networks. However, to the best of our knowledge, few papers were published on the exponential stability of fuzzy BAM neural networks with distributed delays and impulses on time scales. Motivated by the above, in this paper, we study the following fuzzy BAM neural networks with infinitely distributed delays and impulses on time scales: 𝑚

𝑚

∞

0

𝑗=1

𝑗=1

𝑡 ≠ 𝑡𝑘 , 𝑡 ∈ T + , 𝑖 = 1, 2, . . . , 𝑛, Δ𝑥𝑖 (𝑡𝑘 ) = 𝑃𝑘 (𝑥𝑖 (𝑡𝑘 )) ,

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

∞

𝑦𝑗Δ (𝑡) = − 𝑏𝑗 𝑦𝑗 (𝑡) + ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1

𝑛

0

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1 𝑛

0

∞

+ ⋁𝑞𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1

0

𝑛

𝑛

𝑖=1

𝑖=1

+ ⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐽𝑗 , 𝑡 ≠ 𝑡𝑘 , 𝑡 ∈ T + , 𝑗 = 1, 2, . . . , 𝑚, Δ𝑦𝑗 (𝑡𝑘 ) = 𝑄𝑘 (𝑦𝑗 (𝑡𝑘 )) ,

𝑗 = 1, 2, . . . , 𝑚, 𝑘 = 1, 2, . . . , (3)

where T is a time scale that is closed under addition; 𝑛, 𝑚 are the numbers of neurons in layers; 𝑥𝑖 (𝑡) and 𝑦𝑗 (𝑡) denote the activations of the 𝑖th neuron and the 𝑗th neuron at time 𝑡; 𝑎𝑖 > 0 and 𝑏𝑗 > 0 represent the rate with which the 𝑖th neuron and the 𝑗th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs; 𝑓𝑗 , 𝑔𝑖 are the input-output functions (the activation functions); 𝑐𝑗𝑖 , 𝑑𝑖𝑗 are elements of feedback templates; 𝛼𝑗𝑖 , 𝑝𝑖𝑗 denote elements of fuzzy feedback MIN templates and 𝛽𝑗𝑖 , 𝑞𝑖𝑗 are elements of fuzzy feedback MAX templates; 𝑇𝑗𝑖 , 𝐹𝑖𝑗 are fuzzy feed-forward MIN templates and 𝐻𝑗𝑖 , 𝐺𝑖𝑗 are fuzzy feed-forward MAX templates; 𝜇𝑗 , ]𝑖 denote the input of the 𝑖th neuron and the 𝑗th neuron; the delayed feedback 𝑘𝑗𝑖 (𝑠) and ℎ𝑖𝑗 (𝑠) are real valued nonnegative continuous functions defined on T + with ∞ ∞ ∫0 𝑘𝑗𝑖 (𝑠)Δ𝑠 ≤ 𝑘𝑗𝑖 and ∫0 ℎ𝑖𝑗 (𝑠)Δ𝑠 ≤ ℎ𝑖𝑗 , where 𝑘𝑗𝑖 and ℎ𝑖𝑗 are nonnegative constants; 𝐼𝑖 , 𝐽𝑗 denote biases of the 𝑖th neuron and the 𝑗th neuron, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚; ∧ and ∨ denote the fuzzy AND and fuzzy OR operations, respectively; the impulsive moments 𝑡𝑘 satisfy 0 ≤ 𝑡0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ and lim𝑘 → ∞ 𝑡𝑘 = ∞. For 𝑧 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , 𝑦1 , 𝑦2 , . . . , 𝑦𝑚 ) ∈ R𝑛+𝑚 , we define the norm as ‖𝑧‖ = ∑𝑛𝑖=1 |𝑥𝑖 | + ∑𝑚 𝑗=1 |𝑦𝑗 |. For the sake of convenience, we denote the T-interval [𝑎, 𝑏]T as [𝑎, 𝑏]T := {𝑡 ∈ T | 𝑎 ≤ 𝑡 ≤ 𝑏}. The initial condition of (3) is of the form 𝑥𝑖 (𝑠) = 𝜑𝑖 (𝑠) ,

𝑠 ∈ (−∞, 0]T , 𝑖 = 1, 2, . . . , 𝑛,

0

𝑦𝑗 (𝑠) = 𝜓𝑗 (𝑠) ,

𝑠 ∈ (−∞, 0]T , 𝑗 = 1, 2, . . . , 𝑚,

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 𝑗=1

𝑚

∞

𝑥𝑖Δ (𝑡) = − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 𝑗=1

𝑚

+ ⋀𝑇𝑗𝑖 𝜇𝑗 + ⋁𝐻𝑗𝑖 𝜇𝑗 + 𝐼𝑖 ,

𝑗=1

𝑚

0

𝑗=1

𝑖 = 1, 2, . . . , 𝑛.

𝑚

∞

+ ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠

(4)

where 𝜑𝑖 (⋅), 𝜓𝑗 (⋅) denote positive real-valued continuous functions on (−∞, 0]T .

Journal of Applied Mathematics

3

Throughout this paper, we make the following assumptions:

Lemma 4 (see [33]). Assume that 𝑝, 𝑞 : T → R are two regressive functions; then (i) 𝑒0 (𝑡, 𝑠) ≡ 1 and 𝑒𝑝 (𝑡, 𝑡) ≡ 1;

(H 1 ) for 𝑡, 𝑠 ∈ T, 𝑡 ± 𝑠 ∈ T; 𝑓

(H 2 ) 𝑓𝑗 , 𝑔𝑖 ∈ 𝐶(R, R) and there exist positive constants 𝐿 𝑗 , 𝑔 𝐿 𝑖 such that 󵄨 󵄨󵄨 󵄨󵄨𝑓𝑗 (𝑢) − 𝑓𝑗 (V)󵄨󵄨󵄨 ≤ 𝐿𝑓𝑗 |𝑢 − V| , 󵄨 󵄨 𝑔 󵄨󵄨 󵄨 󵄨󵄨𝑔𝑖 (𝑢) − 𝑔𝑖 (V)󵄨󵄨󵄨 ≤ 𝐿 𝑖 |𝑢 − V| ,

(5)

for all 𝑢, V ∈ R, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚;

(ii) 𝑒𝑝 (𝑡, 𝑠) = 1/𝑒𝑝 (𝑠, 𝑡) = 𝑒⊖𝑝 (𝑠, 𝑡); (iii) 𝑒𝑝 (𝑡, 𝑠)𝑒𝑝 (𝑠, 𝑟) = 𝑒𝑝 (𝑡, 𝑟); (iv) (𝑒𝑝 (𝑡, 𝑠))Δ = 𝑝(𝑡)𝑒𝑝 (𝑡, 𝑠). Definition 5 (see [33]). A function 𝐹 : T 𝑘 → R is called a delta antiderivative of 𝑓 : T → R provided 𝐹Δ = 𝑓 holds for all 𝑡 ∈ T 𝑘 . In this case we defined the integral of 𝑓 by 𝑡

(H 3 ) there exists a positive constant 𝑝 such that for 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚, ∞

∫ 𝑓 (𝑠) Δ𝑠 = 𝐹 (𝑡) − 𝐹 (𝑎) , 𝑎

0

∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) ℎ𝑖𝑗 (𝑠) Δ𝑠 < +∞,

(9)

𝑡 ∈ T 𝑘.

(10)

and we have the following formula:

∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) 𝑘𝑗𝑖 (𝑠) Δ𝑠 < +∞, ∞

𝑡 ∈ T,

∫

(6)

0

𝜎(𝑡)

𝑡

𝑓 (𝑠) Δ𝑠 = 𝜇 (𝑡) 𝑓 (𝑡) ,

Lemma 6 (see [33]). Let 𝑓, 𝑔 be Δ-differentiable functions on 𝑇; then

∀𝑡 ∈ T. Remark 1. It is oblivious that if T = R or T = Z, then (𝐻1 ) holds and that (1) and (2) are all special cases of (3). Our methods used in this paper are different from those used in [22, 30–32]. The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the existence and uniqueness of the equilibrium point of (3). In Section 4, we prove the equilibrium point of (3) is exponentially stable. In Section 5, we give an example to illustrate the feasibility of our results obtained in previous sections.

(i) (]1 𝑓 + ]2 𝑔)Δ = ]1 𝑓Δ + ]2 𝑔Δ , for any constants ]1 , ]2 ; (ii) (𝑓𝑔)Δ (𝑡) = 𝑓Δ (𝑡)𝑔(𝑡) + 𝑓(𝜎(𝑡))𝑔Δ (𝑡) = 𝑓(𝑡)𝑔Δ (𝑡) + 𝑓Δ (𝑡)𝑔(𝜎(𝑡)). Lemma 7 (see [33]). Assume that 𝑝(𝑡) ≥ 0 for 𝑡 ≥ 𝑠; then 𝑒𝑝 (𝑡, 𝑠) ≥ 1. Lemma 8 (see [33]). Suppose that 𝑝 ∈ R+ ; then (i) 𝑒𝑝 (𝑡, 𝑠) > 0, for all 𝑡, 𝑠 ∈ T; (ii) if 𝑝(𝑡) ≤ 𝑞(𝑡) for all 𝑡 ≥ 𝑠, 𝑡, 𝑠 ∈ T, then 𝑒𝑝 (𝑡, 𝑠) ≤ 𝑒𝑞 (𝑡, 𝑠) for all 𝑡 ≥ 𝑠. Lemma 9 (see [33]). If 𝑝 ∈ R and 𝑎, 𝑏, 𝑐 ∈ T, then

2. Preliminaries

Δ

𝜎

[𝑒𝑝 (𝑐, ⋅)] = −𝑝[𝑒𝑝 (𝑐, ⋅)] , 𝑏

In this section, we state some preliminary results. Definition 2 (see [33]). Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators 𝜎, 𝜌 : T → T and the graininess 𝜇 : T → R+ are defined, respectively, by 𝜎 (𝑡) = inf {𝑠 ∈ T : 𝑠 > 𝑡} , 𝜌 (𝑡) = sup {𝑠 ∈ T : 𝑠 < 𝑡} ,

(7)

𝜇 (𝑡) = 𝜎 (𝑡) − 𝑡. Definition 3 (see [33]). A function 𝑟 : T → R is called regressive if 1 + 𝜇 (𝑡) 𝑟 (𝑡) ≠ 0,

(8)

for all 𝑡 ∈ T 𝑘 . The set of all regressive and 𝑟𝑑-continuous functions 𝑟 : T → R will be denoted by R. We define the set R+ = {𝑟 ∈ R : 1 + 𝜇(𝑡)𝑟(𝑡) > 0, ∀𝑡 ∈ T}.

