-Almost Periodic Solutions of BAM Neural Networks with Time-Varying ...

Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 727329, 15 pages http://dx.doi.org/10.1155/2015/727329

Research Article 𝐶1-Almost Periodic Solutions of BAM Neural Networks with Time-Varying Delays on Time Scales Yongkun Li, Lili Zhao, and Li Yang Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China Correspondence should be addressed to Yongkun Li; [email protected] Received 5 May 2014; Accepted 5 August 2014 Academic Editor: P. Balasubramaniam Copyright © 2015 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. On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of C1 almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.

1. Introduction It is well known that bidirectional associative memory (BAM) neural networks have been extensively applied within various engineering and scientific fields such as pattern recognition, signal and image processing, artificial intelligence, and combinatorial optimization [1–3]. Since all these applications closely relate to the dynamics, the dynamical behaviors of BAM neural networks have been widely investigated. There have been extensive results on the problem of the existence and stability of equilibrium points, periodic solutions, and antiperiodic solutions of BAM neural networks in the literature. We refer the reader to [4–16] and the references cited therein. Moreover, it is known that the existence and stability of almost periodic solutions play a key role in characterizing the behavior of dynamical systems (see [17–26]) and the 𝐶1 almost periodic function is an important subclass of almost periodic functions. However, to the best of our knowledge, few authors have studied problems of 𝐶1 -almost periodic solutions of BAM neural networks. On the other hand, the theory of calculus on time scales (see [27, 28] and references cited therein) was initiated by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analyses, and it helps avoid proving twice results, once for differential equations and once for difference equations. Therefore, it is significant to study neural networks on time scales (see [5, 29, 30]). In fact, both continuous-time

and discrete-time BAM-type neural networks have equal importance in various applications. But it is troublesome to study the existence and stability of almost periodic and 𝐶1 -almost periodic solutions for continuous and discrete systems, respectively. Motivated by the above, our purpose of this paper is to study the existence and stability of 𝐶1 -almost periodic solutions for the following BAM neural networks on time scales: 𝑚

𝑥𝑖Δ (𝑡) = −𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) + ∑ 𝑝𝑗𝑖 (𝑡) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡))) + 𝐼𝑖 (𝑡) , 𝑗=1

𝑡 ∈ T,

𝑖 = 1, 2, . . . , 𝑛,

𝑛

𝑦𝑗Δ (𝑡) = −𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) + ∑𝑞𝑖𝑗 (𝑡) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))) + 𝐽𝑗 (𝑡) , 𝑖=1

𝑡 ∈ T,

𝑗 = 1, 2, . . . , 𝑚, (1)

where T is an almost periodic time scale which will be defined in the next section; 𝑥𝑖 (𝑡) and 𝑦𝑗 (𝑡) correspond to the activation of the 𝑖th neurons and the 𝑗th neurons at the time 𝑡, respectively; 𝑎𝑖 (𝑡), 𝑏𝑗 (𝑡) are positive functions and they denote the rates with which the cells 𝑖 and 𝑗 reset their potential to the resting state when isolated from the other cells and inputs at time 𝑡; 𝑝𝑗𝑖 (𝑡) and 𝑞𝑖𝑗 (𝑡) are the connection weights at time

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𝑡; 𝛾𝑗𝑖 (𝑡), 𝜌𝑖𝑗 (𝑡) are nonnegative, which correspond to the finite speed of the axonal signal transmission at time 𝑡; 𝐼𝑖 (𝑡), 𝐽𝑗 (𝑡) denote the external inputs at time 𝑡; and 𝑓𝑗 and 𝑔𝑖 are the activation functions of signal transmission. For each interval 𝐽 of R, we denote 𝐽T = 𝐽 ∩ T. Throughout this paper, we assume the following: (𝐻1 ) 𝑓𝑗 , 𝑔𝑖 ∈ 𝐶(R, R) and there exist positive constants 𝛼𝑗 , 𝛽𝑖 such that 󵄨󵄨 󵄨 󵄨󵄨𝑓𝑗 (𝑢) − 𝑓𝑗 (V)󵄨󵄨󵄨 ≤ 𝛼𝑗 |𝑢 − V| , 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨𝑔𝑖 (𝑢) − 𝑔𝑖 (V)󵄨󵄨󵄨 ≤ 𝛽𝑖 |𝑢 − V| ,

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where |𝑢|, |V| ∈ R, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚; (𝐻2 ) 𝑎𝑖 (𝑡) > 0, 𝑏𝑗 (𝑡) > 0, 𝛾𝑗𝑖 (𝑡) ≥ 0, 𝜌𝑖𝑗 (𝑡) ≥ 0, 𝑝𝑗𝑖 (𝑡), 𝑞𝑖𝑗 (𝑡), 𝐼𝑖 (𝑡), 𝐽𝑗 (𝑡) are bounded almost periodic functions on T, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚. System (1) is supplemented with the initial values given by

If T has a right-scattered minimum 𝑚, then T𝑘 = T \ {𝑚}; otherwise, T𝑘 = T. A function 𝑓 : T → R is right-dense continuous provided it is continuous at right-dense point in T and its leftside limits exist at left-dense points in T. If 𝑓 is continuous at each right-dense point and each left-dense point, then 𝑓 is said to be continuous function on T. For 𝑦 : T → R and 𝑡 ∈ T 𝑘 , we define the delta derivative of 𝑦(𝑡), 𝑦Δ (𝑡), to be the number (if it exists) with the property that for a given 𝜀 > 0 there exists a neighborhood 𝑈 of 𝑡 such that 󵄨 󵄨󵄨 󵄨󵄨[𝑦 (𝜎 (𝑡)) − 𝑦 (𝑠)] − 𝑦Δ (𝑡) [𝜎 (𝑡) − 𝑠]󵄨󵄨󵄨 < 𝜀 |𝜎 (𝑡) − 𝑠| (6) 󵄨 󵄨 for all 𝑠 ∈ 𝑈. If 𝑦 is continuous, then 𝑦 is right-dense continuous, and if 𝑦 is delta differentiable at 𝑡, then 𝑦 is continuous at 𝑡. Let 𝑦 be right-dense continuous. If 𝑌Δ (𝑡) = 𝑦(𝑡), then we define the delta integral by 𝑡

∫ 𝑦 (𝑠) Δ𝑠 = 𝑌 (𝑡) − 𝑌 (𝑎) . 𝑎

𝑥𝑖 (𝑠) = 𝜑𝑖 (𝑠) , 𝑦𝑗 (𝑠) = 𝜑𝑛+𝑗 (𝑠) , 𝑖 = 1, 2, . . . , 𝑛,

A function 𝑟 : T → R is called regressive if (3)

𝑗 = 1, 2, . . . , 𝑚,

where 𝜑𝑘 (⋅) denotes a real-valued bounded rd-continuous function defined on [−V, 0]T , and 𝛾𝑖 = max sup 𝛾𝑗𝑖 (𝑡) , 1≤𝑗≤𝑚 𝑡∈T

𝛾 = max 𝛾𝑖 , 1≤𝑖≤𝑛 𝑡∈T

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for all 𝑡 ∈ T 𝑘 . The set of all regressive and rd-continuous functions 𝑟 : T → R will be denoted by R = R(T) = R(T, R). We define the set R+ = R+ (T, R) = {𝑟 ∈ R : 1 + 𝜇(𝑡)𝑟(𝑡) > 0, ∀𝑡 ∈ T}. If 𝑟 is a regressive function, then the generalized exponential function 𝑒𝑟 is defined by 𝑒𝑟 (𝑡, 𝑠) = exp {∫ 𝜉𝜇(𝜏) (𝑟 (𝜏)) Δ𝜏} ,

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𝑠

for 𝑠, 𝑡 ∈ T,

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with the cylinder transformation

𝜌 = max 𝜌𝑗 ,

{ Log (1 + ℎ𝑧) 𝜉ℎ (𝑧) = { ℎ {𝑧

1≤𝑗≤𝑚

V = max {𝛾, 𝜌} .

if ℎ ≠ 0, if ℎ = 0.

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Let 𝑝, 𝑞 : T → R be two regressive functions; we define

2. Preliminaries In this section, we will first recall some basic definitions and lemmas which are used in what follows. 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} ,

1 + 𝜇 (𝑡) 𝑟 (𝑡) ≠ 0

𝑡

1≤𝑖≤𝑛

𝜌𝑗 = max sup 𝜌𝑖𝑗 (𝑡) ,

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𝜇 (𝑡) = 𝜎 (𝑡) − 𝑡. A point 𝑡 ∈ T is called left-dense if 𝑡 > inf T and 𝜌(𝑡) = 𝑡, left-scattered if 𝜌(𝑡) < 𝑡, right-dense if 𝑡 < sup T and 𝜎(𝑡) = 𝑡, and right-scattered if 𝜎(𝑡) > 𝑡. If T has a leftscattered maximum 𝑚, then T 𝑘 = T \ {𝑚}; otherwise, T 𝑘 = T.

𝑝 ⊕ 𝑞 := 𝑝 + 𝑞 + 𝜇𝑝𝑞, ⊖ 𝑝 := −

𝑝 , 1 + 𝜇𝑝

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𝑝 ⊖ 𝑞 := 𝑝 ⊕ (⊖𝑞) . Then the generalized exponential function has the following properties. Lemma 1 (see [31]). Assume that 𝑝, 𝑞 : T → R are two regressive functions; then, (i) 𝑒0 (𝑡, 𝑠) ≡ 1 and 𝑒𝑝 (𝑡, 𝑡) ≡ 1; (ii) 𝑒𝑝 (𝜎(𝑡), 𝑠) = (1 + 𝜇(𝑡)𝑝(𝑡))𝑒𝑝 (𝑡, 𝑠); (iii) 𝑒𝑝 (𝑡, 𝜎(𝑠)) = 𝑒𝑝 (𝑡, 𝑠)/(1 + 𝜇(𝑠)𝑝(𝑠));

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(iv) 1/𝑒𝑝 (𝑡, 𝑠) = 𝑒⊖𝑝 (𝑡, 𝑠);

Definition 5. Let 𝑓(𝑡, 𝑥) ∈ 𝐶(T × 𝐷, E𝑛 ), and if for any given sequence 𝛼󸀠 ⊂ Π and each compact subset 𝑆 of 𝐷 there exists a subsequence 𝛼 ⊂ 𝛼󸀠 such that 𝑇𝛼 𝑓(𝑡, 𝑥) exists uniformly on T × 𝑆, then 𝑓(𝑡, 𝑥) is called an almost periodic function in 𝑡 uniformly for 𝑥 ∈ 𝐷.

