Existence and Exponential Stability of Periodic Solution of High-Order Hopfield Neural Network with Delays on Time Scales

Abstract

On time scales, by using the continuation theorem of coincidence degree theory and constructing some suitable Lyapunov functions, the periodicity and the exponential stability are investigated for a class of delayed high-order Hopfield neural networks (HHNNs), which are new and complement of previously known results. Finally, an example is given to show the effectiveness of the proposed method and results.

1. Introduction

Consider the following HHNNs with time-varying delays: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛, (1.1) where 𝑛 corresponds to the number of units in a neural network, 𝑥_𝑖(𝑡) corresponds to the state vector of the 𝑖th unit at the time 𝑡, 𝑐_𝑖(𝑡) represents the rate with which the 𝑖th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖𝑗𝑙(𝑡) are the first- and second-order connection weights of the neural network, 𝛾_𝑖𝑗(𝑡)≥0,𝜎_𝑖𝑗𝑙(𝑡)≥0, and 𝑣_𝑖𝑗𝑙(𝑡)≥0 correspond to the transmission delays, 𝐼_𝑖(𝑡) denote the external inputs at time 𝑡, and 𝑓_𝑗 and 𝑔_𝑗 are the activation functions of signal transmission.

Due to the fact that high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the object of intensive analysis by numerous authors in the recent years. In particular, there have been extensive results on the problem of the existence and stability of equilibrium points and periodic solutions of HHNNs (1.1) in the literature. We refer the reader to [1–9] and the references cited therein. In fact, both continuous and discrete systems are very important in implementing and applications. The theory of calculus on time scales (see [10, 11] and references cited therein) was initiated by Stefan Hilger in his Ph.D. thesis in 1988 [12] in order to unify continuous and discrete analysis, and it has a tremendous potential for application and has recently received much attention since his foundational work. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations.

Our purpose of this paper is to consider the model 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑡∈𝕋, (1.2) where 𝕋 is an 𝜔-periodic time scale which has the subspace topology inherited from the standard topology on ℝ,𝐼(𝑡)=(𝐼₁(𝑡),𝐼₂(𝑡),…,𝐼_𝑛(𝑡))^𝑇 is an input periodic vector function with period 𝜔; that is, there exists 𝜔>0 such that 𝐼_𝑖(𝑡+𝜔)=𝐼_𝑖(𝑡)(𝑖=1,2,…,𝑛) for all 𝑡∈(0,+∞)⋂𝕋, and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈ℝ^𝑛,𝑓(𝑥)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂),…,𝑓_𝑛(𝑥_𝑛))^𝑇, and 𝑔(𝑥)=(𝑔₁(𝑥₁),𝑔₂(𝑥₂),…,𝑔_𝑛(𝑥_𝑛))^𝑇 is the activation function of the neurons.

System (1.2) is supplemented with initial values given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈[−𝜃,0]𝕋,𝑖=1,2,…,𝑛, (1.3) where 𝜑_𝑖(⋅) denotes continuous 𝜔-periodic function defined on [−𝜃,0]⋂𝕋,𝜃=max{𝛾,𝜎,𝑣},𝛾=max_{1≤𝑖,𝑗≤𝑛}{𝛾⁺_𝑖𝑗},𝜎=max_{1≤𝑖,𝑗,𝑙≤𝑛}{𝜎⁺_𝑖𝑗𝑙},𝑣=max_{1≤𝑖,𝑗,𝑙≤𝑛}{𝑣⁺_𝑖𝑗𝑙},𝛾⁺_𝑖𝑗=max_{⋂𝕋𝑡∈[0,𝜔]}𝛾_𝑖𝑗(𝑡),𝜎⁺_𝑖𝑗𝑙=max_{⋂𝕋𝑡∈[0,𝜔]}𝜎_𝑖𝑗𝑙(𝑡),𝑣⁺_𝑖𝑗𝑙=max_{⋂𝕋𝑡∈[0,𝜔]}𝑣_𝑖𝑗𝑙(𝑡),𝑖,𝑗,𝑙=1,2,…,𝑛. To the best of our knowledge, this is the first paper to study the stability and existence of periodic solutions of (1.2).

Throughout this paper, we assume the following.