∫ 𝑝 (𝑡) 𝑒𝑝 (𝑐, 𝜎 (𝑡)) Δ𝑡 = 𝑒𝑝 (𝑐, 𝑎) − 𝑒𝑝 (𝑐, 𝑏) .

(11)

𝑎

Lemma 10 (see [33]). Let 𝑎 ∈ T 𝑘 , 𝑏 ∈ T and assume that 𝑓 : T × T 𝑘 → R is continuous at (𝑡, 𝑡), where 𝑡 ∈ T 𝑘 with 𝑡 > 𝑎. Also assume that 𝑓Δ (𝑡, ⋅) is 𝑟𝑑-continuous on [𝑎, 𝜎(𝑡)]. Suppose that for each 𝜀 > 0, there exists a neighborhood 𝑈 of 𝜏 ∈ [𝑎, 𝜎(𝑡)] such that 󵄨 󵄨󵄨 󵄨󵄨𝑓 (𝜎 (𝑡) , 𝜏) − 𝑓 (𝑠, 𝜏) − 𝑓Δ (𝑡, 𝜏) (𝜎 (𝑡) − 𝑠)󵄨󵄨󵄨 󵄨 󵄨 (12) ≤ 𝜀 |𝜎 (𝑡) − 𝑠| , ∀𝑠 ∈ 𝑈, where 𝑓Δ denotes the derivative of 𝑓 with respect to the first variable. Then 𝑡

𝑡

𝑏

𝑏

(i) 𝑔(𝑡) := ∫𝑎 𝑓(𝑡, 𝜏)Δ𝜏 implies 𝑔Δ (𝑡) := ∫𝑎 𝑓Δ (𝑡, 𝜏)Δ𝜏 + 𝑓(𝜎(𝑡), 𝑡); (ii) ℎ(𝑡) := ∫𝑡 𝑓(𝑡, 𝜏)Δ𝜏 implies ℎΔ (𝑡) := ∫𝑡 𝑓Δ (𝑡, 𝜏)Δ𝜏 − 𝑓(𝜎(𝑡), 𝑡).

4


Definition 11 (see [33]). For each 𝑡 ∈ T, let 𝑁 be a neighborhood of 𝑡; then we define the generalized derivative (or Dini derivative), 𝐷+ 𝑢Δ (𝑡), to mean that, given 𝜖 > 0, there exists a right neighborhood 𝑁(𝜖) ⊂ 𝑁 of 𝑡 such that 𝑢 (𝜎 (𝑡)) − 𝑢 (𝑠) ≤ 𝐷+ 𝑢Δ (𝑡) + 𝜖, 𝜎 (𝑡) − 𝑠

(13)

for each 𝑠 ∈ 𝑁(𝜖), 𝑠 > 𝑡. In case 𝑡 is right-scattered and 𝑢(𝑡) is continuous at 𝑡, this reduces to 𝑢 (𝜎 (𝑡)) − 𝑢 (𝑡) . 𝐷 𝑢 (𝑡) = 𝜎 (𝑡) − 𝑡 + Δ

𝑏

∫ 𝑓 (𝑡) Δ𝑡 = lim ∫ 𝑓 (𝑡) Δ𝑡,

(15)

𝑏→∞ 𝑎

𝑎

provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges. Definition 13 (see [33]). If 𝑎 ∈ T, inf T = −∞, and 𝑓 is rdcontinuous on (−∞, 𝑎), then we define the improper integral by ∫

𝑎

−∞

𝑎

𝑓 (𝑡) Δ𝑡 = lim ∫ 𝑓 (𝑡) Δ𝑡,

(16)

𝑏 → −∞ 𝑏

provided this limit exists, and we say that the improper integral converges in this case. If this limit does not exist, then we say that the improper integral diverges. Lemma 14 (see [33]). Let 𝑓, 𝑔, ℎ ∈ 𝐶𝑟𝑑 ([𝑎, 𝑏]T , R) and 1/𝑝 + 1/𝑞 = 1 with 𝑝 > 1; then 󵄨 󵄨 ∫ |ℎ (𝑥)| 󵄨󵄨󵄨𝑓 (𝑥) 𝑔 (𝑥)󵄨󵄨󵄨 Δ𝑥 𝑏

(19)

𝑡 ∈ [𝑡0 , ∞)T , 𝑡 ≥ 𝑡0 , 2

2

∗ where |𝑧(𝑡) − 𝑧∗ |1 = ∑𝑛𝑖=1 |𝑥𝑖 (𝑡) − 𝑥𝑖∗ | + ∑𝑚 𝑗=1 |𝑦𝑗 (𝑡) − 𝑦𝑗 | ,

‖𝜙

∑𝑚 𝑗=1

−

𝑧∗ ‖

∑𝑛𝑖=1 sup𝑠∈(−∞,0]T {|𝜑𝑖 (𝑠) −

=

2 𝑦𝑗∗ | }}.

sup𝑠∈(−∞,0]T {|𝜓𝑗 (𝑠) − Then the point 𝑧∗ is said to be exponentially stable.

2 𝑥𝑖∗ |

+

equilibrium

3. Existence and Uniqueness of the Equilibrium Point In this section, we discuss the existence and uniqueness of the equilibrium point of (3). Without loss of generality, we assume that the impulsive jump vectors 𝑃 and 𝑄 satisfy 𝑇

𝑃 (𝑥∗ ) = (𝑃1 (𝑥1∗ ) , 𝑃2 (𝑥2∗ ) , . . . , 𝑃𝑛 (𝑥𝑛∗ )) = 0, 𝑇

𝑄 (𝑦∗ ) = (𝑄1 (𝑦1∗ ) , 𝑄2 (𝑦2∗ ) , . . . , 𝑄𝑛 (𝑦𝑛∗ )) = 0.

(20)

∗ 𝑇 That is, if (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ , 𝑦1∗ , 𝑦2∗ , . . . , 𝑦𝑚 ) is an equilibrium point of the following nonimpulsive system 𝑚

∞

𝑥𝑖Δ (𝑡) = − 𝑎𝑖 𝑥𝑖 (𝑡) + ∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠

𝑏

𝑎

󵄨󵄨 󵄩 ∗󵄨 ∗󵄩 󵄨󵄨𝑧 (𝑡) − 𝑧 󵄨󵄨󵄨1 ≤ 𝑀 󵄩󵄩󵄩𝜙 − 𝑧 󵄩󵄩󵄩 𝑒⊖𝜆 (𝑡, 𝑡0 ) ,

(14)

Definition 12 (see [33]). If 𝑎 ∈ T, sup T = ∞, and 𝑓 is rdcontinuous on [𝑎, ∞), then we define the improper integral by ∞

∗ 𝑇 ) Definition 17. Let 𝑧∗ = (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ , 𝑦1∗ , 𝑦2∗ , . . . , 𝑦𝑚 be an equilibrium point of (3). If there exists a positive constant 𝜆 with −𝜆 ∈ R+ such that, for 𝑡0 ∈ (−∞, 0]T , there exists 𝑀 > 1 such that for an arbitrary solution 𝑧(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), . . . , 𝑥𝑛 (𝑡), 𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑚 (𝑡))𝑇 of (3) with initial value 𝜙(𝑠) = (𝜑1 (𝑠), 𝜑2 (𝑠), . . . , 𝜑𝑛 (𝑠), 𝜓1 (𝑠), 𝜓2 (𝑠), . . . , 𝜓𝑚 (𝑠))𝑇 satisfies

𝑗=1

1/𝑝

󵄨𝑝 󵄨 ≤ (∫ |ℎ (𝑥)| 󵄨󵄨󵄨𝑓 (𝑥)󵄨󵄨󵄨 Δ𝑥) 𝑎

𝑏

1/𝑞

󵄨𝑞 󵄨 (∫ |ℎ (𝑥)| 󵄨󵄨󵄨𝑔 (𝑥)󵄨󵄨󵄨 Δ𝑥) 𝑎

𝑚

. (17)

∗ 𝑇 ) ∈ Definition 15. A point 𝑧∗ = (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ ,𝑦1∗ ,𝑦2∗ ,. . .,𝑦𝑚 𝑛+𝑚 is said to be an equilibrium point of (3) if 𝑧(𝑡) = 𝑧∗ is a R solution of (3).