(v) (𝑒⊖𝑝 (𝑡, 𝑠))Δ = (⊖𝑝)(𝑡)𝑒⊖𝑝 (𝑡, 𝑠). In this section, 𝐸𝑛 denotes R𝑛 or C𝑛 , 𝐷 denotes an open set in 𝐸𝑛 or 𝐷 = 𝐸𝑛 , and 𝑆 denotes an arbitrary compact subset of 𝐷. Definition 2. A time scale T is called an almost periodic time scale if Π := {𝜏 ∈ R : T𝜏 ≠ 𝜙} ≠ {0}

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Definition 6. A function 𝑓 ∈ 𝐶1 (T, R) is said to be a 𝐶1 almost periodic function, if 𝑓, 𝑓Δ are two almost periodic functions on T. Definition 7 (see [31]). Let 𝑥 ∈ R𝑛 , and let 𝐴(𝑡) be an 𝑛 × 𝑛 rd-continuous matrix on T; the linear system

satisfies that, for any 𝜏1 , 𝜏2 ∈ Π, one has 𝜏1 ± 𝜏2 ∈ Π, where T𝜏 = T ∩ {T − 𝜏}. Definition 3. Let T be an almost periodic time scale. For any 𝑡 ∈ T, 𝜏 ∈ Π, we define ̃𝜏 = { 𝑡+

̃ 𝑡 + 𝜏 if 𝑡 ∈ T, ̃ 𝑡 if 𝑡 ∉ T,

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𝑥Δ (𝑡) = 𝐴 (𝑡) 𝑥 (𝑡) ,

󵄩 󵄩󵄩 󵄩󵄩𝑋 (𝑡) 𝑃𝑋−1 (𝜎 (𝑠))󵄩󵄩󵄩 ≤ 𝑘𝑒⊖𝛼 (𝑡, 𝜎 (𝑠)) , 󵄩0 󵄩 𝑠, 𝑡 ∈ T, 𝑡 ≥ 𝜎 (𝑠) ,

Obviously, if T is an almost periodic time scale, then inf T = −∞ and sup T = +∞. If there exists a 𝜏 ∈ Π such that T𝜏 = T, then Definition 2 is equivalent to Definition 3.7 in [31]; otherwise, Definition 2 is more general than Definition 3.7 in [31]. Definition 4. Let T be an almost periodic time scale. A function 𝑓 ∈ 𝐶(T × 𝐷, E𝑛 ) is called an almost periodic function in 𝑡 ∈ T uniformly for 𝑥 ∈ 𝐷 if the 𝜀-translation set of 𝑓 𝐸 {𝜀, 𝑓, 𝑆} 󵄨 ̃ 𝜏, 𝑥) − 𝑓 (𝑡, 𝑥)󵄨󵄨󵄨󵄨 < 𝜀, ∀ (𝑡, 𝑥) ∈ T × 𝑆} = {𝜏 ∈ Π : 󵄨󵄨󵄨𝑓 (𝑡+ (14) is a relatively dense set in T for all 𝜀 > 0 and for each compact subset 𝑆 of 𝐷; that is, for any given 𝜀 > 0 and each compact subset 𝑆 of 𝐷, there exists a constant 𝑙(𝜀, 𝑆) > 0 such that each interval of length 𝑙(𝜀, 𝑆) contains a 𝜏(𝜀, 𝑆) ∈ 𝐸{𝜀, 𝑓, 𝑆} such that ∀𝑡 ∈ T × 𝑆.

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𝜏 is called the 𝜀-translation number of 𝑓 and and 𝑙(𝜀, 𝑆) is called the inclusion length of 𝐸{𝜀, 𝑓, 𝑆}. For convenience, we introduce some notations. Let 𝛼 = {𝛼𝑛 } and 𝛽 = {𝛽𝑛 } be two sequences. Then 𝛽 ⊂ 𝛼 means that 𝛽 is a subsequence of 𝛼. We introduce the translation operator 𝑇, and 𝑇𝛼 𝑓(𝑡, 𝑥) = 𝑔(𝑡, 𝑥) means that 𝑔(𝑡, 𝑥) = ̃ 𝛼𝑛 , 𝑥). From Definitions 2 and 4, one can easily lim𝑛 → +∞ 𝑓(𝑡+ see that all the results obtained in [31] are still valid under the new concepts of almost periodic time scales and almost periodic functions on time scales. For example, similar to Theorems 3.13 and 3.14 in [31], we can obtain the following equivalent definition of uniformly almost periodic functions.

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is said to admit an exponential dichotomy on T if there exist positive constants 𝑘 and 𝛼, projection 𝑃, and the fundamental solution matrix 𝑋(𝑡) of (15), satisfying

where T̃ = ⋂𝜏∈Π T𝜏 .

󵄨󵄨 ̃ 𝜏, 𝑥) − 𝑓 (𝑡, 𝑥)󵄨󵄨󵄨󵄨 < 𝜀, 󵄨󵄨𝑓 (𝑡+

𝑡∈T

󵄩󵄩 󵄩 󵄩󵄩𝑋 (𝑡) (𝐼 − 𝑃) 𝑋−1 (𝜎 (𝑠))󵄩󵄩󵄩 ≤ 𝑘𝑒⊖𝛼 (𝜎 (𝑠) , 𝑡) , 󵄩 󵄩0

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𝑠, 𝑡 ∈ T, 𝑡 ≤ 𝜎 (𝑠) , where ‖ ⋅ ‖0 is a matrix norm on T. Consider the following linear almost periodic system: 𝑥Δ (𝑡) = 𝐴 (𝑡) 𝑥 (𝑡) + 𝑓 (𝑡) ,

𝑡 ∈ T,

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where 𝐴(𝑡) is an almost periodic matrix function and 𝑓(𝑡) is an almost periodic vector function. Lemma 8 (see [31]). If the linear system (15) admits exponential dichotomy, then system (16) has a unique almost periodic solution 𝑥(𝑡): 𝑥 (𝑡) = ∫

𝑡

𝑋 (𝑡) 𝑃𝑋−1 (𝜎 (𝑠)) 𝑓 (𝑠) Δ𝑠

−∞

+∞

−∫

𝑡

(19) −1

𝑋 (𝑡) (𝐼 − 𝑃) 𝑋 (𝜎 (𝑠)) 𝑓 (𝑠) Δ𝑠,

where 𝑋(𝑡) is the fundamental solution matrix of (15). Lemma 9 (see [24]). Let 𝑐𝑖 (𝑡) be an almost periodic function on T, where 𝑐𝑖 (𝑡) > 0, −𝑐𝑖 (𝑡) ∈ R+ , ∀𝑡 ∈ T, and ̃ > 0; min {inf 𝑐𝑖 (𝑡)} = 𝑚

1≤𝑖≤𝑛

𝑡∈T

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then, the linear system 𝑥Δ (𝑡) = diag (−𝑐1 (𝑡) , −𝑐2 (𝑡) , . . . , −𝑐𝑛 (𝑡)) 𝑥 (𝑡) admits an exponential dichotomy on T.

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Lemma 10 (see [27]). Every rd-continuous function has an antiderivative. In particular, if 𝑡0 ∈ T, then 𝐹 defined by 𝑡

𝐹 (𝑡) = ∫ 𝑓 (𝑠) Δ𝑠, 𝑡0

𝑡∈T

where

󵄩 󵄩 Δ 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩1 = max {󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩0 , 󵄩󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩󵄩0 } , 󵄨 󵄩 󵄨 󵄩󵄩 󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩0 = max 󵄨󵄨󵄨𝜑 (𝑡)󵄨󵄨󵄨 , 1≤𝑖≤𝑛+𝑚

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󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩𝜑󵄩󵄩0 = sup 󵄩󵄩󵄩𝜑 (𝑡)󵄩󵄩󵄩0 ,

is an antiderivative of 𝑓.

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𝑡∈T

Lemma 11 (see [27]). If 𝑝 ∈ R and 𝑎, 𝑏, 𝑐 ∈ T, then Δ

𝜑Δ (𝑡)

𝜎

𝑇

[𝑒𝑝 (𝑐, ⋅)] = −𝑝 [𝑒𝑝 (𝑐, ⋅)] ,

Δ Δ = (𝜑1Δ (𝑡) , . . . , 𝜑𝑛Δ (𝑡) , 𝜑𝑛+1 (𝑡) , . . . , 𝜑𝑛+𝑚 (𝑡)) ,

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𝑏

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

Theorem 15. Assume that (𝐻1 ), (𝐻2 ), and the following hold:

By Lemmas 10 and 11, it is easy to get the following lemma. Lemma 12. Suppose that 𝑓(𝑡) is an 𝑟𝑑-continuous function and 𝑐(𝑡) is a positive 𝑟𝑑-continuous function which satisfies that −𝑐(𝑡) ∈ R+ . Let

𝑡0

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where 𝑡0 ∈ T; then, 𝑡

𝑔Δ (𝑡) = 𝑓 (𝑡) + ∫ −𝑐 (𝑡) 𝑒−𝑐 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) Δ𝑠. 𝑡0

(𝐻3 ) −𝑎𝑖 , −𝑏𝑗 ∈ R+ , 𝑡 − 𝛾𝑗𝑖 (𝑡), 𝑡 − 𝜌𝑖𝑗 (𝑡) ∈ T, ∀𝑡 ∈ T, 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚; (𝐻4 ) there exists a constant 𝑟0 such that { 𝑎𝑖 + 𝑎𝑖 𝑏𝑗 + 𝑏𝑗 } 𝜂𝑖 , 𝜂𝑗 } + max {𝐿 1 , 𝐿 2 } ≤ 𝑟0 , 1≤𝑖≤𝑛,1≤𝑗≤𝑚 { 𝑎𝑖 𝑏𝑗 { } 𝑎𝑖 < 𝑎𝑖 , 0 < Π𝑖 < 𝑎𝑖 + 𝑎𝑖 (29) max

𝑡

𝑔 (𝑡) = ∫ 𝑒−𝑐 (𝑡, 𝜎 (𝑠)) 𝑓 (𝑠) Δ𝑠,

then B is a Banach space.