(𝐻₁)For 𝑖,𝑗,𝑙=1,2,…,𝑛,𝑐_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗𝑙(𝑡),𝐼_𝑖(𝑡),𝛾_𝑖𝑗(𝑡),𝜎_𝑖𝑗𝑙(𝑡),𝑣_𝑖𝑗𝑙(𝑡) are positive continuous periodic functions with period 𝜔>0, and 𝑐_𝑖(𝑡) is regressive.(𝐻₂)There exist positive constants 𝑀_𝑗,𝑁_𝑗,𝑗=1,2,…,𝑛 such that |𝑓_𝑗(𝑥)|≤𝑀_𝑗,|𝑔_𝑗(𝑥)|≤𝑁_𝑗 for 𝑗=1,2,…,𝑛,𝑥∈ℝ.(𝐻₃)Functions 𝑓_𝑗(𝑢),𝑔_𝑗(𝑢)(𝑗=1,2,…,𝑛) satisfy the Lipschitz condition; that is, there exist constants 𝐿_𝑗,𝐻_𝑗>0 such that |𝑓_𝑗(𝑢₁)−𝑓_𝑗(𝑢₂)|≤𝐿_𝑗|𝑢₁−𝑢₂|,|𝑔_𝑗(𝑢₁)−𝑔_𝑗(𝑢₂)|≤𝐻_𝑗|𝑢₁−𝑢₂|,𝑗=1,2,…,𝑛.

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},𝜇(𝑡)=𝜎(𝑡)−𝑡. (2.1)

A point 𝑡∈𝕋 is called left-dense if 𝑡>inf𝕋 and 𝜌(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, right-dense if 𝑡<sup𝕋 and 𝜎(𝑡)=𝑡, and right-scattered if 𝜎(𝑡)>𝑡. If 𝕋 has a left-scattered maximum 𝑚, then 𝕋^𝑘=𝕋⧵{𝑚}; otherwise 𝕋^𝑘=𝕋. If 𝕋 has a right-scattered minimum 𝑚, then 𝕋_𝑘=𝕋⧵{𝑚}; otherwise 𝕋_𝑘=𝕋.

A function 𝑓∶𝕋→ℝ is right-dense continuous provided that it is continuous at right-dense point in 𝕋 and its left-side 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), 𝑦^Δ(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)]−𝑦^Δ(𝑡)[𝜎(𝑡)−𝑠]|<𝜀|𝜎(𝑡)−𝑠| (2.2) 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.

A function 𝑟∶𝕋→ℝ is called regressive if 1+𝜇(𝑡)𝑟(𝑡)≠0 (2.3) for all 𝑡∈𝕋^𝑘.

If 𝑟 is regressive function, then the generalized exponential function 𝑒_𝑟 is defined by 𝑒_𝑟(𝑡,𝑠)=exp{^𝑡_𝑠𝜉_𝜇(𝜏)(𝑟(𝜏))Δ𝜏}for𝑠,𝑡∈𝕋, (2.4) with the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ,ifℎ≠0,𝑧,ifℎ=0. (2.5) Let 𝑝,𝑞∶𝕋→ℝ be two regressive functions, we define 𝑝𝑝⊕𝑞∶=𝑝+𝑞+𝜇𝑝𝑞,⊝𝑝∶=−1+𝜇𝑝,𝑝⊝𝑞∶=𝑝⊕(⊝𝑞). (2.6) Then the generalized exponential function has the following properties.

Lemma 2.1 (see [10]). Assume that 𝑝,𝑞∶𝕋→ℝ are two regressive functions, then (i)𝑒₀(𝑡,𝑠)≡1 and 𝑒_𝑝(𝑡,𝑡)≡1;(ii)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(iii)𝑒_𝑝(𝑡,𝜎(𝑠))=𝑒_𝑝(𝑡,𝑠)/(1+𝜇(𝑠)𝑝(𝑠));(iv)1/𝑒_𝑝(𝑡,𝑠)=𝑒_⊝𝑝(𝑡,𝑠);(v)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊝𝑝(𝑠,𝑡);(vi)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);(vii)𝑒_𝑝(𝑡,𝑠)𝑒_𝑞(𝑡,𝑠)=𝑒_𝑝⊕𝑞(𝑡,𝑠);(viii)𝑒_𝑝(𝑡,𝑠)/𝑒_𝑞(𝑡,𝑠)=𝑒_𝑝⊝𝑞(𝑡,𝑠).