Lemma 16 (see [19]). Let 𝑓𝑗 be defined on R, 𝑗 = 1, 2, . . . , 𝑚. Then for any 𝑎𝑖𝑗 ∈ R, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚, we have the following estimations: 󵄨󵄨 𝑚 󵄨󵄨󵄨 𝑚 𝑚 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 󵄨 󵄨󵄨⋀𝑎𝑖𝑗 𝑓𝑗 (𝑢𝑗 ) − ⋀𝑎𝑖𝑗 𝑓𝑗 (V𝑗 )󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨𝑎𝑖𝑗 󵄨󵄨󵄨 󵄨󵄨󵄨𝑓𝑗 (𝑢𝑗 ) − 𝑓𝑗 (V𝑗 )󵄨󵄨󵄨 , 󵄨 󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨󵄨𝑗=1 󵄨󵄨 𝑗=1 𝑗=1 󵄨 󵄨󵄨 𝑚 󵄨 󵄨󵄨 𝑚 𝑚 󵄨󵄨 󵄨󵄨⋁𝑎 𝑓 (𝑢 ) − ⋁𝑎 𝑓 (V )󵄨󵄨󵄨 ≤ ∑ 󵄨󵄨󵄨𝑎 󵄨󵄨󵄨 󵄨󵄨󵄨𝑓 (𝑢 ) − 𝑓 (V )󵄨󵄨󵄨 , 󵄨󵄨 󵄨 󵄨󵄨 𝑖𝑗 𝑗 𝑗 󵄨󵄨 𝑗 𝑗 󵄨󵄨 󵄨󵄨𝑗=1 𝑖𝑗 𝑗 𝑗 󵄨󵄨 𝑗=1 󵄨 𝑖𝑗 󵄨 󵄨 𝑗 𝑗 𝑗=1 󵄨 󵄨 (18) where 𝑢𝑗 , V𝑗 ∈ R, 𝑗 = 1, 2, . . . , 𝑚.

0

∞

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 0

𝑗=1 𝑚

∞

+ ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 0

𝑗=1 𝑚

𝑚

+ ⋀𝑇𝑗𝑖 𝜇𝑗 + ⋁𝐻𝑗𝑖 𝜇𝑗 + 𝐼𝑖 , 𝑗=1

𝑗=1

+

𝑡 ∈ T , 𝑖 = 1, 2, . . . , 𝑛, 𝑛

∞

𝑦𝑗Δ (𝑡) = − 𝑏𝑗 𝑦𝑗 (𝑡) + ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1

𝑛

0

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1 𝑛

0

∞


0


5

𝑛

where

+ ⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐽𝑗 , 𝑖=1

𝑖=1

+

𝑡∈T ,

𝑗 = 1, 2, . . . , 𝑚,

𝑚

(21) then it is also the equilibrium point of impulsive system (3).

(22)

𝑛

∗ 𝑇 Proof. If 𝑧∗ = (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ , 𝑦1∗ , 𝑦2∗ , . . . , 𝑦𝑚 ) is an equilibrium point of (3), then we have

𝑛

𝑖=1 𝑛

∞

𝑖=1

0

𝑛

0

𝑚

𝑗=1

𝑗=1

+ ⋀𝑇𝑗𝑖 𝜇𝑗 + ⋁𝐻𝑗𝑖 𝜇𝑗 + 𝐼𝑖 , ∞

0

󵄨󵄨 󵄨 󵄨󵄨Φ𝑖 (ℎ𝑖 ) − Φ𝑖 (ℎ𝑖 )󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝑚 󵄨󵄨 ∞ 󵄨 = 󵄨󵄨󵄨𝑎𝑖−1 [∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (V𝑗 ) − 𝑓𝑗 (V𝑗 )) Δ𝑠] 󵄨󵄨 0 󵄨󵄨 [𝑗=1 ]

0

∞

+ ⋁𝑞𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖∗ ) Δ𝑠 𝑖=1

𝑚

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖∗ ) Δ𝑠 𝑛

0

where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. Obviously, we need to show that Φ is a contraction mapping on R𝑛+𝑚 . In fact, for any 𝜗 = (ℎ1 , ℎ2 , . . .,ℎ𝑛 , V1 , V2 , . . . , V𝑚 ) and 𝜗 = (ℎ1 , ℎ2 , . . .,ℎ𝑛 , V1 , V2 , . . . , V𝑚 ) ∈ R𝑛+𝑚 , we have

𝑏𝑗 𝑦𝑗∗ = ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖∗ ) Δ𝑠

𝑖=1

∞

(25)

𝑚

𝑛

0

+ 𝑏𝑗−1 [⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐵𝑗 ] , 𝑗=1 [ 𝑖=1 ]

∞

+ ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗∗ ) Δ𝑠

𝑖=1

∞

+ 𝑏𝑗−1 [∑𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 ) Δ𝑠]

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗∗ ) Δ𝑠

𝑛

0

+ 𝑏𝑗−1 [⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 ) Δ𝑠]

0

𝑗=1

∞

𝑖=1

𝑎𝑖 𝑥𝑖∗ = ∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗∗ ) Δ𝑠

𝑚

𝑚

Φ𝑛+𝑗 (𝑦𝑗 ) = 𝑏𝑗−1 [∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 ) Δ𝑠]

∞

𝑗=1

∞

+ 𝑎𝑖−1 [⋀𝑇𝑗𝑖 𝜇𝑗 + ⋁𝐻𝑗𝑖 𝜇𝑗 + 𝐼𝑖 ] , 𝑗=1 [𝑗=1 ]

then (3) has one unique equilibrium point.

𝑚

𝑚

𝑚

{ 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 }} 𝑔 max ∑𝑏𝑗−1 𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)}} ; 1≤𝑖≤𝑛 { }} {𝑗=1

𝑗=1

∞

+ 𝑎𝑖−1 [⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 ) Δ𝑠] 0 ] [𝑗=1

𝑚

𝑚

𝑚

+ 𝑎𝑖−1 [⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 ) Δ𝑠] 0 ] [𝑗=1

Theorem 18. Let (𝐻1 ) and (𝐻2 ) hold. Suppose further that (𝐻4 ) 𝜃 < 1, where 𝑛 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 𝜃 = max { max {∑𝑎𝑖−1 𝐿 𝑗 𝑘𝑗𝑖 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)} , 1≤𝑗≤𝑚 𝑖=1 {

∞

Φ𝑖 (𝑥𝑖 ) = 𝑎𝑖−1 [ ∑𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 ) Δ𝑠] 0 ] [𝑗=1

0

𝑛

𝑛

𝑖=1

𝑖=1

+ ⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐽𝑗 ,

𝑚

(23) where 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. To finish the proof, it suffices to prove that (23) has a unique solution. Define a mapping Φ : R𝑛+𝑚 → R𝑛+𝑚 as follows: Φ (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , 𝑦1 , 𝑦2 , . . . , 𝑦𝑚 ) = (Φ1 (𝑥1 ) , Φ2 (𝑥2 ) , . . . , Φ𝑛 (𝑥𝑛 ) , 𝑇

Φ𝑛+1 (𝑦1 ), . . . , Φ𝑛+𝑚 (𝑦𝑚 )) ,

(24)

∞

+ 𝑎𝑖−1 [⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (V𝑗 ) − 𝑓𝑗 (V𝑗 )) Δ𝑠] 0 ] [𝑗=1 󵄨󵄨 𝑚 󵄨󵄨 ∞ 󵄨 −1 [ +𝑎𝑖 ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (V𝑗 ) − 𝑓𝑗 (V𝑗 )) Δ𝑠]󵄨󵄨󵄨 󵄨󵄨 0 ]󵄨󵄨 [𝑗=1 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 ≤ 𝑎𝑖−1 ∑ 𝐿 𝑗 𝑘𝑗𝑖 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨V𝑗 − V𝑗 󵄨󵄨󵄨󵄨 , 𝑗=1

𝑖 = 1, 2, . . . , 𝑛,

6

Journal of Applied Mathematics 󵄨󵄨 󵄨 󵄨󵄨Φ𝑛+𝑗 (V𝑗 ) − Φ𝑛+𝑗 (V𝑗 )󵄨󵄨󵄨 󵄨 󵄨 󵄨󵄨 𝑛 ∞ 󵄨󵄨 = 󵄨󵄨󵄨𝑏𝑗−1 [∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (ℎ𝑖 ) − 𝑔𝑖 (ℎ𝑖 )) Δ𝑠] 󵄨󵄨 0 𝑖=1 󵄨 𝑛

Therefore, (3) has exactly one equilibrium point. The proof of Theorem 18 is completed.