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0 < Π𝑗
0, it follows from Lemma 10 that the linear system 𝑥𝑖Δ (𝑡) = −𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) ,

𝑖 = 1, 2, . . . , 𝑛,

𝑦𝑗Δ (𝑡) = −𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡) ,

𝑗 = 1, 2, . . . , 𝑚,

(33)

𝑗 = 1, 2, . . . , 𝑚, (35) are almost periodic functions on T; that is, (34) is not only an almost periodic solution of system (32), but also a 𝐶1 -almost periodic solution of system (32). First, we define a nonlinear operator on B by Φ (𝜑) (𝑡)

admits an exponential dichotomy on T. Thus, by Lemma 9, we know that system (32) has exactly one almost periodic solution:

𝑇

= (𝑥𝜑1 (𝑡) , 𝑥𝜑2 (𝑡) , . . . , 𝑥𝜑𝑛 (𝑡) , 𝑦𝜑𝑛+1 (𝑡) , . . . , 𝑦𝜑𝑛+𝑚 (𝑡)) , ∀𝜑 ∈ B. (36)

𝑥𝜑𝑖 (𝑡) =∫

𝑡

−∞

𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 𝑚

⋅ ( ∑𝑝𝑗𝑖 (𝑠) 𝑓𝑗 (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))) + 𝐼𝑖 (𝑠)) Δ𝑠, 𝑗=1

𝑦𝜑𝑛+𝑗 (𝑡) =∫

𝑡

−∞

Next, we check that Φ(𝐸) ⊂ 𝐸. For any given 𝜑 ∈ 𝐸, it suffices to prove that ‖Φ(𝜑)‖B ≤ 𝑟0 . By conditions (𝐻1 )–(𝐻4 ), we have 󵄨 󵄨 sup 󵄨󵄨󵄨󵄨𝑥𝜑𝑖 (𝑡)󵄨󵄨󵄨󵄨 𝑡∈T 󵄨 {󵄨󵄨󵄨󵄨 𝑡 = sup {󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨 𝑚

𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))

⋅ ( ∑ 𝑝𝑗𝑖 (𝑠) 𝑓𝑗 (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))) 𝑗=1

𝑛

⋅ (∑𝑞𝑖𝑗 (𝑠) 𝑔𝑖 (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))) + 𝐽𝑗 (𝑠)) Δ𝑠. 𝑖=1

(34)

󵄨󵄨 󵄨󵄨} +𝐼𝑖 (𝑠)) Δ𝑠󵄨󵄨󵄨󵄨} 󵄨󵄨 󵄨}

6

The Scientific World Journal 󵄨󵄨 𝑡 󵄨󵄨 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨 −∞ 𝑡∈T 󵄨

󵄨 {󵄨󵄨󵄨󵄨 𝑡 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨

󵄨󵄨 𝑛 𝐽𝑗 󵄨󵄨 󵄨 󵄨 ⋅ (∑𝑞𝑖𝑗 (󵄨󵄨󵄨𝑔𝑖 (0)󵄨󵄨󵄨 + 𝛽𝑖 𝑟0 )) Δ𝑠󵄨󵄨󵄨} + 󵄨󵄨 𝑏𝑗 𝑖=1 󵄨

󵄨󵄨 󵄨󵄨} 𝐼 ⋅ ( ∑ 𝑝𝑗𝑖 𝑓𝑗 (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)))) Δ𝑠󵄨󵄨󵄨󵄨} + 𝑖 𝑎𝑖 󵄨󵄨 𝑗=1 󵄨} 𝑚

󵄨 {󵄨󵄨󵄨󵄨 𝑡 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨

𝜂𝑗

≤

𝑏𝑗

+

𝑏𝑗 + 𝑏𝑗 𝑏𝑗

𝑏𝑗 + 𝑏𝑗

𝐽𝑗 ≤

𝑏𝑗

𝜂𝑗 + 𝐿 2 ≤ 𝑟0 , 𝑗 = 1, 2, . . . , 𝑚,

𝑚

󵄨 󵄨 ⋅ ( ∑𝑝𝑗𝑖 (󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 𝑗=1

󵄨󵄨 󵄨󵄨} 󵄨󵄨 󵄨󵄨 󵄨 󵄨 +𝛼𝑗 󵄨󵄨𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))󵄨󵄨)) Δ𝑠󵄨󵄨󵄨󵄨} 󵄨󵄨 󵄨} +

𝐼𝑖 𝑎𝑖

󵄨 󵄨 sup 󵄨󵄨󵄨󵄨𝑥𝜑Δ𝑖 (𝑡)󵄨󵄨󵄨󵄨 𝑡∈T

󵄨 {󵄨󵄨󵄨󵄨 𝑚 = sup {󵄨󵄨󵄨( ∑𝑝𝑗𝑖 (𝑡) 𝑓𝑗 (𝜑𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡))) + 𝐼𝑖 (𝑡)) 󵄨 𝑡∈T {󵄨󵄨 𝑗=1 − 𝑎𝑖 (𝑡) ∫

𝑡

−∞

󵄨 {󵄨󵄨󵄨󵄨 𝑡 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨

𝑚

⋅ ( ∑𝑝𝑗𝑖 (𝑠) 𝑓𝑗 𝑗=1

󵄨󵄨 𝑚 󵄨󵄨} 𝐼 󵄨 󵄨 ⋅ ( ∑ 𝑝𝑗𝑖 (󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 𝛼𝑗 𝑟0 )) Δ𝑠󵄨󵄨󵄨󵄨} + 𝑖 𝑎𝑖 󵄨󵄨 𝑗=1 󵄨} ≤

𝜂𝑖 + 𝑎𝑖

𝑎𝑖 + 𝑎𝑖 𝑎𝑖

𝐼𝑖 ≤

𝑎𝑖 + 𝑎𝑖 𝑎𝑖

⋅ (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))) 󵄨󵄨 󵄨󵄨} +𝐼𝑖 (𝑠) ) Δ𝑠󵄨󵄨󵄨󵄨} 󵄨󵄨 󵄨}

𝜂𝑖 + 𝐿 1 ≤ 𝑟0 , 𝑖 = 1, 2, . . . , 𝑛,

󵄨󵄨 󵄨󵄨 sup 󵄨󵄨󵄨𝑦𝜑𝑛+𝑗 (𝑡)󵄨󵄨󵄨 󵄨 𝑡∈T 󵄨

{𝑚 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ sup { ∑𝑝𝑗𝑖 (󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 𝛼𝑗 󵄨󵄨󵄨󵄨𝜑𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡))󵄨󵄨󵄨󵄨) + 󵄨󵄨󵄨𝐼𝑖 (𝑡)󵄨󵄨󵄨 𝑡∈T 𝑗=1 {

󵄨󵄨 𝑡 󵄨󵄨 = sup {󵄨󵄨󵄨 ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨 −∞ 𝑡∈T 󵄨

+ 𝑎𝑖 [∫

𝑚

𝑗=1

󵄨 󵄨 +𝛼𝑗 󵄨󵄨󵄨󵄨𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))󵄨󵄨󵄨󵄨)

󵄨󵄨 𝑡 󵄨󵄨 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨 −∞ 𝑡∈T 󵄨

󵄨󵄨 𝑡 󵄨󵄨 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨 −∞ 𝑡∈T 󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 ⋅ (∑𝑞𝑖𝑗 (󵄨󵄨𝑔𝑖 (0)󵄨󵄨 + 𝛽𝑖 󵄨󵄨𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))󵄨󵄨)) Δ𝑠󵄨󵄨󵄨} 󵄨󵄨 𝑖=1 󵄨

+

𝐽𝑗 𝑏𝑗

𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 󵄨 ⋅ ( ∑𝑝𝑗𝑖 ( 󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨

󵄨󵄨 󵄨󵄨 ⋅ (∑𝑞𝑖𝑗 (𝑠) 𝑔𝑖 (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))) + 𝐽𝑗 (𝑠)) Δ𝑠󵄨󵄨󵄨} 󵄨󵄨 𝑖=1 󵄨

󵄨󵄨 𝑛 𝐽𝑗 󵄨󵄨 ⋅ (∑𝑞𝑖𝑗 𝑔𝑖 (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)))) Δ𝑠󵄨󵄨󵄨} + 󵄨󵄨 𝑏𝑗 𝑖=1 󵄨

𝑡

−∞

𝑛

𝑛

𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠))

} 󵄨 󵄨 + 󵄨󵄨󵄨𝐼𝑖 (𝑠)󵄨󵄨󵄨) Δ𝑠]} ]} 𝑚 󵄨 󵄨 ≤ ∑𝑝𝑗𝑖 (󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 𝛼𝑗 𝑟0 ) + 𝐼𝑖 𝑗=1

𝑡

+ 𝑎𝑖 [ ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) −∞ [ 𝑚 𝐼 󵄨 󵄨 ⋅ ( ∑𝑝𝑗𝑖 (󵄨󵄨󵄨󵄨𝑓𝑗 (0)󵄨󵄨󵄨󵄨 + 𝛼𝑗 𝑟0 )) Δ𝑠 + 𝑖 ] 𝑎𝑖 𝑗=1 ]

The Scientific World Journal ≤

𝑎𝑖 + 𝑎𝑖 𝑎𝑖

𝜂𝑖 + max {

𝑎𝑖 + 𝑎𝑖 𝑎𝑖

1≤𝑖≤𝑛

7 𝐼𝑖 } ≤

𝑎𝑖 + 𝑎𝑖 𝑎𝑖

Taking 𝜑, 𝜓 ∈ 𝐸 and combining conditions (𝐻1 ) and (𝐻4 ), we obtain that

𝜂𝑖 + 𝐿 1 ≤ 𝑟0 ,

󵄨 󵄨 sup 󵄨󵄨󵄨󵄨𝑥𝜑𝑖 (𝑡) − 𝑥𝜓𝑖 (𝑡)󵄨󵄨󵄨󵄨

𝑖 = 1, 2, . . . , 𝑛,

𝑡∈T

󵄨󵄨 󵄨󵄨 sup 󵄨󵄨󵄨𝑦𝜑Δ𝑛+𝑗 (𝑡)󵄨󵄨󵄨 󵄨 𝑡∈T 󵄨

󵄨 {󵄨󵄨󵄨󵄨 𝑡 = sup {󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨

󵄨󵄨 𝑛 󵄨󵄨 = sup {󵄨󵄨󵄨(∑𝑞𝑖𝑗 (𝑡) 𝑔𝑖 (𝜑𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))) + 𝐽𝑗 (𝑡)) 󵄨󵄨 𝑖=1 𝑡∈T 󵄨 − 𝑏𝑗 (𝑡) ∫