Lemma 2.2 (see [10]). Assume that 𝑓,𝑔∶𝕋→ℝ are delta differentiable at 𝑡∈𝕋^𝑘, then (𝑓𝑔)^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝜎(𝑡)). (2.7)

Let 𝑦 be right-dense continuous. If 𝑌^Δ(𝑡)=𝑦(𝑡), then one defines the delta integral by ^𝑡_𝑎𝑦(𝑠)Δ𝑠=𝑌(𝑡)−𝑌(𝑎). (2.8)

Lemma 2.3. If 𝑎,𝑏∈𝕋,𝛼,𝛽∈ℝ and 𝑓,𝑔∈𝐶(𝕋,ℝ), then (1)∫^𝑏_𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽∫^𝑏_𝑎𝑔(𝑡)Δ𝑡;(2)If 𝑓(𝑡)≥0 for all 𝑎≤𝑡<𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0;(3)If |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)∶={𝑡∈𝕋∶𝑎≤𝑡<𝑏}, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡.

In this paper, one assumes that 𝑘=min{[0,+∞)⋂𝕋}. Clearly, from Lemma 2.3, one can obtain Lemma 2.4.

Lemma 2.4. If 𝑓,𝑔∈𝐶(𝕋,ℝ), and 𝑓(𝑡)≤𝑔(𝑡) on 𝑡∈[𝑘,𝑘+𝜔), then ∫^𝑘+𝜔_𝑘𝑓(𝑡)Δ𝑡≤∫^𝑘+𝜔_𝑘𝑔(𝑡)Δ𝑡.

In the proof of our main result, one will use the following three lemmas which can be found in [13, 14].

Lemma 2.5 (see [13]). Let 𝑡₁,𝑡₂∈[𝑘,𝑘+𝜔]⋂𝕋, and 𝑡∈𝕋. If 𝑓∶𝕋→ℝ is 𝜔-periodic, then 𝑓(𝑡)≤𝑓(𝑡₁)+∫^𝑘+𝜔_𝑘|𝑓^Δ(𝑠)|Δ𝑠, and 𝑓(𝑡)≥𝑓(𝑡₂)−∫^𝑘+𝜔_𝑘|𝑓^Δ(𝑠)|Δ𝑠.

Lemma 2.6 ([14], Cauchy-Schwarz inequality on time scales). Let 𝑎,𝑏∈𝕋. For 𝑟𝑑-continuous functions 𝑓,𝑔∶[𝑎,𝑏]→ℝ, one has (∫^𝑏_𝑎|𝑓(𝑡)𝑔(𝑡)|Δ𝑡)²≤∫^𝑏_𝑎|𝑓(𝑡)|²Δ𝑡∫^𝑏_𝑎|𝑔(𝑡)|²Δ𝑡.

Lemma 2.7. Let 𝑟∶𝕋→ℝ be right-dense continuous and regressive. 𝑎∈𝕋, and 𝑦_𝑎∈ℝ. The unique solution of the initial value problem 𝑦^Δ(𝑡)=𝑟(𝑡)𝑦(𝑡)+ℎ(𝑡),𝑦(𝑎)=𝑦_𝑎 (2.9) is given by 𝑦(𝑡)=𝑒_𝑟(𝑡,𝑎)𝑦_𝑎+^𝑡_𝑎𝑒_𝑟(𝑡,𝜎(𝑠))ℎ(𝑠)Δ𝑠. (2.10)

For the sake of convenience, one introduces the following notations:

𝑐_𝑖=1𝜔^𝑘+𝜔_𝑘𝑐_𝑖(𝑡)Δ𝑡,𝑎_𝑖𝑗=1𝜔^𝑘+𝜔_𝑘𝑎_𝑖𝑗(𝑡)Δ𝑡,𝐼_𝑖=1𝜔^𝑘+𝜔_𝑘𝐼_𝑖(𝑡)Δ𝑡,𝑏_𝑖𝑗𝑙=1𝜔^𝑘+𝜔_𝑘𝑏_𝑖𝑗𝑙(𝑡)Δ𝑡,𝑐⁺_𝑖=max_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝑐_𝑖(𝑡)|,𝑐⁻_𝑖=min_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝑐_𝑖𝑎(𝑡)|,_+𝑖𝑗=max_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝑎_𝑖𝑗(𝑡)|,𝑏⁺_𝑖𝑗𝑙=max_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝑏_𝑖𝑗𝑙(𝑡)|,𝐼⁺_𝑖=max_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝐼_𝑖𝐴(𝑡)|,_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑖,𝑗,𝑙=1,2,…𝑛. (2.11) Obliviously, for system (1.2), finding the periodic solutions is equivalent to finding those of the following boundary-value problem:

𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑡∈[𝑘,𝑥𝑘+𝜔]∩𝕋,𝑖=1,2,…,𝑛,_𝑖(𝑘)=𝑥_𝑖(𝑘+𝜔),𝑖=1,2,…,𝑛. (2.12) Now, one states Mawhin's continuous theorem [15].