4. Exponential Stability of Equilibrium Point

∞

+ 𝑏𝑗−1 [⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (ℎ𝑖 ) − 𝑔𝑖 (ℎ𝑖 )) Δ𝑠] 𝑖=1

0

󵄨󵄨 󵄨󵄨 −1 +𝑏𝑗 [⋁𝑞𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (ℎ𝑖 ) − 𝑔𝑖 (ℎ𝑖 )) Δ𝑠]󵄨󵄨󵄨 󵄨󵄨 0 𝑖=1 󵄨 𝑛

≤

𝑛 𝑔 𝑏𝑗−1 ∑𝐿 𝑖 ℎ𝑖𝑗 𝑖=1

∞

In this section, we consider the impulsive fuzzy BAM neural networks of the following type: 𝑥𝑖Δ (𝑡) = −𝑎𝑖 𝑥𝑖 (𝑡) 𝑚

𝑚

∞

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 0

𝑗=1 𝑚

(26)

∞


𝑗=1

Therefore, we have 󵄩󵄩 󵄩 󵄩󵄩Φ (𝜗) − Φ (𝜗)󵄩󵄩󵄩 󵄩 󵄩 󵄨 󵄨 = ∑ 󵄨󵄨󵄨󵄨Φ𝑖 (ℎ𝑖 ) − Φ𝑖 (ℎ𝑖 )󵄨󵄨󵄨󵄨 󵄨 󵄨 + ∑ 󵄨󵄨󵄨󵄨Φ𝑛+𝑗 (V𝑗 ) − Φ𝑛+𝑗 (V𝑗 )󵄨󵄨󵄨󵄨

𝑛

𝑗=1

𝑖=1

𝑗=1

𝑛

𝑘 = 1, 2, . . . , 𝑖 = 1, 2, . . . , 𝑛, ∞

𝑦𝑗Δ (𝑡) = −𝑏𝑗 𝑦𝑗 (𝑡) + ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠

𝑗=1

𝑚

𝑗=1

Δ𝑥𝑖 (𝑡𝑘 ) = −𝛾𝑖𝑘 (𝑥𝑖 (𝑡𝑘 ) − 𝑥𝑖∗ ) ,

𝑚

𝑓 ∑𝑎𝑖−1 ∑ 𝐿 𝑗 𝑘𝑗𝑖 𝑖=1 𝑗=1

𝑚

𝑡 ∈ T + , 𝑖 = 1, 2, . . . , 𝑛,

𝑖=1

𝑚

𝑚


𝑛

≤

0

𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨ℎ𝑖 − ℎ𝑖 󵄨󵄨󵄨󵄨 , 𝑗 = 1, 2, . . . , 𝑚.

𝑛

∞

+ ∑𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠

𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨V𝑗 − V𝑗 󵄨󵄨󵄨󵄨

𝑛

0

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑𝑏𝑗−1 ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨) 󵄨󵄨󵄨󵄨ℎ𝑖 − ℎ𝑖 󵄨󵄨󵄨󵄨

𝑛

0

∞


𝑛

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 ≤ max {∑𝑎𝑖−1 𝐿 𝑗 𝑘𝑗𝑖 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)} 1≤𝑗≤𝑚

0

𝑛

𝑛

𝑖=1

𝑖=1

+ ⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐽𝑗 ,

𝑖=1

𝑚

󵄨 󵄨 × ∑ 󵄨󵄨󵄨󵄨V𝑗 − V𝑗 󵄨󵄨󵄨󵄨

𝑡 ∈ T + , 𝑗 = 1, 2, . . . , 𝑚,

𝑗=1

Δ𝑦𝑗 (𝑡𝑘 ) = −𝛾𝑗𝑘 (𝑦𝑗 (𝑡𝑘 ) − 𝑦𝑗∗ ) ,

{𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } 𝑔 + max { ∑𝑎𝑖−1 𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)} 1≤𝑖≤𝑛 } {𝑗=1

𝑘 = 1, 2, . . . , 𝑗 = 1, 2, . . . , 𝑚, (28)

𝑛

󵄨 󵄨 × ∑ 󵄨󵄨󵄨󵄨ℎ𝑖 − ℎ𝑖 󵄨󵄨󵄨󵄨 𝑛 󵄨 󵄨 𝑚 󵄨 󵄨 ≤ 𝜃 (∑ 󵄨󵄨󵄨󵄨ℎ𝑖 − ℎ𝑖 󵄨󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨󵄨V𝑗 − V𝑗 󵄨󵄨󵄨󵄨)

where 𝑎𝑖 , 𝑏𝑗 , 𝑐𝑗𝑖 , 𝛽𝑗𝑖 , 𝛼𝑗𝑖 , 𝑑𝑖𝑗 , 𝑝𝑖𝑗 , 𝑞𝑖𝑗 , 𝑘𝑗𝑖 , ℎ𝑖𝑗 , 𝑓𝑗 , 𝑔𝑖 are defined as those in (3). The initial conditions associated with (28) are given by (4). In the following, we study the exponential stability of the unique equilibrium point for (28) on time scales by using Lyapunov method.

󵄩 󵄩 = 𝜃 󵄩󵄩󵄩󵄩𝜗 − 𝜗󵄩󵄩󵄩󵄩 .

Theorem 19. Let (𝐻1 )–(𝐻4 ) hold. Suppose further that (𝐻4 ) there exists a constant 𝑝 > 0 such that

𝑖=1

𝑖=1

𝑗=1

(27) 𝑛+𝑚

𝑛+𝑚

By (𝐻3 ), we obtain that Φ : R → R is a contraction mapping. By the fixed point theorem of Banach space, there exists a unique fixed point of Φ, which is a solution of (21).

𝑝 + (1 + 𝑝𝜇) 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 × (4𝑎𝑖2 𝜇 − 2𝑎𝑖 + ∑ 𝐿 𝑗 𝑘𝑗𝑖 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)) 𝑗=1

Journal of Applied Mathematics 𝑚

7 𝑛

𝑔

∞

+ ⋁𝑞𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) − 𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠,

+ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 𝑗=1

0

𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨 + 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) ∞

× ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) ℎ𝑖𝑗 (𝑠) Δ𝑠 < 0, 0

𝑡 ∈ T + , 𝑗 = 1, 2, . . . , 𝑚, ΔV𝑗 (𝑡𝑘 ) = −𝛾𝑗𝑘 (V𝑗 (𝑡𝑘 )) ,

𝑖 = 1, 2, . . . , 𝑛,

(30)

𝑝 + (1 + 𝑝𝜇)

For 𝑝 > 0, construct Lyapunov functional 𝑉(𝑡) = 𝑉1 (𝑡) + 𝑉2 (𝑡) + 𝑉3 (𝑡) + 𝑉4 (𝑡), where

𝑛

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 × (4𝑏𝑗2 𝜇 − 2𝑏𝑗 + ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨))

𝑛

𝑖=1

𝑛

𝑉1 (𝑡) = ∑𝑢𝑖2 (𝑡) 𝑒𝑝 (𝑡, 0) , 𝑖=1 𝑚

𝑓

+ ∑𝐿 𝑗 (1 + 𝑝𝜇) 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 + 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) ∞

× ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) 𝑘𝑗𝑖 (𝑠) Δ𝑠 < 0, 0

𝑗 = 1, 2, . . . , 𝑚,

𝑉2 (𝑡) = ∑ V𝑗2 (𝑡) 𝑒𝑝 (𝑡, 0) , 𝑗=1

𝑛 𝑚

𝑖=1 𝑗=1

where 𝜇 = sup𝑡∈T 𝜇(𝑡) < ∞. (𝐻5 ) 0 < 𝛾𝑖𝑘 < 2 and 0 < 𝛾𝑗𝑘 < 2, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚, 𝑘 = 1, 2, . . .. Then the equilibrium point 𝑧∗ = (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ , 𝑦1∗ , ∗ ∗ 𝑇 ) of (28) is exponentially stable. 𝑦2 , . . . , 𝑦𝑚

𝑓

𝑚

∞

0

∞

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠 𝑗=1 𝑚

0

0

𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠,

𝑡 ∈ T + , 𝑖 = 1, 2, . . . , 𝑛, Δ𝑢𝑖 (𝑡𝑘 ) = −𝛾𝑖𝑘 (𝑢𝑖 (𝑡𝑘 )) ,

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

𝑚 𝑛

∞

+ ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) − 𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠 𝑖=1 𝑛

0

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) − 𝑖=1

0

0

𝑡−𝑠

𝑔

𝑉4 (𝑡) = ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 𝑗=1 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × ( 󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨 𝑔

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 ))

∞

𝑡

0

𝑡−𝑠

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑗𝑖 (𝑠) 𝑢𝑖2 (𝑟) Δ𝑟 Δ𝑠. (31)

𝑛

Δ

𝑉1Δ (𝑡) = ∑ [𝑒𝑝 (𝜎 (𝑡) , 0) (𝑢𝑖2 (𝑡)) + 𝑢𝑖2 (𝑡) 𝑒𝑝Δ (𝑡, 0)] 𝑖=1 𝑛

2

= ∑ [𝑒𝑝 (𝜎 (𝑡) , 0) (2𝑢𝑖 (𝑡) 𝑢𝑖Δ (𝑡) + 𝜇 (𝑡) (𝑢𝑖Δ (𝑡)) ) + 𝑝𝑢𝑖2 (𝑡) 𝑒𝑝 (𝑡, 0) ] 𝑛 { = ∑ {𝑒𝑝 (𝜎 (𝑡) , 0) 𝑖=1 {

× [2𝑢𝑖 (𝑡) [

V𝑗Δ (𝑡) = − 𝑏𝑗 V𝑗 (𝑡) 𝑛

𝑡

𝑖=1

∞

+ ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑗=1

∞

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑟) Δ𝑟 Δ𝑠,

Calculating the Δ-derivative 𝑉Δ (𝑡) of 𝑉(𝑡) along the solution of (30), we have

+ ∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × ( 󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

+ 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 ))

Proof. By Theorem 18, (28) has one unique equilibrium point ∗ 𝑇 𝑧∗ = (𝑥1∗ , 𝑥2∗ , . . . , 𝑥𝑛∗ ,𝑦1∗ , 𝑦2∗ ,. . . , 𝑦𝑚 ) . Let 𝑧(𝑡) = (𝑥1 (𝑡), 𝑥2 (𝑡), 𝑇 . . . , 𝑥𝑛 (𝑡), 𝑦1 (𝑡), 𝑦2 (𝑡), . . . , 𝑦𝑚 (𝑡)) be an arbitrary solution of (28). Denote 𝑢𝑖 (𝑡) = 𝑥𝑖 (𝑡) − 𝑥𝑖∗ , V𝑗 (𝑡) = 𝑦𝑗 (𝑡) − 𝑦𝑗∗ , 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. Then from (28), we have the following: 𝑢𝑖Δ (𝑡) = − 𝑎𝑖 𝑢𝑖 (𝑡)

𝑓

𝑉3 (𝑡) = ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

(29)

𝑚

𝑘 = 1, 2, . . . , 𝑗 = 1, 2, . . . , 𝑚.

𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠

× (− 𝑎𝑖 𝑢𝑖 (𝑡) 𝑚

∞

+ ∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑗=1

0

8

Journal of Applied Mathematics ×(𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) −𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠 𝑚

∞

󵄨 󵄨 × ∫ 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) 0

∞

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 0

𝑗=1

×(𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 𝑚

𝑚

+ 4 (∑

−𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠

𝑗=1

∞

󵄨󵄨 󵄨󵄨 𝑓 󵄨󵄨𝛼𝑗𝑖 󵄨󵄨 𝐿 𝑗 󵄨 󵄨 2

∞

󵄨 󵄨 × ∫ 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) 0

+ ⋁𝛽𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 0

𝑗=1

−𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠)

𝑚 󵄨 󵄨 𝑓 + 4 ( ∑ 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 𝐿 𝑗

𝑚

+ 𝜇 (𝑡) (−𝑎𝑖 𝑢𝑖 (𝑡) + ∑ 𝑐𝑗𝑖

𝑗=1

𝑗=1

∞

∞

× ∫ 𝑘𝑗𝑖 (𝑠)

× ∫ 𝑘𝑗𝑖 (𝑠)

0

0

2

× (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) − 𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠

} 󵄨 󵄨 × 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) )]} ]}

𝑚

+ ⋀𝛼𝑗𝑖 𝑗=1 ∞

𝑛 { ≤ ∑ {𝑝𝑒𝑝 (𝑡, 0) 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝜎 (𝑡) , 0) 𝑖=1 {

× ∫ 𝑘𝑗𝑖 (𝑠) 0

× (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠))

𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 × [ − 2𝑎𝑖 𝑢𝑖2 (𝑡) + ∑ (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 𝐿 𝑗 𝑗=1 [

−𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠 𝑚

+ ⋁𝛽𝑗𝑖

∞

× ∫ 𝑘𝑗𝑖 (𝑠) (V𝑗2 (𝑡 − 𝑠) + 𝑢𝑖2 (𝑡)) Δ𝑠

𝑗=1 ∞

0

× ∫ 𝑘𝑗𝑖 (𝑠) 0

𝑚

× (𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) 2

−𝑓𝑗 (𝑦𝑗∗ )) Δ𝑠 + 𝑝𝑢𝑖2

)] ]

} (𝑡) 𝑒𝑝 (𝑡, 0) } }

{ ≤ ∑ {𝑝𝑒𝑝 (𝑡, 0) 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝜎 (𝑡) , 0) 𝑖=1 { × [ − 2𝑎𝑖 𝑢𝑖2 (𝑡) [ 𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 + 2 ∑ (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 𝐿 𝑗 𝑗=1 ∞

󵄨 󵄨󵄨 󵄨 × ∫ 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 Δ𝑠 0

+

(𝑡)

𝑚 󵄨 󵄨 𝑓 + 4 ( ∑ 󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 𝐿 𝑗 𝑗=1

𝑓 2

+ 𝜇 (𝑡) ( 4𝑎𝑖2 𝑢𝑖2 (𝑡) + 4𝑚∑ 𝑐𝑗𝑖2 (𝐿 𝑗 ) 𝑗=1

2 ∞ 󵄨 󵄨 × (∫ 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) 0

𝑚

𝑓 2

+ 4𝑚∑ 𝛼𝑗𝑖2 (𝐿 𝑗 ) 𝑗=1

𝑛

𝜇 (𝑡) (4𝑎𝑖2 𝑢𝑖2

2

× (∫

∞

0 𝑚

2 󵄨 󵄨 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) 𝑓 2

+ 4𝑚∑ 𝛽𝑗𝑖2 (𝐿 𝑗 ) 𝑗=1

2 ∞ } 󵄨 󵄨 × (∫ 𝑘𝑗𝑖 (𝑠) 󵄨󵄨󵄨󵄨V𝑗 (𝑡 − 𝑠)󵄨󵄨󵄨󵄨 Δ𝑠) )]} 0 ]} 𝑛 { ≤ ∑ {𝑝𝑒𝑝 (𝑡, 0) 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝜎 (𝑡) , 0) 𝑖=1 {

× [ − 2𝑎𝑖 𝑢𝑖2 (𝑡) [


9

𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 + ∑ (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 𝐿 𝑗

𝑚

𝑓

+ ∑𝐿 𝑗 𝑘𝑗𝑖

𝑗=1 ∞

𝑗=1

× ∫ 𝑘𝑗𝑖 (𝑠) (V𝑗2 (𝑡 − 𝑠) + 𝑢𝑖2 (𝑡)) Δ𝑠+𝜇 (𝑡)

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)]} ]}

0

× (4𝑎𝑖2 𝑢𝑖2 (𝑡)

𝑛 𝑚

𝑚

𝑓 2

∞

+ 4𝑚∑ 𝑐𝑗𝑖2 (𝐿 𝑗 ) ∫ 𝑘𝑗𝑖 (𝑠) Δ𝑠 ×∫ + × +

𝑘𝑗𝑖 (𝑠) V𝑗2

󵄨 󵄨 󵄨 𝑓 󵄨 󵄨 󵄨 × [𝐿 𝑗 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)

(𝑡 − 𝑠) Δ𝑠

𝑓 2

+ 4𝑚𝜇 (𝑡) (𝐿 𝑗 ) 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )]

𝑚

∞ 𝑓 2 4𝑚∑ 𝛼𝑗𝑖2 (𝐿 𝑗 ) ∫ 𝑘𝑗𝑖 (𝑠) Δ𝑠 0 𝑗=1 ∞ ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠 0 𝑚 𝑓 2 4𝑚∑ 𝛽𝑗𝑖2 (𝐿 𝑗 ) 𝑗=1 ∞

× ∫ 𝑘𝑗𝑖 (𝑠) Δ𝑠 0

∞ } × ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠)]} 0 ]}

𝑛

𝑖=1 𝑗=1

0

𝑗=1 ∞

0

× 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝑡, 0) ∑ ∑ (1 + 𝑝𝜇 (𝑡))

∞

× ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠 0

𝑛

≤ ∑𝑒𝑝 (𝑡, 0) 𝑖=1

{ × {𝑝 + (1 + 𝑝𝜇) { × [4𝑎𝑖2 𝜇 − 2𝑎𝑖 [

≤ ∑ {𝑝𝑒𝑝 (𝑡, 0) 𝑢𝑖2 (𝑡) + (1 + 𝑝𝜇 (𝑡)) 𝑒𝑝 (𝑡, 0)

𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } 𝑓 + ∑ 𝐿 𝑗 𝑘𝑗𝑖 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)]} 𝑗=1 ]}

𝑖=1

× [− 2𝑎𝑖 𝑢𝑖2 (𝑡)

𝑛 𝑚

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 + ∑ (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 𝐿 𝑗 𝑘𝑗𝑖 𝑢𝑖2 (𝑡)

× 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝑡, 0) ∑ ∑ (1 + 𝑝𝜇)

𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 + ∑ (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) 𝐿 𝑗

󵄨 󵄨 󵄨 𝑓 󵄨 󵄨 󵄨 𝑓 2 × { [𝐿 𝑗 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) + 4𝑚𝜇(𝐿 𝑗 )

𝑚

𝑗=1

𝑖=1 𝑗=1

𝑗=1 ∞

× 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )]

× ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠 0

∞

× ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠}.

+ 4𝑎𝑖2 𝜇 (𝑡) 𝑢𝑖2 (𝑡) + 4𝑚𝜇 (𝑡) 𝑚

0

(32)

𝑓 2

× ∑ (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 ) (𝐿 𝑗 ) 𝑘𝑗𝑖 𝑗=1

∞

×∫

0

Similarly, we have that 𝑘𝑗𝑖 (𝑠) V𝑗2

(𝑡 − 𝑠) Δ𝑠]}

𝑛

= ∑𝑒𝑝 (𝑡, 0) 𝑖=1

{ × {𝑝 + (1 + 𝑝𝜇 (𝑡)) { × [4𝑎𝑖2 𝜇 (𝑡) − 2𝑎𝑖 [

𝑚

Δ

𝑉2Δ (𝑡) = ∑ [𝑒𝑝 (𝜎 (𝑡) , 0) (V𝑗2 (𝑡)) + V𝑗2 (𝑡) 𝑒𝑝Δ (𝑡, 0)] 𝑗=1 𝑚

2

= ∑ [𝑒𝑝 (𝜎 (𝑡) , 0) (2V𝑗 (𝑡) V𝑗Δ (𝑡) + 𝜇 (𝑡) (V𝑗Δ (𝑡)) ) 𝑗=1

+ 𝑝V𝑖2 (𝑡) 𝑒𝑝 (𝑡, 0)] 𝑚

= ∑𝑒𝑝 (𝜎 (𝑡) , 0) 𝑗=1

10

Journal of Applied Mathematics × V𝑗2 (𝑡) + 𝑒𝑝 (𝑡, 0)