𝑡

−∞

𝑚

⋅ ( ∑ 𝑝𝑗𝑖 (𝑠)

𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))

𝑗=1

⋅ [𝑓𝑗 (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)))

𝑛

⋅ (∑𝑞𝑖𝑗 (𝑠) 𝑔𝑖

󵄨󵄨 󵄨󵄨} − 𝑓𝑗 (𝜓𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)))]) Δ𝑠󵄨󵄨󵄨󵄨} 󵄨󵄨 󵄨}

𝑖=1

󵄨󵄨 󵄨󵄨 ⋅ (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))) + 𝐽𝑗 (𝑠)) Δ𝑠󵄨󵄨󵄨} 󵄨󵄨 󵄨 𝑛

󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ≤ sup {∑𝑞𝑖𝑗 (󵄨󵄨󵄨𝑔𝑖 (0)󵄨󵄨󵄨 + 𝛽𝑖 󵄨󵄨󵄨󵄨𝜑𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨) + 󵄨󵄨󵄨󵄨𝐽𝑗 (𝑡)󵄨󵄨󵄨󵄨 + 𝑏𝑗 𝑡∈T

𝑖=1

⋅ [∫

𝑡

−∞

𝑚 󵄨 ⋅ ( ∑ 𝑝𝑗𝑖 (𝑠) 𝛼𝑗 󵄨󵄨󵄨󵄨𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))

𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))

𝑗=1

󵄨󵄨 󵄨󵄨} 󵄨 − 𝜓𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))󵄨󵄨󵄨󵄨 ) Δ𝑠󵄨󵄨󵄨󵄨} 󵄨󵄨 󵄨} 󵄨 󵄨󵄨 𝑚 󵄨󵄨} 󵄩 {󵄨󵄨󵄨󵄨 𝑡 󵄩 ≤ sup {󵄨󵄨󵄨∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) ( ∑𝑝𝑗𝑖 𝛼𝑗 ) Δ𝑠󵄨󵄨󵄨󵄨} 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B 󵄨󵄨 −∞ 󵄨 𝑡∈T 󵄨 𝑗=1 󵄨} {󵄨

𝑛 󵄨 󵄨 󵄨 󵄨 ⋅ (∑𝑞𝑖𝑗 (󵄨󵄨󵄨𝑔𝑖 (0)󵄨󵄨󵄨 + 𝛽𝑖 󵄨󵄨󵄨󵄨𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))󵄨󵄨󵄨󵄨) 𝑖=1

󵄨 󵄨 + 󵄨󵄨󵄨󵄨𝐽𝑗 (𝑠)󵄨󵄨󵄨󵄨 ) Δ𝑠]} 𝑛

󵄨 󵄨 ≤ ∑𝑞𝑖𝑗 (󵄨󵄨󵄨𝑔𝑖 (0)󵄨󵄨󵄨 + 𝛽𝑖 𝑟0 ) + 𝐽𝑗

≤

𝑖=1

𝑡

+ 𝑏𝑗 [ ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) −∞ [

󵄨󵄨 󵄨󵄨 sup 󵄨󵄨󵄨𝑦𝜑𝑛+𝑗 (𝑡) − 𝑦𝜓𝑛+𝑗 (𝑡)󵄨󵄨󵄨 󵄨 󵄨 𝑡∈T

𝐽𝑗 󵄨 󵄨 ⋅ (∑𝑞𝑖𝑗 (󵄨󵄨󵄨𝑔𝑖 (0)󵄨󵄨󵄨 + 𝛽𝑖 𝑟0 )) Δ𝑠 + ] 𝑏𝑗 𝑖=1 ] 𝑏𝑗 + 𝑏𝑗 𝑏𝑗

𝑎𝑖 + 𝑎𝑖 󵄩 Π𝑖 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 Π𝑖 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B < 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B , 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B ≤ 𝑎𝑖 𝑎𝑖 𝑖 = 1, 2, . . . , 𝑛,

𝑛

≤

󵄨 {󵄨󵄨󵄨 𝑡 ≤ sup {󵄨󵄨󵄨󵄨 ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 󵄨 −∞ 𝑡∈T {󵄨󵄨

󵄨󵄨 𝑡 󵄨 = sup {󵄨󵄨󵄨∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 𝑡∈T 󵄨󵄨 −∞

{ 𝑏𝑗 + 𝑏𝑗 } 𝑏𝑗 + 𝑏𝑗 𝜂𝑗 + max { 𝐽 ≤ 𝜂𝑗 + 𝐿 2 ≤ 𝑟0 , 1≤𝑗≤𝑚 𝑏𝑗 𝑗 } 𝑏𝑗 { }

𝑛

⋅ (∑𝑞𝑖𝑗 (𝑠) [𝑔𝑖 (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))) 𝑖=1

𝑗 = 1, 2, . . . , 𝑚; (37) then, it follows from (37) that 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄩 󵄩󵄩 󵄩󵄩Φ (𝜑)󵄩󵄩󵄩B = max {sup 󵄨󵄨󵄨󵄨𝑥𝜑𝑖 (𝑡)󵄨󵄨󵄨󵄨 , sup 󵄨󵄨󵄨𝑦𝜑𝑛+𝑗 (𝑡)󵄨󵄨󵄨 , 󵄨 󵄨 1≤𝑖≤𝑛, 1≤𝑗≤𝑚 𝑡∈T

𝑡∈T

󵄨󵄨 󵄨󵄨 󵄨 󵄨 sup 󵄨󵄨󵄨󵄨𝑥𝜑Δ𝑖 (𝑡)󵄨󵄨󵄨󵄨 , sup 󵄨󵄨󵄨𝑦𝜑Δ𝑛+𝑗 (𝑡)󵄨󵄨󵄨} ≤ 𝑟0 . 󵄨 𝑡∈T 𝑡∈T 󵄨 (38) Therefore, Φ(𝐸) ⊂ 𝐸.

󵄨󵄨 𝑡 󵄨󵄨 ≤ sup {󵄨󵄨󵄨 ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 󵄨󵄨 −∞ 𝑡∈T 󵄨

󵄨󵄨 󵄨󵄨 − 𝑔𝑖 (𝜓𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)))]) Δ𝑠󵄨󵄨󵄨} 󵄨󵄨 󵄨

𝑛

󵄨 ⋅ (∑𝑞𝑖𝑗 (𝑠) 𝛽𝑖 󵄨󵄨󵄨󵄨𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)) 𝑖=1

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 − 𝜓𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))󵄨󵄨) Δ𝑠󵄨󵄨󵄨} 󵄨󵄨 󵄨

8

The Scientific World Journal 󵄨󵄨 𝑡 󵄨󵄨 𝑛 󵄨󵄨 󵄨󵄨 󵄩 󵄩 ≤ sup {󵄨󵄨󵄨∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) (∑𝑞𝑖𝑗 𝛽𝑖 ) Δ𝑠󵄨󵄨󵄨} 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B 󵄨 󵄨 −∞ 𝑡∈T 󵄨󵄨 󵄨󵄨 𝑖=1 ≤

𝑏𝑗 + 𝑏𝑗 Π𝑗 󵄩 󵄩 󵄩 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩 ≤ Π𝑗 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B 󵄩 󵄩 B 𝑏𝑗 𝑏𝑗

󵄩 󵄩 < 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B ,

𝑗 = 1, 2, . . . , 𝑚,

󵄨󵄨 Δ 󵄨󵄨 sup 󵄨󵄨󵄨󵄨(𝑦𝜑𝑛+𝑗 (𝑡) − 𝑦𝜓𝑛+𝑗 (𝑡)) 󵄨󵄨󵄨󵄨 󵄨 𝑡∈T 󵄨 󵄨󵄨 󵄨󵄨 = sup 󵄨󵄨󵄨𝑦𝜑Δ𝑛+𝑗 (𝑡) − 𝑦𝜓Δ𝑛+𝑗 (𝑡)󵄨󵄨󵄨 󵄨 󵄨 𝑡∈T 𝑛 󵄨󵄨 󵄨 󵄨 ≤ sup {∑ 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑔𝑖 (𝜑𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))) − 𝑔𝑖 (𝜓𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡)))󵄨󵄨󵄨󵄨 𝑡∈T

𝑖=1

󵄨󵄨 Δ 󵄨󵄨 sup 󵄨󵄨󵄨(𝑥𝜑𝑖 (𝑡) − 𝑥𝜓𝑖 (𝑡)) 󵄨󵄨󵄨 󵄨 𝑡∈T 󵄨

𝑡

+ 𝑏𝑗 [ ∫

−∞

󵄨 󵄨 = sup 󵄨󵄨󵄨󵄨𝑥𝜑Δ𝑖 (𝑡) − 𝑥𝜓Δ𝑖 (𝑡)󵄨󵄨󵄨󵄨

𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))

𝑛 󵄨 󵄨 ⋅ (∑ 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨

𝑡∈T

𝑖=1

{𝑚 󵄨 󵄨󵄨 ≤ sup { ∑ 󵄨󵄨󵄨󵄨𝑝𝑗𝑖 (𝑡)󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝑓𝑗 (𝜑𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡))) 𝑡∈T {𝑗=1

󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑔𝑖 (𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)))

󵄨 −𝑓𝑗 (𝜓𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡)))󵄨󵄨󵄨󵄨

𝑡

󵄨 −𝑔𝑖 (𝜓𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)))󵄨󵄨󵄨󵄨 ) Δ𝑠]} 𝑛 󵄨 󵄨 󵄨 󵄨 ≤ sup {∑ 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 (𝑡)󵄨󵄨󵄨󵄨 𝛽𝑖 󵄨󵄨󵄨󵄨𝜑𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡)) − 𝜓𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))󵄨󵄨󵄨󵄨