Theorem 2.8. Let 𝑋 and 𝑍 be two Banach spaces and let 𝐿 be a Fredholm mapping of index zero. let Ω⊂𝕏 be an open bounded set and let 𝑁∶Ω→ℤ be a continuous operator which is 𝐿-compact on Ω. Assume that (1)for each 𝜆∈(0,1),𝑥∈𝜕Ω∩Dom𝐿,𝐿𝑥≠𝜆𝑁𝑥;(2)for each 𝑥∈𝜕Ω∩Ker𝐿,𝑄𝑁𝑥≠0;(3)deg(𝐽𝑁𝑄𝑥,Ω∩Ker𝐿,0)≠0, where 𝐽𝑄𝑁∶Ker𝐿→Ker𝐿.Then 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩Dom𝐿.

In order to apply Theorem 2.8 to system (2.12), we first define

𝑋=𝑍={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝐶(𝕋,ℝ^𝑛)∶𝑥(𝑡+𝜔)=𝑥(𝑡),𝑡∈𝕋},‖𝑥‖=‖(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇‖=^𝑛_𝑖=1max_{𝑡∈[𝑘,⋂𝕋𝑘+𝜔]}|𝑥_𝑖(𝑡)| (2.13) for any 𝑥∈𝑋(or𝑍). Then 𝑋 and 𝑍 are Banach spaces with the norm ‖⋅‖. Let 𝑁𝑥=(𝐴₁(𝑡),𝐴₂(𝑡),…,𝐴_𝑛(𝑡))^𝑇,𝑥∈𝑋𝐿𝑥=(𝑥^Δ₁,𝑥^Δ₂,…,𝑥^Δ_𝑛)^𝑇1𝑃𝑥=(𝜔^𝑘+𝜔_𝑘𝑥₁1(𝑡)Δ𝑡,𝜔^𝑘+𝜔_𝑘𝑥₂1(𝑡)Δ𝑡,…,𝜔^𝑘+𝜔_𝑘𝑥_𝑛(𝑡)Δ𝑡)^𝑇,1𝑄𝑧=(𝜔^𝑘+𝜔_𝑘𝑧₁1(𝑡)Δ𝑡,𝜔^𝑘+𝜔_𝑘𝑧₂1(𝑡)Δ𝑡,…,𝜔^𝑘+𝜔_𝑘𝑧_𝑛(𝑡)Δ𝑡)^𝑇,𝑧∈𝑍. (2.14)(2.15)(2.16) Then it follows that

Ker𝐿={(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝕏∶(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇∈ℝ^𝑛1,𝑡∈𝕋},Im𝐿={𝑧∈ℤ∶(𝜔^𝑘+𝜔_𝑘𝑧₁1(𝑡)Δ𝑡,𝜔^𝑘+𝜔_𝑘𝑧₂1(𝑡)Δ𝑡,…,𝜔^𝑘+𝜔_𝑘𝑧_𝑛(𝑡)Δ𝑡)^𝑇=0} (2.17) is closed in 𝑍, dim Ker𝐿=𝑛=codimIm𝐿. It is not difficult to show that 𝑃 and 𝑄 are continuous and satisfy Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im(𝐼−𝑄). It is easy to see that Im𝐿 is closed in 𝑍, which leads to the following lemma.

Lemma 2.9. Let 𝐿 and 𝑁 be defined by (2.14) and (2.15), respectively, then 𝐿 is a Fredholm operator of index zero.

Lemma 2.10. Let 𝐿 and 𝑁 be defined by (2.14) and (2.15), respectively, suppose that Ω is an open bounded subset of Dom𝐿, then 𝑁 is 𝐿-compact on Ω.