× [ 2V𝑖 (𝑡) [

𝑚 𝑛

× ∑ ∑ (1 + 𝑝𝜇 (𝑡)) 𝑗=1 𝑖=1

× (− 𝑏𝑗 V𝑖 (𝑡) 𝑚

󵄨 󵄨 󵄨 𝑔 󵄨 󵄨 󵄨 × [𝐿 𝑖 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨) 𝑔 2

∞

+ 4𝑛𝜇 (𝑡) (𝐿 𝑖 ) (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )]

+ ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 0

𝑗=1

∞

× (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) −


𝑛

× ∫ ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑡 − 𝑠) Δ𝑠 0

𝑚 { ≤ ∑𝑒𝑝 (𝑡, 0) { 𝑝 + (1 + 𝑝𝜇) 𝑗=1 {

+ ⋀𝑝𝑖𝑗 𝑖=1

∞

× ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) − 𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠

× [ 4𝑏𝑗2 𝜇 − 2𝑏𝑗 [

0

𝑛

∞

+ ⋁𝛽𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) 0

𝑖=1

𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } 𝑔 + ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)]} 𝑖=1 ]}

−𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠)

× V𝑗2 (𝑡) + 𝑒𝑝 (𝑡, 0)

+ 𝜇 (𝑡)

𝑚 𝑛

× ∑ ∑ (1 + 𝑝𝜇)

𝑛

𝑗=1 𝑖=1

× (−𝑏𝑗 V𝑗 (𝑡) + ∑ 𝑑𝑖𝑗

󵄨 󵄨 󵄨 𝑔 󵄨 󵄨 󵄨 × [𝐿 𝑖 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)

𝑖=1

∞

× ∫ ℎ𝑖𝑗 (𝑠) (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) − 𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠

𝑔 2

+ 4𝑛𝜇(𝐿 𝑖 ) ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )]

0

𝑛

∞

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠)

× ∫ ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑡 − 𝑠) Δ𝑠.

0

𝑖=1

0

× (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) −


𝑛

(33) Besides, we can obtain that

+ ⋁𝑞𝑖𝑗

𝑛 𝑚

𝑖=1

∞

× ∫ ℎ𝑖𝑗 (𝑠)

𝑖=1 𝑗=1

0

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

2

× (𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) −𝑔𝑖 (𝑥𝑖∗ )) Δ𝑠 ) ] ] + 𝑝V𝑗2 (𝑡) 𝑒𝑝 (𝑡, 0)

∞

0

{ ≤ ∑ 𝑒𝑝 (𝑡, 0) {𝑝 + (1 + 𝑝𝜇 (𝑡)) 𝑗=1 { ×

𝑓

+ 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) × ∫ 𝑒𝑝 (𝑡 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡) Δ𝑠

𝑚

[4𝑏2 𝜇 (𝑡) 𝑗

𝑓

𝑉3Δ (𝑡) = ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

𝑛 𝑚

𝑖=1 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

− 2𝑏𝑗

𝑓


[

𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } 𝑔 + ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨) ] } 𝑖=1 ]}

𝑓

− ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

∞

× ∫ 𝑒𝑝 (𝑡, 0) 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠 0

Journal of Applied Mathematics 𝑛 𝑚

11 Hence, we have that

𝑓

= ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇) 𝑖=1 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 𝑓

+ 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) ∞

× 𝑒𝑝 (𝑡, 0) ∫ 𝑒𝑝 (𝑡 + 0

𝑠, 𝑡) 𝑘𝑗𝑖 (𝑠) Δ𝑠V𝑗2

𝑉Δ (𝑡) = 𝑉1Δ (𝑡) + 𝑉2Δ (𝑡) + 𝑉3Δ (𝑡) + 𝑉4Δ (𝑡)

(𝑡)

𝑛

≤ ∑𝑒𝑝 (𝑡, 0)

− 𝑒𝑝 (𝑡, 0) (1 + 𝑝𝜇)

𝑖=1

𝑛 𝑚 󵄨 󵄨 󵄨 𝑓 󵄨 󵄨 󵄨 × ∑ ∑ 𝐿 𝑗 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

{ × {𝑝 + (1 + 𝑝𝜇) {

𝑖=1 𝑗=1

𝑓

+ 4𝑚𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) ×∫

∞

0

𝑉4Δ

(𝑡) =

𝑘𝑗𝑖 (𝑠) V𝑗2

𝑚 𝑛

𝑔 ∑ ∑𝐿 𝑖 𝑗=1 𝑖=1

× [4𝑎𝑖2 𝜇 − 2𝑎𝑖 [

(𝑡 − 𝑠) Δ𝑠,

𝑚

(1 + 𝑝𝜇)

𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 } × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨) ]} ]}

𝑔


𝑛 𝑚

∞

× ∫ 𝑒𝑝 (𝑡 + 0

𝑚 𝑛

𝑠, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2

× 𝑢𝑖2 (𝑡) + 𝑒𝑝 (𝑡, 0) ∑ ∑ (1 + 𝑝𝜇)

(𝑡) Δ𝑠

𝑖=1 𝑗=1

󵄨 󵄨 󵄨 𝑓 󵄨 󵄨 󵄨 × { [𝐿 𝑗 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨)

𝑔

− ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 𝑗=1 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

𝑓 2

+ 4𝑚𝜇(𝐿 𝑗 ) 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )]

𝑔


∞

× ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠}

∞

0

× ∫ 𝑒𝑝 (𝑡, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑡 − 𝑠) Δ𝑠

𝑚

0

=

𝑓

+ ∑𝐿 𝑗 𝑘𝑗𝑖

𝑚 𝑛

𝑔 ∑ ∑𝐿 𝑖 𝑗=1 𝑖=1

+ ∑𝑒𝑝 (𝑡, 0) 𝑗=1

(1 + 𝑝𝜇)

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

× {𝑝 + (1 + 𝑝𝜇)

𝑔

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) ∞

× 𝑒𝑝 (𝑡, 0) ∫ 𝑒𝑝 (𝑡 + 0

𝑠, 𝑡) ℎ𝑖𝑗 (𝑠) Δ𝑠𝑢𝑖2

× [4𝑏𝑗2 𝜇 − 2𝑏𝑗

(𝑡)

𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)]} V𝑗2 (𝑡)

− 𝑒𝑝 (𝑡, 0) (1 + 𝑝𝜇) ×

𝑚 𝑛

𝑔 ∑ ∑𝐿 𝑖 𝑗=1 𝑖=1

𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

𝑚 𝑛

+ 𝑒𝑝 (𝑡, 0) ∑ ∑ (1 + 𝑝𝜇) 𝑗=1 𝑖=1

𝑔

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) ×∫

∞

0

ℎ𝑗𝑖 (𝑠) 𝑢𝑖2

󵄨 󵄨 󵄨 𝑔 󵄨 󵄨 󵄨 × [𝐿 𝑖 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨)

(𝑡 − 𝑠) Δ𝑠. (34)

𝑔 2

+ 4𝑛𝜇(𝐿 𝑖 ) ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )]

12

Journal of Applied Mathematics ∞

∞ } × ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) ℎ𝑖𝑗 (𝑠) Δ𝑠} 𝑢𝑖2 (𝑡) 0 }

× ∫ ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑡 − 𝑠) Δ𝑠 0

𝑛 𝑚

𝑓

+ ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

𝑚

𝑖=1 𝑗=1

+ ∑𝑒𝑝 (𝑡, 0) {[𝑝 + (1 + 𝑝𝜇)

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

𝑗=1

× (4𝑏𝑗2 𝜇 − 2𝑏𝑗

𝑓

+ 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) 𝑒𝑝 (𝑡, 0) ∞

× ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) 𝑘𝑗𝑖 (𝑠) Δ𝑠V𝑗2 (𝑡) − 𝑒𝑝 (𝑡, 0) (1 + 𝑝𝜇)

𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑𝐿 𝑖 ℎ𝑖𝑗 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨))]

0

×

𝑛 𝑚

𝑓 ∑∑𝐿𝑗 𝑖=1 𝑗=1

𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

𝑛

𝑓

+ ∑𝐿 𝑗 (1 + 𝑝𝜇) 𝑖=1

𝑓

+ 4𝑚𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 ))

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

∞

× ∫ 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑡 − 𝑠) Δ𝑠

𝑓

+ 4𝑚𝜇𝐿 𝑗

0

𝑚 𝑛 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

×𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 ))

𝑗=1 𝑖=1

∞

+

𝑔 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗

(𝑑𝑖𝑗2

+

𝑝𝑖𝑗2

+

× ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) 𝑘𝑗𝑖 (𝑠) Δ𝑠 } V𝑗2 (𝑡) .