+ 𝑎𝑖 [ ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) −∞ [

𝑡∈T

𝑖=1

𝑚

󵄨 󵄨 ⋅ ( ∑ 󵄨󵄨󵄨󵄨𝑝𝑗𝑖 (𝑠)󵄨󵄨󵄨󵄨

+ 𝑏𝑗 [ ∫

𝑡

−∞

𝑗=1

󵄨 ⋅ 󵄨󵄨󵄨󵄨𝑓𝑗 (𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)))

𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠))

𝑛 󵄨 󵄨 ⋅ (∑ 󵄨󵄨󵄨󵄨𝑞𝑖𝑗 (𝑠)󵄨󵄨󵄨󵄨 𝛽𝑖 𝑖=1

− 𝑓𝑗 (𝜓𝑛+𝑗

󵄨 ⋅ 󵄨󵄨󵄨󵄨𝜑𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))

} 󵄨 ⋅ (𝑠 − 𝛾𝑗𝑖 (𝑠)))󵄨󵄨󵄨󵄨 ) Δ𝑠]} ]} {𝑚 󵄨 󵄨 ≤ sup { ∑ 󵄨󵄨󵄨󵄨𝑝𝑗𝑖 (𝑡)󵄨󵄨󵄨󵄨 𝛼𝑗 𝑡∈T {𝑗=1 󵄨 󵄨 ⋅ 󵄨󵄨󵄨󵄨𝜑𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡)) − 𝜓𝑛+𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡))󵄨󵄨󵄨󵄨 + 𝑎𝑖 [ ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) −∞ [

𝑗=1

󵄨 ⋅ 󵄨󵄨󵄨󵄨𝜑𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)) } 󵄨 −𝜓𝑛+𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠))󵄨󵄨󵄨󵄨 ) Δ𝑠]} ]}

𝑗=1

≤ Π𝑖

𝑎𝑖 + 𝑎𝑖 󵄩 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩 󵄩B 𝑎𝑖 󵄩

𝑎𝑖 + 𝑎𝑖 󵄩 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩 < 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩 , 󵄩B 󵄩 󵄩B 𝑎𝑖 󵄩

𝑏𝑗 + 𝑏𝑗

𝑖=1

𝑏𝑗

𝑏𝑗 + 𝑏𝑗 𝑏𝑗

󵄩 󵄩󵄩 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B

󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B < 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B , 𝑗 = 1, 2, . . . , 𝑚. (39)

𝑚 󵄨 󵄨 ⋅ ( ∑ 󵄨󵄨󵄨󵄨𝑝𝑗𝑖 (𝑠)󵄨󵄨󵄨󵄨 𝛼𝑗

𝑚

𝑛

≤ (∑𝑞𝑖𝑗 𝛽𝑖 )

≤ Π𝑗

𝑡

≤ ( ∑ 𝑝𝑗𝑖 𝛼𝑗 )

󵄨 −𝜓𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))󵄨󵄨󵄨󵄨) Δ𝑠]}

Similarly, from (39) it follows that 󵄩󵄩󵄩Φ (𝜑) − Φ (𝜓)󵄩󵄩󵄩 󵄩B 󵄩 =

max

1≤𝑖≤𝑛,1≤𝑗≤𝑚

󵄩 󵄩 {sup 󵄩󵄩󵄩󵄩𝑥𝜑𝑖 (𝑡) − 𝑥𝜓𝑖 (𝑡)󵄩󵄩󵄩󵄩1 , 𝑡∈T

󵄩󵄩 󵄩󵄩 sup 󵄩󵄩󵄩𝑦𝜑𝑛+𝑗 (𝑡) − 𝑦𝜓𝑛+𝑗 (𝑡)󵄩󵄩󵄩 } 󵄩1 󵄩 𝑡∈T

(40)

󵄩 󵄩 < 󵄩󵄩󵄩𝜑 − 𝜓󵄩󵄩󵄩B . 𝑖 = 1, 2, . . . , 𝑛,

By (40), we obtain that Φ is a contraction mapping from 𝐸 to 𝐸. Since 𝐸 is a closed subset of B, Φ has a fixed

The Scientific World Journal

9

point in 𝐸, which means that (32) has a unique 𝐶1 -almost periodic solution in 𝐸. Then system (1) has a unique 𝐶1 almost periodic solution in the region 󵄩 󵄩 𝐸 = {𝜑 ∈ B : 󵄩󵄩󵄩𝜑󵄩󵄩󵄩B ≤ 𝑟0 } .

(41)

Then it follows from system (1) that 𝑢𝑖Δ (𝑠) + 𝑎𝑖 (𝑠) 𝑢𝑖 (𝑠) 𝑚

= ∑ 𝑝𝑗𝑖 (𝑠) [𝑓𝑗 (V𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)) + 𝑦𝑗∗ (𝑠 − 𝛾𝑗𝑖 (𝑠))) 𝑗=1

This completes the proof.

−𝑓𝑗 (𝑦𝑗∗ (𝑠 − 𝛾𝑗𝑖 (𝑠)))] ,

1

4. Exponential Stability of the 𝐶 -Almost Periodic Solution

V𝑗Δ (𝑠) + 𝑏𝑗 (𝑠) V𝑗 (𝑠)

Definition 16. The 𝐶1 -almost periodic solution 𝑧∗ (𝑡) = ∗ (𝑥1∗ (𝑡), 𝑥2∗ (𝑡), . . . , 𝑥𝑛∗ (𝑡), 𝑦1∗ (𝑡), . . . , 𝑦𝑚 (𝑡))𝑇 of system (1) with ∗ ∗ ∗ ∗ (𝑡), . . . , initial value 𝜑 (𝑡) = (𝜑1 (𝑡), 𝜑2 (𝑡), . . . , 𝜑𝑛∗ (𝑡), 𝜑𝑛+1 𝑇 ∗ 𝜑𝑛+𝑚 (𝑡)) is said to be globally exponentially stable. There exist a positive constant 𝜆 with ⊖𝜆 ∈ R+ and 𝑀 > 1 such that every solution 𝑧 (𝑡) = (𝑥1 (𝑡) , 𝑥2 (𝑡) , . . . , 𝑥𝑛 (𝑡) , 𝑦1 (𝑡) , . . . , 𝑦𝑚 (𝑡))

𝑇

(42)

𝑛

= ∑𝑞𝑖𝑗 (𝑠) [𝑔𝑖 (𝑢𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠)) + 𝑥𝑖∗ (𝑠 − 𝜌𝑖𝑗 (𝑠)))

(50)

𝑖=1

−𝑔𝑖 (𝑥𝑖∗ (𝑠 − 𝜌𝑖𝑗 (𝑠)))] , where 𝑢𝑖 (𝑠) = 𝑥𝑖 (𝑠) − 𝑥𝑖∗ (𝑠), V𝑗 (𝑠) = 𝑦𝑗 (𝑠) − 𝑦𝑗∗ (𝑠) and 𝑖 = 1, 2, . . . , 𝑛, 𝑗 = 1, 2, . . . , 𝑚, and the initial conditions of (49) and (50) are 𝜓𝑖 (𝑠) = 𝜑𝑖 (𝑠) − 𝜑𝑖∗ (𝑠) ,

of system (1) with any initial value

∗ 𝜓𝑛+𝑗 (𝑠) = 𝜑𝑛+𝑗 (𝑠) − 𝜑𝑛+𝑗 (𝑠) ,

𝜑 (𝑡) = (𝜑1 (𝑡) , 𝜑2 (𝑡) , . . . , 𝜑𝑛 (𝑡) , 𝜑𝑛+1 (𝑡) , . . . , 𝜑𝑛+𝑚 (𝑡))

𝑇

(43)

𝑠 ∈ [−V, 0]T ,

𝑖 = 1, 2, . . . , 𝑛,

(51)

𝑗 = 1, 2, . . . , 𝑚.

Let 𝐻𝑖 and 𝐻𝑗 be defined by

satisfies 󵄩󵄩 󵄩 󵄩 󵄩 ∗ 󵄩󵄩𝑧 (𝑡) − 𝑧 (𝑡)󵄩󵄩󵄩 ≤ 𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ,

∀𝑡 ∈ (0, +∞)T , (44)

𝐻𝑖 (𝜖) 𝑚

where 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜓󵄩󵄩 = sup

= 𝑎𝑖 − 𝜖 − ∑𝑝𝑗𝑖 𝛼𝑗 exp (𝜖 (𝛾 + sup𝜇 (𝑠))) ,

󵄨 󵄨 max 󵄨󵄨𝜑𝑖 (𝑡) − 𝜑𝑖∗ (𝑡)󵄨󵄨󵄨 ,

Proof. According to Theorem 15, we know that (1) has a 𝐶1 almost periodic solution ∗

𝑧 (𝑡) (𝑡) , . . . , 𝑥𝑛∗

(𝑡) , 𝑦1∗

𝜖 ∈ [0, +∞) ,

𝐻𝑗 (𝜖)

Theorem 17. Suppose that (𝐻1 )–(𝐻4 ) hold and sup𝑡∈T 𝜇(𝑡) < +∞; then, system (1) has a unique 𝐶1 -almost periodic solution which is globally exponentially stable.

(𝑡) , 𝑥2∗

𝑖 = 1, 2, . . . , 𝑛,

(45)

𝑡0 = max {[−V, 0]T } .

(𝑥1∗

𝑠∈T

𝑗=1

󵄨 𝑡∈[−V,0]T 1≤𝑖≤𝑛+𝑚

=

(49)

∗ (𝑡) , . . . , 𝑦𝑚

𝑇

(𝑡))

(46)

𝑇

𝑛

= 𝑏𝑗 − 𝜖 − ∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜖 (𝜌 + sup𝜇 (𝑠))) , 𝑠∈T

𝑖=1

𝑗 = 1, 2, . . . , 𝑚,

𝜖 ∈ [0, +∞) .