Proof. Through an easy computation, we find that the inverse 𝐾_𝑃∶Im𝐿→Ker𝑃⋂Dom𝐿 of 𝐿_𝑃 has the form (𝐾_𝑃𝑧)(𝑡)=^𝑡_𝑘1𝑧(𝑠)Δ𝑠−𝜔^𝑘+𝜔_𝑘^𝑡_𝑘𝑧(𝑠)Δ𝑠Δ𝑡. (2.18) Thus, the expression of 𝑄𝑁𝑥 is 1𝑄𝑁𝑥=(𝜔^𝑘+𝜔_𝑘𝐴₁1(𝑡)Δ𝑡,𝜔^𝑘+𝜔_𝑘𝐴₂1(𝑡)Δ𝑡,…,𝜔^𝑘+𝜔_𝑘𝐴_𝑛(𝑡)Δ𝑡)^𝑇, (2.19) and then 𝐾_𝑃(𝐼−𝑄)𝑁𝑥=^𝑡𝑘𝐴₁⋮(𝑠)Δ𝑠_𝑡𝑘𝐴_𝑛−1(𝑠)Δ𝑠𝜔^𝑘+𝜔_𝑘^𝑡_𝑘𝐴₁⋮1(𝑠)Δ𝑠Δ𝑡𝜔^𝑘+𝜔_𝑘^𝑡_𝑘𝐴_𝑛−((𝑠)Δ𝑠Δ𝑡𝑡−𝑘𝜔−1(𝜔)^{2𝑘+𝜔𝑘}(𝑡−𝑘)Δ𝑡)^𝑘+𝜔_𝑘𝐴₁⋮((𝑠)Δ𝑠𝑡−𝑘𝜔−1(𝜔)^{2𝑘+𝜔𝑘}(𝑡−𝑘)Δ𝑡)^𝑘+𝜔_𝑘𝐴_𝑛.(𝑠)Δ𝑠 (2.20) Thus, 𝑄𝑁 and 𝐾_𝑃(𝐼−𝑄)𝑁 are continuous. Since 𝑋 is a Banach space, it is not difficult to show that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact. Moreover, 𝑄𝑁(Ω) is bounded. Thus, 𝑁 is 𝐿-compact on Ω for any open bounded set Ω⊂𝑋. The proof of Lemma 2.10 is completed.

3. Existence of Periodic Solution

In this section, we study the existence of periodic solution of (1.2) based on Mawhin's continuation theorem.

Theorem 3.1. Assume that (𝐻₁)-(𝐻₂) hold, then system (1.2) has at least one 𝜔-periodic solution.