𝑞𝑖𝑗2 ))

0

(35)

∞

× 𝑒𝑝 (𝑡, 0) ∫ 𝑒𝑝 (𝑡 + 𝑠, 𝑡) ℎ𝑖𝑗 (𝑠) Δ𝑠𝑢𝑖2 (𝑡) 0

By (𝐻4 ), we can conclude that 𝑉Δ (𝑡) ≤ 0, for 𝑡 ∈ T + , 𝑡 ≠ 𝑡𝑘 , 𝑘 = 1, 2, . . ., which implies that 𝑉(𝑡) ≤ 𝑉(0), for 𝑡 ∈ T + , 𝑡 ≠ 𝑡𝑘 , 𝑘 = 1, 2, . . .. For 𝑡 = 𝑡𝑘 , 𝑘 = 1, 2, . . ., we have

− 𝑒𝑝 (𝑡, 0) (1 + 𝑝𝜇) 𝑚 𝑛 󵄨 󵄨 󵄨 𝑔 󵄨 󵄨 󵄨 × ∑ ∑ 𝐿 𝑖 (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨 𝑗=1 𝑖=1

𝑉 (𝑡𝑘+ ) = 𝑉1 (𝑡𝑘+ ) + 𝑉2 (𝑡𝑘+ ) + 𝑉3 (𝑡𝑘+ ) + 𝑉4 (𝑡𝑘+ )

𝑔


𝑛

𝑚

𝑖=1

𝑗=1

= ∑𝑢𝑖2 (𝑡𝑘+ ) 𝑒𝑝 (𝑡𝑘+ , 0) + ∑ V𝑗2 (𝑡𝑘+ ) 𝑒𝑝 (𝑡𝑘+ , 0)

∞

× ∫ ℎ𝑗𝑖 (𝑠) 𝑢𝑖2 (𝑡 − 𝑠) Δ𝑠 0

𝑛 𝑚

𝑛

{ ≤ ∑𝑒𝑝 (𝑡, 0) {[𝑝 + (1 + 𝑝𝜇) 𝑖=1 {[

𝑓

+ ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇) 𝑖=1 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

× ( 4𝑎𝑖2 𝜇 − 2𝑎𝑖 𝑚

𝑓


𝑓

+ ∑ 𝐿 𝑗 𝑘𝑗𝑖 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨))] ] +

𝑚

𝑔 ∑𝐿 𝑖 𝑗=1

∞

𝑡𝑘+

0

𝑡𝑘+ −𝑠

×∫ ∫ 𝑚 𝑛

𝑔

+ ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 𝑗=1 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

(1 + 𝑝𝜇)

𝑔


󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × ( 󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨 +


(𝑑𝑖𝑗2

+

𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑟) Δ𝑟 Δ𝑠

𝑝𝑖𝑗2

+

𝑞𝑖𝑗2 ))

∞

𝑡𝑘+

0

𝑡𝑘+ −𝑠

×∫ ∫

𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑟) Δ𝑟 Δ𝑠


13

2

𝑖=1

𝑚

𝑚

𝑗=1

𝑗=1

𝑚

2

≤ ∑ sup V𝑗2 (𝑡) ,

+ ∑((1 − 𝛾𝑗𝑘 ) V𝑗 (𝑡𝑘 )) 𝑒𝑝 (𝑡𝑘 , 0) 𝑗=1

𝑛 𝑚

𝑚

𝑉2 (0) = ∑V𝑗2 (0) 𝑒𝑝 (0, 0) = ∑ V𝑗2 (0)

≤ ∑((1 − 𝛾𝑖𝑘 ) 𝑢𝑖 (𝑡𝑘 )) 𝑒𝑝 (𝑡𝑘 , 0)

𝑗=1 𝑡∈(−∞,0]T 𝑛 𝑚

𝑓

+ ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

𝑓

𝑉3 (0) = ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

𝑖=1 𝑗=1

𝑖=1 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

𝑓


𝑡𝑘

0

𝑡𝑘 −𝑠

×∫ ∫ 𝑚 𝑛

𝑓



∞

0

0

−𝑠

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) V𝑗2 (𝑟) Δ𝑟 Δ𝑠 𝑛 𝑚

𝑔

+ ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇)

𝑓

≤ ∑ ∑ 𝐿 𝑗 (1 + 𝑝𝜇)

𝑗=1 𝑖=1

𝑖=1 𝑗=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

𝑔

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) ∞

𝑡𝑘

×∫ ∫ 0

𝑡𝑘 −𝑠

𝑓


𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑟) Δ𝑟 Δ𝑠

𝑛

𝑚

×

≤ ∑𝑢𝑖2 (𝑡𝑘 ) 𝑒𝑝 (𝑡𝑘 , 0) + ∑V𝑗2 (𝑡𝑘 ) 𝑒𝑝 (𝑡𝑘 , 0) 𝑖=1

+

𝑗=1

𝑛 𝑚

0

0

−𝑠

sup V𝑗2 (𝑡)

𝑡∈(−∞,0]T 𝑛

1≤𝑗≤𝑚

𝑖=1

𝑓


𝑡𝑘

0

𝑡𝑘 −𝑠

×∫ ∫ +

𝑚 𝑛

𝑔 ∑ ∑𝐿 𝑖 𝑗=1 𝑖=1

0

𝑡𝑘 −𝑠

𝑒𝑝 (𝑟 +

𝑠, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2

𝑖=1

𝑖=1

𝑛

≤ ∑ sup 𝑢𝑖2 (𝑡) , 𝑖=1 𝑡∈(−∞,0]T

−𝑠

𝑚 𝑛

𝑔

𝑉4 (0) = ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇)

(𝑟) Δ𝑟 Δ𝑠

𝑗=1 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

So we get 𝑉(𝑡) ≤ 𝑉(0), for all 𝑡 ∈ T + . Now, we estimate the value of 𝑉(0). We have that 𝑛

0

𝑗=1𝑡∈(−∞,0]T

𝑔


(36)

𝑛

0

𝑚

= 𝑉 (𝑡𝑘 ) .

𝑉1 (0) = ∑𝑢𝑖2 (0) 𝑒𝑝 (0, 0) = ∑𝑢𝑖2 (0)

∞

× ∑ sup V𝑗2 (𝑡) ,

𝑔

×∫ ∫

𝑓

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠}

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) 𝑡𝑘

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 + 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 ))


󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (1 + 𝑝𝜇) (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

∞

𝑓

≤ max {∑𝐿 𝑗 (1 + 𝑝𝜇)

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (1 + 𝑝𝜇) (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨

𝑓 ∑∑𝐿𝑗 𝑖=1 𝑗=1

∞

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠

∞

0

0

−𝑠

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑖𝑗 (𝑠) 𝑢𝑖2 (𝑟) Δ𝑟 Δ𝑠 𝑚 𝑛

𝑔

≤ ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 𝑗=1 𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨 𝑔


14

Journal of Applied Mathematics ∞

0

0

−𝑠

So we have

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑖𝑗 (𝑠) Δ𝑟 Δ𝑠 ×

𝑢𝑖2

sup

𝑡∈(−∞,0]T

𝑛

𝑚

𝑖=1

𝑗=1

∑𝑢𝑖2 (𝑡) + ∑ V𝑗2 (𝑡)

(𝑡)

𝑛

{𝑚 𝑛 𝑔 ≤ max {∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 1≤𝑖≤𝑛 {𝑗=1 𝑖=1 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

𝑚

≤ 𝑀𝑒⊖𝑝 (𝑡, 0) (∑ sup 𝑢𝑖2 (𝑡) + ∑ sup V𝑗2 (𝑡)) , 𝑖=1 𝑡∈(−∞,0]

𝑗=1𝑡∈(−∞,0]

∀𝑡 ∈ T + , (40)

𝑔

+ 4𝑛𝜇𝐿 𝑖 ℎ𝑖𝑗 (𝑑𝑖𝑗2 + 𝑝𝑖𝑗2 + 𝑞𝑖𝑗2 )) } × ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠 } 0 −𝑠 } ∞

0

where 𝑚 𝑛 { { 𝑔 𝑀 = max {max {1 + ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 1≤𝑖≤𝑛 𝑗=1 𝑖=1 { { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

𝑛

× ∑ sup 𝑢𝑖2 (𝑡) . 𝑖=1 𝑡∈(−∞,0]T

𝑔


(37) It follows that

∞ 0 } × ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠 } , 0 −𝑠 }

𝑚 𝑛 { 𝑔 𝑉 (0) ≤ max {1 + ∑ ∑ 𝐿 𝑖 (1 + 𝑝𝜇) 1≤𝑖≤𝑛 𝑗=1 𝑖=1 { 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑑𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝𝑖𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 󵄨󵄨󵄨󵄨

+


(𝑑𝑖𝑗2

+

𝑛

1≤𝑗≤𝑚

𝑝𝑖𝑗2

+

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 + 4𝑚𝜇𝐿 𝑗 𝑘𝑗𝑖 × (𝑐𝑗𝑖2 + 𝛼𝑗𝑖2 + 𝛽𝑗𝑖2 )) ∞ 0 } × ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠 } } . 0 −𝑠 } (41)

𝑛

× ∑ sup 𝑢𝑖2 (𝑡) 𝑖=1 𝑡∈(−∞,0]T

It is obvious that 𝑀 > 1. Therefore, the equilibrium point 𝑧∗ of (28) is exponentially stable. This completes the proof of Theorem 19.

𝑓

+ max {1 + ∑ 𝐿 𝑗 (1 + 𝑝𝜇) 1≤𝑗≤𝑚

𝑖=1

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 × (󵄨󵄨󵄨󵄨𝑐𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗𝑖 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗𝑖 󵄨󵄨󵄨󵄨 𝑓


0

0

−𝑠

𝑖=1

𝑞𝑖𝑗2 ))

∞ 0 } × ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) ℎ𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠 } 0 −𝑠 }

𝑛

𝑓

max {1 + ∑𝐿 𝑗 (1 + 𝑝𝜇)

5. An Example In this section, an example is given to verify the feasibility of our results obtained in previous sections.