By (𝐻4 ), we get 𝑚

𝐻𝑖 (0) = 𝑎𝑖 − ∑𝑝𝑗𝑖 𝛼𝑗 = 𝑎𝑖 − Π𝑖 > 0, 𝑗=1

∗ (𝑡), . . . , with initial value 𝜑∗ (𝑡) = (𝜑1∗ (𝑡), 𝜑2∗ (𝑡), . . . , 𝜑𝑛∗ (𝑡), 𝜑𝑛+1 𝑇 ∗ 𝜑𝑛+𝑚 (𝑡)) . Suppose that

𝑧 (𝑡) = (𝑥1 (𝑡) , 𝑥2 (𝑡) , . . . , 𝑥𝑛 (𝑡) , 𝑦1 (𝑡) , . . . , 𝑦𝑚 (𝑡))

(52)

(47)

𝑖 = 1, 2, . . . , 𝑛, 𝑛

(53)

𝐻𝑗 (0) = 𝑏𝑗 − ∑𝑞𝑖𝑗 𝛽𝑖 = 𝑏𝑗 − Π𝑗 > 0, 𝑖=1

𝑗 = 1, 2, . . . , 𝑚.

is an arbitrary solution of (1) with initial value 𝜑 (𝑡) 𝑇

= (𝜑1 (𝑡) , 𝜑2 (𝑡) , . . . , 𝜑𝑛 (𝑡) , 𝜑𝑛+1 (𝑡) , . . . , 𝜑𝑛+𝑚 (𝑡)) .

(48)

Since 𝐻𝑖 , 𝐻𝑗 are continuous on [0, +∞) and 𝐻𝑖 (𝜖) → −∞, 𝐻𝑗 (𝜖) → −∞ as 𝜖 → +∞, there exist 𝜖𝑖 , 𝜖𝑗 > 0 such that 𝐻𝑖 (𝜖𝑖 ) = 0, 𝐻𝑗 (𝜖𝑗 ) = 0, and 𝐻𝑖 (𝜖) > 0 for 𝜖 ∈ (0, 𝜖𝑖 )

10

The Scientific World Journal

and 𝐻𝑗 (𝜖) > 0 for 𝜖 ∈ (0, 𝜖𝑗 ). By choosing 𝜀 = min{𝜖1 , 𝜖2 , . . . , 𝜖𝑛 , 𝜖1 , . . . , 𝜖𝑚 }, we have

Similarly, multiplying (50) by 𝑒−𝑏𝑗 (𝑡0 , 𝜎(𝑠)) and integrating on [𝑡0 , 𝑡]T , we have V𝑗 (𝑡)

𝐻𝑖 (𝜀) ≥ 0,

= V𝑗 (𝑡0 ) 𝑒−𝑏𝑗 (𝑡, 𝑡0 )

𝐻𝑗 (𝜀) ≥ 0, 𝑖 = 1, 2, . . . , 𝑛,

(54)

𝑡

+ ∫ 𝑒−𝑏𝑗 (𝑡, 𝜎 (𝑠)) 𝑡0

𝑗 = 1, 2, . . . , 𝑚.

𝑛

So, we can choose a positive constant 0 min{𝜀, min1≤𝑖≤𝑛 {𝑎𝑖 }, min1≤𝑗≤𝑚 {𝑏𝑗 }} such that


0, 𝑖 = 1, 2, . . . , 𝑛,

⋅ (∑𝑞𝑖𝑗 (𝑠) [𝑔𝑖 (𝑢𝑖 (𝑠 − 𝜌𝑖𝑗 (𝑠))

𝑗 = 1, 2, . . . , 𝑚.

(55) max1≤𝑖≤𝑛,1≤𝑗≤𝑚 {𝑎𝑖 / ∑𝑚 𝑗=1

Take 𝑀 > 𝑝𝑗𝑖 𝛼𝑗 , 𝑏𝑗 / ∑𝑛𝑖=1 𝑞𝑖𝑗 𝛽𝑖 }; then by (𝐻4 ) we have 𝑀 > 1. Thus, there exists 0 < 𝜆 0 < min{𝜖1 , 𝜖2 , . . . , 𝜖𝑛 , 𝜖1 , . . . , 𝜖𝑚 } such that, for 0 < 𝜆 ≤ 𝜆 0 ,

𝑗 = 1, 2, . . . , 𝑚,

which imply that

1 1 [𝑚 − ∑𝑝 𝛼 exp (𝜆 (𝛾 + sup𝜇 (𝑠)))] < 0, 𝑀 𝑎𝑖 − 𝜆 𝑗=1 𝑗𝑖 𝑗 𝑠∈T ] [

1 [𝑚 ∑𝑝 𝛼 exp (𝜆 (𝛾 + sup𝜇 (𝑠)))] < 1, 𝑎𝑖 − 𝜆 𝑗=1 𝑗𝑖 𝑗 𝑠∈T ] [ 𝑖 = 1, 2, . . . , 𝑛,

𝑖 = 1, 2, . . . , 𝑛,

(56)

𝑛

1 [∑𝑞 𝛽 exp (𝜆 (𝜌 + sup𝜇 (𝑠)))] < 1, 𝑏𝑗 − 𝜆 𝑖=1 𝑖𝑗 𝑖 𝑠∈T 𝑗 = 1, 2, . . . , 𝑚.

𝑗 = 1, 2, . . . , 𝑚. +

It is easy to see that ⊖𝜆 ∈ R and 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨󵄨 󵄩 󵄩 󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝜓𝑖 (𝑡)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ≤ 𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 , 𝑡 ∈ [−V, 0]T ,

Multiplying (49) by 𝑒−𝑎𝑖 (𝑡0 , 𝜎(𝑠)) and integrating on [𝑡0 , 𝑡]T , by Lemma 12, we get

𝑖 = 1, 2, . . . , 𝑛,

󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨V𝑗 (𝑡)󵄨󵄨󵄨 = 󵄨󵄨󵄨𝜓𝑛+𝑗 (𝑡)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩 ≤ 𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩󵄩󵄩𝜓󵄩󵄩󵄩󵄩 , 󵄨 󵄨 󵄨 󵄨 𝑡 ∈ [−V, 0]T ,

= 𝑢𝑖 (𝑡0 ) 𝑒−𝑎𝑖 (𝑡, 𝑡0 )

󵄩 󵄩 ≤ 𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ,

𝑡

+ ∫ 𝑒−𝑎𝑖 (𝑡, 𝜎 (𝑠)) 𝑡0

𝑚

⋅ ( ∑𝑝𝑗𝑖 (𝑠) [𝑓𝑗 (V𝑗 (𝑠 − 𝛾𝑗𝑖 (𝑠)) + 𝑦𝑗∗ (𝑠 − 𝛾𝑗𝑖 (𝑠))) 𝑗=1

−𝑓𝑗 (𝑦𝑗∗ (𝑠 − 𝛾𝑗𝑖 (𝑠)))]) Δ𝑠, 𝑖 = 1, 2, . . . , 𝑛. (57)

(60)

𝑗 = 1, 2, . . . , 𝑚,

which imply that 󵄨 󵄩 󵄨 󵄨 󵄨 󵄩󵄩 ∗ 󵄩󵄩𝑧 (𝑡) − 𝑧 (𝑡)󵄩󵄩󵄩 = max {󵄨󵄨󵄨𝑢𝑖 (𝑡)󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨V𝑗 (𝑡)󵄨󵄨󵄨󵄨} 1≤𝑖≤𝑛,1≤𝑗≤𝑚

𝑢𝑖 (𝑡)

(59)

𝑛 1 1 − [∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜆 (𝜌 + sup𝜇 (𝑠)))] < 0, 𝑀 𝑏𝑗 − 𝜆 𝑖=1 𝑠∈T

(61)

𝑡 ∈ [−V, 0]T .

Next, we claim that 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ 󵄩󵄩𝑧 (𝑡) − 𝑧 (𝑡)󵄩󵄩 ≤ 𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩 , ∀𝑡 ∈ (0, +∞)T . (62) If (62) is not true, then there must be some 𝑡1 ∈ (0, +∞)T , 𝑝 > 1 and some 𝑘 such that 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 ∗ ∗ 󵄩󵄩𝑧 (𝑡1 ) − 𝑧 (𝑡1 )󵄩󵄩 = 󵄨󵄨𝑧𝑘 (𝑡1 ) − 𝑧𝑘 (𝑡1 )󵄨󵄨 = 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩 , (63) 󵄩󵄩 󵄩 󵄩 󵄩 ∗ 󵄩󵄩𝑧 (𝑡) − 𝑧 (𝑡)󵄩󵄩󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡, 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 , ∀𝑡 ∈ [−V, 𝑡1 ]T .

(64)

The Scientific World Journal

11

By (57)–(64) and (𝐻2 )–(𝐻4 ), we obtain 󵄨󵄨 󵄨 󵄨󵄨𝑢𝑖 (𝑡1 )󵄨󵄨󵄨

󵄩 󵄩 ≤ 𝑒−𝑎𝑖 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 𝑡

1 󵄩 󵄩 + ∫ 𝑝𝑀 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠)) 𝑡 0

𝑚 } ⋅ ( ∑ 𝑝𝑗𝑖 𝛼𝑗 exp (𝜆 (𝛾 + sup𝜇 (𝑠))))} 𝑠∈T 𝑗=1 } 󵄩󵄩 󵄩󵄩 < 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩 , 󵄨󵄨 󵄨 󵄨󵄨V𝑗 (𝑡1 )󵄨󵄨󵄨 󵄨 󵄨 󵄩 󵄩 ≤ 𝑒−𝑏𝑗 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 𝑡1

𝑚

⋅ ( ∑𝑝𝑗𝑖 𝛼𝑗 𝑒⊖𝜆 (𝑠 − 𝛾𝑗𝑖 (𝑠) , 𝑡0 )) Δ𝑠 𝑗=1

󵄩 󵄩 + ∫ 𝑝𝑀 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 𝑒−𝑏𝑗 (𝑡1 , 𝜎 (𝑠)) 𝑡 0

𝑛

󵄩 󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩

⋅ (∑𝑞𝑖𝑗 𝛽𝑖 𝑒⊖𝜆 (𝑠 − 𝜌𝑖𝑗 (𝑠) , 𝑡0 )) Δ𝑠 𝑖=1

󵄩 󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩

{ 1 ⋅{ 𝑒 (𝑡 , 𝑡 ) 𝑒 (𝑡 , 𝑡 ) 𝑝𝑀 −𝑎𝑖 1 0 ⊖𝜆 0 1 {

⋅{

𝑡1

+ ∫ 𝑒−𝑎𝑖 (𝑡1 , 𝜎 (𝑠)) 𝑒𝜆 (𝑡1 , 𝜎 (𝑠)) 𝑡0

𝑚 } ⋅ ( ∑𝑝𝑗𝑖 𝛼𝑗 𝑒⊖𝜆 (𝑠 − 𝛾, 𝜎 (𝑠))) Δ𝑠} 𝑗=1 } 󵄩󵄩 󵄩󵄩 < 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩

{1 ⋅ { 𝑒−𝑎𝑖 ⊕𝜆 (𝑡1 , 𝑡0 ) 𝑀 {

𝑡1

+ ∫ 𝑒−𝑏𝑗 (𝑡1 , 𝜎 (𝑠)) 𝑒𝜆 (𝑡1 , 𝜎 (𝑠)) 𝑡0

𝑛

⋅ (∑𝑞𝑖𝑗 𝛽𝑖 𝑒⊖𝜆 (𝑠 − 𝜌, 𝜎 (𝑠))) Δ𝑠} 𝑖=1

󵄩 󵄩 < 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 ⋅{

𝑚

+ ( ∑𝑝𝑗𝑖 𝛼𝑗 exp (𝜆 (𝛾 + sup𝜇 (𝑠)))) 𝑗=1

𝑠∈T

𝑛

𝑠∈T

𝑖=1

} ⋅ ∫ 𝑒−𝑎𝑖 ⊕𝜆 (𝑡1 , 𝜎 (𝑠)) Δ𝑠} 𝑡0 } 󵄩󵄩 󵄩󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩

𝑡1

⋅ ∫ 𝑒−𝑏𝑗 ⊕𝜆 (𝑡1 , 𝜎 (𝑠)) Δ𝑠} 𝑡0

󵄩 󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩

{1 ⋅ { 𝑒−𝑎𝑖 ⊕𝜆 (𝑡1 , 𝑡0 ) 𝑀 {

⋅{

1 (𝑡 , 𝑡 ) 𝑒 𝑀 −𝑏𝑗 ⊕𝜆 1 0 𝑛

𝑚

+ ( ∑𝑝𝑗𝑖 𝛼𝑗 exp (𝜆 (𝛾 + sup𝜇 (𝑠))))

+ (∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜆 (𝜌 + sup𝜇 (𝑠))))

𝑚 { 1 1 ⋅ {[ − ( ∑ 𝑝𝑗𝑖 𝛼𝑗 exp (𝜆 (𝛾 + sup𝜇 (𝑠))))] 𝑀 𝑎𝑖 − 𝜆 𝑗=1 𝑠∈T {[ ]

𝑠∈T

𝑖=1

𝑠∈T

1 − 𝑒−𝑎𝑖 ⊕𝜆 (𝑡1 , 𝑡0 ) } ⋅ } 𝑎𝑖 − 𝜆 } 󵄩󵄩 󵄩󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩

1 ⋅ 𝑒−𝑎𝑖 ⊕𝜆 (𝑡1 , 𝑡0 ) + 𝑎𝑖 − 𝜆

1 (𝑡 , 𝑡 ) 𝑒 𝑀 −𝑏𝑗 ⊕𝜆 1 0

+ (∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜆 (𝜌 + sup𝜇 (𝑠))))

𝑡1

𝑗=1

1 𝑒 (𝑡 , 𝑡 ) 𝑒 (𝑡 , 𝑡 ) 𝑝𝑀 −𝑏𝑗 1 0 ⊖𝜆 0 1

1 − 𝑒−𝑏𝑗 ⊕𝜆 (𝑡1 , 𝑡0 ) } } 𝑏𝑗 − 𝜆 } 󵄩󵄩 󵄩󵄩 ≤ 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩𝜓󵄩󵄩 ⋅

𝑛 { 1 1 ⋅ {[ − (∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜆 (𝜌 + sup𝜇 (𝑠))))] 𝑀 𝑏𝑗 − 𝜆 𝑖=1 𝑠∈T {[ ]

⋅ 𝑒−𝑏𝑗 ⊕𝜆 (𝑡1 , 𝑡0 ) +

1 𝑏𝑗 − 𝜆

12

The Scientific World Journal 𝑛

Example 1. In (67), take T = R: 󵄨 󵄨 𝑎1 (𝑡) = 11 + 󵄨󵄨󵄨󵄨cos (√2𝑡)󵄨󵄨󵄨󵄨 ,

⋅ (∑𝑞𝑖𝑗 𝛽𝑖 exp (𝜆 (𝜌 + sup𝜇 (𝑠))))} 𝑠∈T

𝑖=1

󵄩 󵄩 < 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 . (65)

𝑏1 (𝑡) = 9 − |cos 𝑡| ,

Equations in (65) imply that 󵄨󵄨 󵄨 󵄩 󵄩 ∗ 󵄨󵄨𝑧𝑘 (𝑡1 ) − 𝑧𝑘 (𝑡1 )󵄨󵄨󵄨 < 𝑝𝑀𝑒⊖𝜆 (𝑡1 , 𝑡0 ) 󵄩󵄩󵄩𝜓󵄩󵄩󵄩 , ∀𝑘 ∈ {1, 2, . . . , 𝑛 + 𝑚} ,

𝑏2 (𝑡) = 8 + sin 𝑡2 , (66)

which contradicts (63), and so (62) holds. Hence, the 𝐶1 almost periodic solution of system (1) is globally exponentially stable. Global exponential stability implies that the 𝐶1 almost periodic solution is unique. Remark 18. In [17, 25, 26, 29], the existence and stability of almost periodic solutions are studied for several classes of neural networks on almost periodic time scales. However, the almost periodic time scales used in [17, 25, 26, 29] are a kind of periodic time scales. So, the methods and the results of this paper are essentially new.

5. Some Examples

2

𝑥𝑖Δ (𝑡) = − 𝑎𝑖 (𝑡) 𝑥𝑖 (𝑡) + ∑𝑝𝑗𝑖 (𝑡) 𝑓𝑗 (𝑦𝑗 (𝑡 − 𝛾𝑗𝑖 (𝑡)))

𝐼2 (𝑡) = 4𝐽2 (𝑡) =

𝑡 ∈ T, 𝑖 = 1, 2, (67)

2

𝑖=1

(𝑝𝑗𝑖 (𝑡))2×2

1 |sin 𝑡| 14 ), 1 |sin 𝑡| 28

(𝑞𝑖𝑗 (𝑡))2×2

1 |sin 𝑡| 6 =( 1 |sin 𝑡| 12

1 |cos 𝑡| 12 ). 1 |sin 𝑡| 24

+ max {𝐿 1 , 𝐿 2 } =

where 3

cos 𝑥 + 5 , 18

2 − sin4 𝑥 𝑔1 (𝑥) = , 16 𝑔2 (𝑥) =

3 − sin6 𝑥 . 24

4

(69) ,

23 23 + ≈ 0.784 < 1 = 𝑟0 , 88 44

𝑎1 3 11 < 11 = 𝑎1 , < = 56 23 𝑎1 + 𝑎1

𝑎2 3 11 0 < Π2 = < 11 = 𝑎2 , < = 112 23 𝑎2 + 𝑎2

𝑡 ∈ T, 𝑗 = 1, 2,

cos3 𝑥 + 3 , 𝑓2 (𝑥) = 12

sin (√2𝑡) + cos (√2𝑡)

1 |cos 𝑡| 7 =( 1 |cos 𝑡| 14

0 < Π1 =

+ ∑𝑞𝑖𝑗 (𝑡) 𝑔𝑖 (𝑥𝑖 (𝑡 − 𝜌𝑖𝑗 (𝑡))) + 𝐽𝑗 (𝑡) ,

𝑓1 (𝑥) =

cos 𝑡 + √3 sin 𝑡 , 8

𝑎2 + 𝑎2 𝑏1 + 𝑏1 𝑏2 + 𝑏2 } { 𝑎1 + 𝑎1 𝜂1 , 𝜂2 , 𝜂1 , 𝜂2 } max { 𝑎1 𝑎2 𝑏1 𝑏2 } {

𝑗=1

𝑦𝑗Δ (𝑡) = −𝑏𝑗 (𝑡) 𝑦𝑗 (𝑡)

𝐼1 (𝑡) = 2𝐽1 (𝑡) =

Let 𝛾𝑗𝑖 , 𝜌𝑖𝑗 (𝑖, 𝑗 = 1, 2) : R → R be arbitrary almost periodic functions; then, (𝐻2 )-(𝐻3 ) hold. Let 𝛼1 = 𝛼2 = 𝛽1 = 𝛽2 = 1/4; then, (𝐻1 ) holds. Next, let us check (𝐻4 ); if we take 𝑟0 = 1, then

Consider the following neural network:

+ 𝐼𝑖 (𝑡) ,

𝑎2 (𝑡) = 12 − |sin 𝑡| ,

(68)

0 < Π1 =

𝑏1 1 8 < 8 = 𝑏1 , < = 16 17 𝑏1 + 𝑏1

0 < Π2 =

𝑏2 1 8 < 8 = 𝑏2 . < = 32 17 𝑏2 + 𝑏2

(70)

Thus, (𝐻4 ) holds for 𝑟0 = 1. By Theorems 15 and 17, system (67) has a unique 𝐶1 -almost periodic solution in the region 󵄩 󵄩 𝐸 = {𝜑 ∈ B : 󵄩󵄩󵄩𝜑󵄩󵄩󵄩B ≤ 1} , (71) which is globally exponentially stable (see Figures 1–4).

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13 0.3

2.5

0.2 y3 -axis

3

States

2

0.1

1.5

0

1

−0.1 0.3

0.2

x2 0.1 -ax is

0.5 0 −0.5

0

−0.1 −0.1

0

0.1 is x 1-ax

0.2

0.3

Figure 3: Responses of 𝑥1 , 𝑥2 , 𝑦2 with time 𝑡. 0

20

40

60

80

100

Time t

y1 y2

0.2

Figure 1: Responses of 𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 with continuous time 𝑡.

y2 -axis

x1 x2

0.3

0.1 0 −0.1 0.1

0.1

0 y1 axi −0.05 s

y1 -axis

0.05 0

−0.1 −0.1

0

0.2 0.1 s i x 2-ax

0.3

Figure 4: Responses of 𝑥2 , 𝑦1 , 𝑦2 .

−0.05 −0.1 0.3

0.05

0.2

0.1 x2 axis

0

−0.1 −0.1

0

0.1 is x 1-ax

0.2

0.3

Figure 2: Responses of 𝑥1 , 𝑥2 , 𝑦1 .