Proof. Based on the Lemma 2.9 and Lemma 2.10, now, what we need to do is just to search for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation 𝐿𝑥=𝜆𝑁𝑥,𝜆∈(0,1), we have 𝑥^Δ_𝑖(𝑡)=𝜆[−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑡∈𝕋[𝑘,𝑥𝑘+𝜔]],𝑖=1,2,…,𝑛,_𝑖(𝑘)=𝑥_𝑖(𝑘+𝜔),𝑖=1,2,…,𝑛. (3.1) For the sake of convenience, defined ‖𝑥‖₂ by ‖𝑥‖₂=(^𝑘+𝜔_𝑘|𝑥(𝑡)|²Δ𝑡)^1/2 (3.2) for 𝑥∈𝐶(𝕋,ℝ). Suppose that (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑋 is a solution of system (3.1) for a certain 𝜆∈(0,1). Integrating (3.1) over [𝑘,𝑘+𝜔], we obtain ^𝑘+𝜔_𝑘𝐴_𝑖(𝑡)Δ𝑡=0. (3.3) Hence ^𝑘+𝜔_𝑘𝑐_𝑖(𝑠)𝑥_𝑖(𝑠)Δ𝑠=[^𝑘+𝜔_𝑘^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡)))+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡)]Δ𝑡,𝑖=1,2,…,𝑛. (3.4) Let 𝜁_𝑖,𝜂_𝑖∈𝕋⋂[𝑘,𝑘+𝜔], such that 𝑥_𝑖(𝜁_𝑖)=inf_{⋂[𝑡∈𝕋𝑘,𝑘+𝜔]}𝑥_𝑖(𝑡),𝑥_𝑖(𝜂_𝑖)=sup_{⋂[𝑡∈𝕋𝑘,𝑘+𝜔]}𝑥_𝑖(𝑡). Then by (3.4) andLemma 2.4, we have 𝜔𝑐_𝑖𝑥_𝑖(𝜁_𝑖)≤^𝑘+𝜔_𝑘[|^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗+(𝑠)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑠)𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙(𝑠)))+𝐼_𝑖≤(𝑠)|]Δ𝑠_{𝑘+𝜔𝑘𝑛𝑗=1}|𝑎_𝑖𝑗(𝑠)||𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗+(𝑠)))|Δ𝑠_{𝑘+𝜔𝑘𝑛𝑛𝑗=1𝑙=1}|𝑏_𝑖𝑗𝑙(𝑠)‖𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))‖𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙(𝑠)))|Δ𝑠+^𝑘+𝜔_𝑘𝐼_𝑖(𝑠)Δ𝑠≤𝜔[^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖],𝑖=1,2,…,𝑛. (3.5) Hence 𝑥_𝑖(𝜁_𝑖)≤(1/𝑐_𝑖){[∑^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖]}∶=𝐵_𝑖,𝑖=1,2,…,𝑛. By (3.4) and Lemma 2.4, we can also have 𝜔𝑐_𝑖𝑥_𝑖(𝜂_𝑖)≥−^𝑘+𝜔_𝑘[|^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗+(𝑠)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑠)𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙(𝑠)))+𝐼_𝑖(𝑠)|]Δ𝑠≥−^𝑘+𝜔_𝑘^𝑛_𝑗=1|𝑎_𝑖𝑗(𝑠)||𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗−(𝑠)))|Δ𝑠_{𝑘+𝜔𝑘𝑛𝑛𝑗=1𝑙=1}|𝑏_𝑖𝑗𝑙(𝑠)||𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))||𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙(𝑠)))|Δ𝑠−^𝑘+𝜔_𝑘|𝐼_𝑖(𝑠)|Δ𝑠≥−𝜔[^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖],𝑖=1,2,…,𝑛. (3.6) Hence 𝑥_𝑖(𝜂_𝑖)≥(−1/𝑐_𝑖){[∑^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗+∑^𝑛_𝑗=1∑^𝑛_𝑙=1𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖]}=−𝐵_𝑖,𝑖=1,2,…,𝑛. From (3.1), (3.4), and Lemma 2.6, we have ^𝑘+𝜔_𝑘|𝑥^Δ_𝑖≤(𝑡)|Δ𝑡_{𝑘+𝜔𝑘}|𝑐_𝑖(𝑡)||𝑥_𝑖(𝑡)|Δ𝑡+^𝑘+𝜔_𝑘^𝑛_𝑗=1|𝑎_𝑖𝑗(𝑡)||𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))|Δ𝑡_{𝑘+𝜔𝑘𝑛𝑛𝑗=1𝑙=1}|𝑏_𝑖𝑗𝑙(𝑡)||𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))||𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))|Δ𝑡+^𝑘+𝜔_𝑘|𝐼_𝑖(𝑡)|Δ𝑡≤(^𝑘+𝜔_𝑘|𝑐_𝑖(𝑡)|²Δ𝑡)^1/2(^𝑘+𝜔_𝑘|𝑥_𝑖(𝑡)|²Δ𝑡)^1/2+^𝑛_𝑗=1(^𝑘+𝜔_𝑘|𝑎_𝑖𝑗(𝑡)|²Δ𝑡)^1/2(^𝑘+𝜔_𝑘|𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡)))|²Δ𝑡)^1/2+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙(^𝑘+𝜔_𝑘|𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))|²Δ𝑡)^1/2(^𝑘+𝜔_𝑘|𝑔_𝑙(𝑥_𝑙(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))|²Δ𝑡)^1/2+𝐼⁺_𝑖𝜔≤(𝜔)^1/2𝑐⁺_𝑖‖𝑥_𝑖‖₂+^𝑛_𝑗=1𝜔𝑎⁺_𝑖𝑗𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1(𝜔)𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖𝜔. (3.7) From Lemma 2.7 and (3.1), for 𝑖=1,2,…,𝑛, we can obtain 