× ∫ ∫ 𝑒𝑝 (𝑟 + 𝑠, 0) 𝑘𝑗𝑖 (𝑠) Δ𝑟 Δ𝑠 }

Example 20. Let 𝑛 = 𝑚 = 2. Consider the following fuzzy BAM neural networks without impulses on time scale T = R or Z:

𝑚

× ∑ sup V𝑗2 (𝑡) . 𝑗=1𝑡∈(−∞,0]T

(38) Observe that

𝑥𝑖Δ (𝑡) = − 𝑎𝑖 𝑥𝑖 (𝑡) 2

𝑉 (𝑡) ≥ 𝑉1 (𝑡) + 𝑉2 (𝑡) 𝑛

𝑗=1

𝑚

≥ 𝑒𝑝 (𝑡, 0) (∑𝑢𝑖2 (𝑡) + ∑V𝑗2 (𝑡)) . 𝑖=1

∞

+ ∑ 𝑐𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠

𝑗=1

(39)

2

0

∞

+ ⋀𝛼𝑗𝑖 ∫ 𝑘𝑗𝑖 (𝑠) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝑠)) Δ𝑠 𝑗=1

0

Journal of Applied Mathematics 2

15 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 − 2𝑎2 + ∑𝐿 𝑗 𝑘𝑗2 (󵄨󵄨󵄨󵄨𝑐𝑗2 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗2 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗2 󵄨󵄨󵄨󵄨)

∞


𝑗=1 2

2

𝑗=1

𝑗=1


𝑗=1

𝑗=1

𝑦𝑗Δ (𝑡) = − 𝑏𝑗 𝑦𝑗 (𝑡) 2

= −0.37 < 0, 2

𝑔 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 − 2𝑏1 + ∑𝐿 𝑖 ℎ𝑖1 (󵄨󵄨󵄨𝑑𝑖1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝𝑖1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞𝑖1 󵄨󵄨󵄨)

∞

+ ∑𝑑𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠

𝑖=1

0

𝑖=1 2

2 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 󵄨 + ∑𝐿 1 (󵄨󵄨󵄨󵄨𝑑2𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝2𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞2𝑗 󵄨󵄨󵄨󵄨 + ℎ2𝑗 )

𝑡 ∈ T + , 𝑖 = 1, 2,

2

𝑓 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑𝐿 𝑗 (󵄨󵄨󵄨𝑐1𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛼1𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛽1𝑖 󵄨󵄨󵄨 + 𝑘1𝑖 )

∞

+ ⋀𝑝𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠

𝑖=1

0

𝑖=1 2

= −0.17 < 0,

∞

+ ⋁𝑞𝑖𝑗 ∫ ℎ𝑖𝑗 (𝑠) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝑠)) Δ𝑠

2

0

𝑖=1 2

2

𝑖=1

𝑖=1

+ ⋀𝐹𝑖𝑗 ]𝑖 + ⋁𝐺𝑖𝑗 ]𝑖 + 𝐽𝑗 ,

𝑔 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 − 2𝑏2 + ∑𝐿 𝑖 ℎ𝑖2 (󵄨󵄨󵄨𝑑𝑖2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝𝑖2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞𝑖2 󵄨󵄨󵄨) 𝑖=1

𝑡 ∈ T + , 𝑗 = 1, 2,

2

𝑓 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑𝐿 𝑗 (󵄨󵄨󵄨𝑐2𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛼2𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛽2𝑖 󵄨󵄨󵄨 + 𝑘2𝑖 )

(42)

= −0.77 < 0.

where (𝑎1 , 𝑎2 )𝑇 = (0.2, 0.3)𝑇 , (𝑏1 , 𝑏2 )𝑇 = (0.2, 0.5)𝑇 ,

(45)

𝑐 𝑐 𝛼 𝛼 𝛽 𝛽 0 0.01 ( 11 12 ) = ( 11 12 ) = ( 11 12 ) = ( ), 𝑐21 𝑐22 𝛼21 𝛼22 𝛽21 𝛽22 0.02 0 𝑝 𝑝 𝑞 𝑞 0 0.03 𝑑 𝑑 ), ( 11 12 ) = ( 11 12 ) = ( 11 12 ) = ( 𝑑21 𝑑22 𝑝21 𝑝22 𝑞21 𝑞22 0.01 0 𝑓1 (𝑢) = 𝑓2 (𝑢) = 𝑔1 (𝑢) = 𝑔2 (𝑢) = 0.1 |𝑢| , 𝑘𝑗𝑖 (𝑠) = ℎ𝑖𝑗 (𝑠) = 𝑒2 (0, 𝑠) ,

𝑖=1

If 𝜇(𝑡) = 1, we have that 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 4𝑎12 − 2𝑎1 + ∑ 𝐿 𝑗 𝑘𝑗1 (󵄨󵄨󵄨󵄨𝑐𝑗1 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗1 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗1 󵄨󵄨󵄨󵄨) 𝑗=1

2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑𝐿 1 ℎ1𝑗 (󵄨󵄨󵄨󵄨𝑑1𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝1𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞1𝑗 󵄨󵄨󵄨󵄨 𝑗=1

𝑖, 𝑗 = 1, 2.

𝑔

2 2 2 + 4𝑛𝐿 1 (𝑑1𝑗 + 𝑝1𝑗 + 𝑞1𝑗 ) + ℎ1𝑗 )

(43) ≈ −0.0336 < 0, 𝑓 𝐿1

𝑓 𝐿2

𝑔 𝐿1

𝑔 𝐿2

= = = = 0.1. If T = R, We can easily see that then 𝜇(𝑡) = 0; if T = Z then 𝜇(𝑡) = 1. Hence, we have

2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 4𝑎22 − 2𝑎2 + ∑ 𝐿 𝑗 𝑘𝑗2 (󵄨󵄨󵄨󵄨𝑐𝑗2 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗2 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗2 󵄨󵄨󵄨󵄨) 𝑗=1

1 { if T = R, { , ∫ 𝑒2 (0, 𝑠) Δ𝑠 = { 2 1 { 0 , if T = Z. { 3 ln 3 ∞

(44)

So, we can take 𝑘𝑗𝑖 = ℎ𝑖𝑗 = 1, 𝑖, 𝑗 = 1, 2. We can obtain that 𝜃 = 0.06 < 1. Moreover, if 𝜇(𝑡) = 0, we can obtain that 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑓 − 2𝑎1 + ∑𝐿 𝑗 𝑘𝑗1 (󵄨󵄨󵄨󵄨𝑐𝑗1 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛼𝑗1 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝛽𝑗1 󵄨󵄨󵄨󵄨) 𝑗=1

2 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 󵄨 + ∑𝐿 1 (󵄨󵄨󵄨󵄨𝑑1𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝1𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞1𝑗 󵄨󵄨󵄨󵄨 + ℎ1𝑗 )

2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 𝑔 + ∑𝐿 2 ℎ2𝑗 (󵄨󵄨󵄨󵄨𝑑2𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑝2𝑗 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨𝑞2𝑗 󵄨󵄨󵄨󵄨 𝑗=1

𝑔

2 2 2 + 4𝑛𝐿 2 (𝑑2𝑗 + 𝑝2𝑗 + 𝑞2𝑗 ) + ℎ2𝑗 )

≈ −0.0374 < 0, 2

𝑔 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 4𝑏12 − 2𝑏1 + ∑𝐿 𝑖 ℎ𝑖1 (󵄨󵄨󵄨𝑑𝑖1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝𝑖1 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞𝑖1 󵄨󵄨󵄨) 𝑖=1

2

𝑓 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑𝐿 1 𝑘1𝑖 (󵄨󵄨󵄨𝑐1𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛼1𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛽1𝑖 󵄨󵄨󵄨 𝑖=1

𝑓

2 2 + 4𝑚𝐿 1 𝑘1𝑖 (𝑐1𝑖2 + 𝛼1𝑖 + 𝛽1𝑖 ) + 𝑘1𝑖 )

𝑗=1

= −0.17 < 0,

≈ −0.0336 < 0,

16

Journal of Applied Mathematics 2

𝑔 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 4𝑏22 − 2𝑏2 + ∑𝐿 𝑖 ℎ𝑖12 (󵄨󵄨󵄨𝑑𝑖2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑝𝑖2 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑞𝑖2 󵄨󵄨󵄨) 𝑖=1

2

𝑓 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 + ∑𝐿 2 𝑘2𝑖 (󵄨󵄨󵄨𝑐2𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛼2𝑖 󵄨󵄨󵄨 + 󵄨󵄨󵄨𝛽2𝑖 󵄨󵄨󵄨 𝑖=1

𝑓

2 2 + 4𝑚𝐿 2 𝑘2𝑖 (𝑐2𝑖2 + 𝛼2𝑖 + 𝛽2𝑖 ) + 𝑘2𝑖 )

≈ −0.0381 < 0, (46) which imply that all assumptions in Theorems 18 and 19 are satisfied. Thus, it follows from Theorems 18 and 19 that (42) has one unique equilibrium point, which is exponentially stable. Remark 21. The conclusion of Example 1 cannot be obtained by the results obtained in [22, 30–32].

6. Conclusion In this paper, using the time scale calculus theory, the fixed point theory, and Lyapunov functional method, some sufficient conditions are obtained to ensure the existence and the exponential stability of unique equilibrium point of a class of fuzzy BAM neural networks with continuously distributed delays and impulses on time scales. Since it is troublesome to study the existence and stability of equilibrium points for continuous and discrete delay systems, respectively, it is significant to study that on time scales which can unify the continuous and discrete time situations. The sufficient conditions we obtained can easily be checked in practice by simple algebraic methods.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment This work is supported by the National Natural Sciences Foundation of China under Grant no. 11361072.

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