𝑎2 + 𝑎2 𝑏1 + 𝑏1 𝑏2 + 𝑏2 } { 𝑎1 + 𝑎1 𝜂1 , 𝜂2 , 𝜂1 , 𝜂2 } max { 𝑎1 𝑎2 𝑏1 𝑏2 } { 231 11 + max {𝐿 1 , 𝐿 2 } = + ≈ 0.395 < 1 = 𝑟0 , 1920 40 𝑎1 5 8 0 < Π1 = < 0.8 = 𝑎1 , < = 112 17 𝑎1 + 𝑎1

Example 2. In (67), take T = Z: 󵄨 󵄨 𝑎1 (𝑡) = 0.9 − 0.1 󵄨󵄨󵄨󵄨sin (√3𝑡)󵄨󵄨󵄨󵄨 ,

𝑎2 (𝑡) = 0.8 + 0.1cos2 𝑡,

𝑏1 (𝑡) = 0.6 − 0.1 |sin 𝑡| , 4

𝑏2 (𝑡) = 0.5 + 0.1 cos 𝑡 , sin 𝑡 + √3 cos 𝑡 , 16 √2 sin 𝑡 + √2 cos 𝑡 , 𝐼2 (𝑡) = 2𝐽2 (𝑡) = 32 1 2 1 sin 𝑡 |sin 𝑡| 7 (𝑝𝑗𝑖 (𝑡))2×2 = ( 17 1 󵄨󵄨 󵄨) , 󵄨󵄨sin (√2𝑡)󵄨󵄨󵄨 |cos 𝑡| 󵄨 󵄨 28 14 1 1 cos2 𝑡 |sin 𝑡| 8 24 (𝑞𝑖𝑗 (𝑡))2×2 = ( 1 ). 1 |sin 𝑡| |cos 𝑡| 48 16

Let 𝛾𝑗𝑖 , 𝜌𝑖𝑗 (𝑖, 𝑗 = 1, 2) : Z → Z be arbitrary almost periodic functions; then, (𝐻2 )-(𝐻3 ) hold. Let 𝛼1 = 𝛼2 = 𝛽1 = 𝛽2 = 1/4; then, (𝐻1 ) holds. Next, let us check (𝐻4 ); if we take 𝑟0 = 1, then

𝐼1 (𝑡) = 𝐽1 (𝑡) =

0 < Π2 =

𝑎2 3 8 < 0.8 = 𝑎2 , < = 56 17 𝑎2 + 𝑎2

0 < Π1 =

𝑏1 7 5 < 0.5 = 𝑏1 , < = 192 11 𝑏1 + 𝑏1

0 < Π2 =

𝑏2 5 5 < 0.5 = 𝑏2 . < = 192 11 𝑏2 + 𝑏2

(72)

(73)

Thus, (𝐻4 ) holds for 𝑟0 = 1. By Theorems 15 and 17, system (67) has a unique 𝐶1 -almost periodic solution in the region 󵄩 󵄩 𝐸 = {𝜑 ∈ B : 󵄩󵄩󵄩𝜑󵄩󵄩󵄩B ≤ 1} , (74) which is globally exponentially stable (see Figures 5–8).

14

The Scientific World Journal 0.3

2.5

0.2 y2 -axis

3

States

2

0.1 0

1.5

−0.1 0.3

1

0.2

0.5 0 −0.5

0.1 x2 axis

0

−0.1 −0.1

0

0.1 is x 1-ax

0.2

0.3

Figure 7: Responses of 𝑥1 , 𝑥2 , 𝑦2 with time 𝑡. 0

20

40

60

80

100

Time t

0.3 y1 y2

0.2 y2 -axis

x1 x2

Figure 5: Responses of 𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 with discrete time 𝑡.

0.1 0 −0.1 0.1

0.05

0.1

0 y1 axi −0.05 s −0.1 −0.1

y1 -axis

0.05 0

0

0.1 is x 2-ax

0.2

0.3

Figure 8: Responses of 𝑥2 , 𝑦1 , 𝑦2 .

−0.05 −0.1 0.3

Acknowledgments 0.2

x2 - 0.1 axis

0

−0.1 −0.1

0

0.1 is x 1-ax

0.2

0.3

Figure 6: Responses of 𝑥1 , 𝑥2 , 𝑦1 .

The authors thank the referees for their careful reading of the paper, support, and insightful comments. This work is supported by the National Natural Sciences Foundation of China under Grant 11361072.

References 6. Conclusion In this paper, by using calculus theory on time scales, a fixed point theorem, and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability of 𝐶1 -almost periodic solutions for a class of neural networks with time-varying delays on a new type of almost periodic time scales are established. To the best of our knowledge, this is the first time to study the existence of 𝐶1 -almost periodic solutions of BAM neural networks on time scales. Our methods that are used in this paper can be used to study other types of neural networks, such as Cohen-Grossberg neural networks and fuzzy cellular neural networks.

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

[1] J. Cao and L. Wang, “Exponential stability and periodic oscillatory solution in BAM networks with delays,” IEEE Transactions on Neural Networks, vol. 13, no. 2, pp. 457–463, 2002. [2] X. Liao, K. Wong, and S. Yang, “Convergence dynamics of hybrid bidirectional associative memory neural networks with distributed delays,” Physics Letters A, vol. 316, no. 1-2, pp. 55–64, 2003. [3] X. F. Liao and J. B. Yu, “Qualitative analysis of bi-directional associative memory with time delay,” International Journal of Circuit Theory and Applications, vol. 26, no. 3, pp. 219–229, 1998. [4] Y. K. Li, “Global exponential stability of BAM neural networks with delays and impulses,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 279–285, 2005. [5] Y. Li, X. Chen, and L. Zhao, “Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales,” Neurocomputing, vol. 72, no. 7–9, pp. 1621–1630, 2009. [6] H. Zhao, “Global stability of bidirectional associative memory neural networks with distributed delays,” Physics Letters A, vol. 297, no. 3-4, pp. 182–190, 2002.

The Scientific World Journal [7] J. Liang and J. Cao, “Exponential stability of continuous-time and discrete-time bidirectional associative memory networks with delays,” Chaos, Solitons & Fractals, vol. 22, no. 4, pp. 773– 785, 2004. [8] Y. Wang, P. Lin, and L. Wang, “Exponential stability of reactiondiffusion high-order Markovian jump Hopfield neural networks with time-varying delays,” Nonlinear Analysis. Real World Applications, vol. 13, no. 3, pp. 1353–1361, 2012. [9] X. F. Liao and J. B. Yu, “Qualitative analysis of bidrectional associative memory with time delays,” International Journal of Circuit Theory and Applications, vol. 26, no. 3, pp. 219–229, 1998. [10] J. Zhang and Z. Gui, “Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 602–613, 2009. [11] Y. Li and C. Wang, “Existence and global exponential stability of equilibrium for discrete-time fuzzy BAM neural networks with variable delays and impulses,” Fuzzy Sets and Systems, vol. 217, pp. 62–79, 2013. [12] X. Yang, Q. Song, Y. Liu, and Z. Zhao, “Uniform stability analysis of fractional-order BAM neural networks with delays in the leakage terms,” Abstract and Applied Analysis, vol. 2014, Article ID 261930, 16 pages, 2014. [13] S. Lakshmanan, J. H. Park, T. H. Lee, H. Y. Jung, and R. Rakkiyappan, “Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 9408–9423, 2013. [14] A. Zhang, J. Qiu, and J. She, “Existence and global exponential stability of periodic solution for high-order discrete-time BAM neural networks,” Neural Networks, vol. 50, pp. 98–109, 2014. [15] X. Li and J. Jia, “Global robust stability analysis for BAM neural networks with time-varying delays,” Neurocomputing, vol. 120, pp. 499–503, 2013. [16] J. Thipcha and P. Niamsup, “Global exponential stability criteria for bidirectional associative memory neural networks with time-varying delays,” Abstract and Applied Analysis, vol. 2013, Article ID 576721, 13 pages, 2013. [17] C. Wang and Y. Li, “Almost periodic solutions to cohen-grossberg neural networks on time scales,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, vol. 20, no. 3, pp. 359–377, 2013. [18] Y. Yu and M. Cai, “Existence and exponential stability of almostperiodic solutions for high-order Hopfield neural networks,” Mathematical and Computer Modelling, vol. 47, no. 9-10, pp. 943–951, 2008. [19] W. Ding and L. Wang, “2𝑁 almost periodic attractors for CohenGrossberg-type BAM neural networks with variable coefficients and distributed delays,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 322–342, 2011. [20] B. Liu and L. Huang, “Positive almost periodic solutions for recurrent neural networks,” Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 830–841, 2008. [21] B. Xu and R. Yuan, “The existence of positive almost periodic type solutions for some neutral nonlinear integral equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 4, pp. 669–684, 2005. [22] Z. Teng, “On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays,” Journal of Mathematical Analysis and Applications, vol. 249, no. 2, pp. 433–444, 2000.

15 [23] B. Liu, “New results on the positive almost periodic solutions for a model of hematopoiesis,” Nonlinear Analysis: Real World Applications, vol. 17, pp. 252–264, 2014. [24] Y. K. Li and C. Wang, “Almost periodic functions on time scales and applications,” Discrete Dyamics in Nature and Society, vol. 2011, Article ID 727068, 20 pages, 2011. [25] Y. Li, C. Wang, and X. Li, “ Existence and global exponential stability of almost periodic solution for high-order BAM neural networks with delays on time scales,” Neural Processing Letters, vol. 39, pp. 247–268, 2014. [26] C. Wang, “Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 8, pp. 2828–2842, 2014. [27] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨auser, Boston, Mass, USA, 2001. [28] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2003. [29] T. Liang, Y. Yang, Y. Liu, and L. Li, “Existence and global exponential stability of almost periodic solutions to Cohen-Grossberg neural networks with distributed delays on time scales,” Neurocomputing, vol. 123, pp. 207–215, 2014. [30] L. Yang, Y. Li, and W. Wu, “𝐶𝑛 -almost periodic functions and an application to a Lasota-Wazewska model on time scales,” Journal of Applied Mathematics, vol. 2014, Article ID 321328, 10 pages, 2014. [31] Y. K. Li and C. Wang, “Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales,” Abstract and Applied Analysis, vol. 2011, Article ID 341520, 22 pages, 2011.