𝑥_𝑖(𝑡)=𝑒_{−𝜆𝑐𝑖(𝑡)}(𝑡,𝑘)𝑥_𝑖(+𝑘)_𝑡𝑘𝜆𝑒_{−𝜆𝑐𝑖(𝑡)}(𝑡,𝜎(𝑠))[^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗+(𝑠)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑠)𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙(𝑠)))+𝐼_𝑖(𝑠)]Δ𝑠. (3.8) Hence |𝑥_𝑖(𝑡)|≤|𝑥_𝑖(𝑘)|+^𝑛_𝑗=1^𝑘+𝜔_𝑘|𝑎_𝑖𝑗(𝑠)||𝑓_𝑗(𝑥_𝑗(𝑠−𝛾_𝑖𝑗+(𝑠)))|Δ𝑠_{𝑛𝑛𝑗=1𝑙=1𝑘+𝜔𝑘}|𝑏_𝑖𝑗𝑙(𝑠)||𝑔_𝑗(𝑥_𝑗(𝑠−𝜎_𝑖𝑗𝑙(𝑠)))||𝑔_𝑙(𝑥_𝑙(𝑠−𝑣_𝑖𝑗𝑙+(𝑠)))|Δ𝑠_{𝑘+𝜔𝑘}|𝐼_𝑖(𝑠)|Δ𝑠≤|𝑥_𝑖(𝑘)|+^𝑛_𝑗=1𝑎_𝑖𝑗𝑁_𝑗𝜔+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙𝜔+𝐼_𝑖𝜔=∶𝑢_𝑖,𝑖=1,2,…,𝑛, (3.9) that is, ‖𝑥_𝑖‖₂=(^𝑘+𝜔_𝑘|𝑥_𝑖(𝑡)|²Δ𝑡)^1/2≤𝑢_𝑖(𝜔)^1/2,𝑖=1,2,…,𝑛. (3.10) Substituting (3.10) into (3.7), we have ^𝑘+𝜔_𝑘|𝑥^Δ_𝑖(𝑡)|Δ𝑡≤𝜔𝑐⁺_𝑖𝑢_𝑖+^𝑛_𝑗=1𝜔𝑎⁺_𝑖𝑗𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1(𝜔)𝑏⁺_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙+𝐼⁺_𝑖𝜔,𝑖=1,2,…,𝑛. (3.11) From Lemma 2.5, we have 𝑥_𝑖(𝑡)≤𝑥_𝑖(𝜁_𝑖)+^𝑘+𝜔_𝑘|𝑥^Δ_𝑖𝑥(𝑡)|Δ𝑡,𝑖=1,2,…,𝑛,_𝑖(𝑡)≥𝑥_𝑖(𝜂_𝑖)−^𝑘+𝜔_𝑘|𝑥^Δ_𝑖(𝑡)|Δ𝑡,𝑖=1,2,…,𝑛. (3.12) From (3.5), (3.6) and (3.11), there exist positive constants 𝜉_𝑖(𝑖=1,2,…,𝑛) such that for 𝑡∈[𝑘,𝑘+𝜔]⋂𝕋,|𝑥_𝑖(𝑡)|≤𝜉_𝑖,𝑖=1,2,…,𝑛. Clearly, 𝜉_𝑖(𝑖=1,2,…,𝑛) is independent of 𝜆. Denote 𝐻^∗=∑^𝑛_𝑖=1𝜉_𝑖+𝐶, where 𝐶>0 is taken sufficiently large so that min_{1≤𝑖≤𝑛}𝑐_𝑖𝐻^∗>𝑛max_{1≤𝑖≤𝑛}(|𝐼_𝑖|+^𝑛_𝑗=1|𝑎_𝑖𝑗|𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1|𝑏_𝑖𝑗𝑙|𝑁_𝑗𝑁_𝑙). (3.13)
Now we take Ω={(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∶‖(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇‖<𝐻^∗}. Thus (1) of Theorem 2.8 is satisfied. When (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝜕Ω⋂ℝ^𝑛,(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is a constant vector in ℝ^𝑛 with |𝑥₁|+|𝑥₂|+⋯+|𝑥_𝑛|=𝐻^∗, then
𝑄𝑁(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇=(−𝑐_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖)_𝑛×1. (3.14) Therefore ‖𝑄𝑁(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇‖=^𝑛_𝑖=1|𝑐_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗−(𝑡)))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))−𝐼_𝑖|≥^𝑛_𝑖=1𝑐_𝑖|𝑥_𝑖(𝑡)|−^𝑛𝑛_𝑖=1𝑗=1|𝑎_𝑖𝑗|𝑀_𝑗−^𝑛𝑛_𝑖=1^𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙−^𝑛_𝑖=1|𝐼_𝑖|≥^𝑛_𝑖=1(𝑐_𝑖|𝑥_𝑖−(𝑡)|)_𝑛𝑖=1(|𝐼_𝑖|+^𝑛_𝑗=1|𝑎_𝑖𝑗|𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏_𝑖𝑗𝑙𝑁_𝑗𝑁_𝑙)≥min_{1≤𝑖≤𝑛}(𝑐_𝑖)^𝑛_𝑖=1|𝑥_𝑖(𝑡)|−𝑛max_{1≤𝑖≤𝑛}(|𝐼_𝑖|+^𝑛_𝑗=1|𝑎_𝑖𝑗|𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1|𝑏_𝑖𝑗𝑙|𝑁_𝑗𝑁_𝑙)>0. (3.15) Consequently, 𝑄𝑁(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇≠(0,0,…,0)^𝑇, (3.16) for (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝜕Ω⋂Ker𝐿. This satisfies condition (2) of Theorem 2.8. Define Ψ∶Ker𝐿×[0,1]→𝑋 by Ψ(𝑥₁,𝑥₂,…,𝑥_𝑛,𝜇)=−𝜇(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇+(1−𝜇)𝑄𝑁(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 (3.17) When (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝜕Ω⋂Ker𝐿,(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 is a constant vector in ℝ^𝑛 with ∑^𝑛_𝑖=1|𝑥_𝑖|=𝐻^∗, we easily have Ψ(𝑥₁,𝑥₂,…,𝑥_𝑛,𝜇)≠(0,0,…,0)^𝑇. Therefore deg(𝑄𝑁(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇,ΩKer𝐿,(0,0,…,0)^𝑇)=deg((−𝑥₁(𝑡),−𝑥₂(𝑡),…,−𝑥_𝑛(𝑡)^𝑇,ΩKer𝐿,(0,0,…,0)^𝑇)≠0. (3.18) Condition (3) of Theorem 2.8 is also satisfied. Thus, by Theorem 2.8 we can obtain that 𝐿𝑥=𝑁𝑥 has at least one solution in 𝑋. That is, system (1.2) has at least one 𝜔-periodic solution. The proof is complete.

4. Global Exponential Stability of Periodic Solution

In this section, we will construct suitable Lyapunov functions to study the global exponential stability of the periodic solution of (1.2) on time scales. So first we will introduce some definitions.

Definition 4.1 (see [12]). A function 𝑓 from 𝕋 to ℝ is positively regressive if 1+𝜇(𝑡)𝑓(𝑡)>0 for every 𝑡∈𝕋.

Denote ℜ⁺ is the set of positively regressive functions from 𝕋 to ℝ, and denote 𝕋⁺=[0,+∞)⋂𝕋.

Definition 4.2. The periodic solution 𝑥^∗(𝑡) of system (1.2) is said to be exponentially stable if there exists a positive constant 𝛼 with −𝛼∈ℜ⁺ such that for every 𝛿∈𝕋, there exists 𝑁=𝑁(𝛿)≥1 such that the solution 𝑥(𝑡) of (1.2) through (𝛿,𝜑(𝛿)) satisfies ‖𝑥(𝑡)−𝑥^∗(𝑡)‖≤𝑁‖𝜑(𝛿)−𝑥^∗(𝛿)‖𝑒_−𝛼(𝑡,𝛿),𝑡∈𝕋⁺ where ‖𝜑(𝛿)−𝑥^∗(𝛿)‖=∑^𝑛_𝑖=1max_{⋂𝕋𝛿∈[−𝜃,0]}|𝜑_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)|.

Theorem 4.3. Assume that (𝐻₁)-(𝐻₃) hold. Suppose further that there exists 𝑛 positive constants 𝜖_𝑖>0,𝑖=1,2,…,𝑛 such that (𝐻₄)−𝜖_𝑖𝑐⁻_𝑖+^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗𝜖_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙(𝐻_𝑗𝑁_𝑙𝜖_𝑗+𝐻_𝑙𝑁_𝑗𝜖_𝑙)<0,𝑖=1,2,…,𝑛, (4.1)then the 𝜔-periodic solution of system (1.2) is globally exponentially stable.

Proof. According to Theorem 3.1, we know that (1.2) has an 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇. Suppose that 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is an arbitrary solution of (1.2). Then it follows from system (1.2) that (𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡))^Δ=−𝑐_𝑖(𝑡)(𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡))+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)(𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡)))−𝑓^∗_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡))))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)(𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))))−𝑔_𝑗(𝑥^∗_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥^∗_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))),𝑖=1,2,…,𝑛, (4.2) with initial values given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈[−𝜃,0]𝕋,𝑖=1,2,…,𝑛, (4.3) where 𝜃 is defined as in(1.3). If (𝐻₄) holds, it can always find a smallenough constant 𝛼>0 satisfying ∀𝑡∈𝕋,1−𝜇(𝑡)𝛼>0, namely, −𝛼∈ℜ⁺ such that (−𝑐⁻_𝑖+𝛼)𝜖_𝑖+^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗𝑒_𝛼(𝑡,𝑡−𝛾_𝑖𝑗(𝑡))𝜖_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙[𝐻_𝑗𝑁_𝑙𝜖_𝑗𝑒_𝛼(𝑡,𝑡−𝜎_𝑖𝑗𝑙(𝑡))+𝐻_𝑙𝑁_𝑗𝜖_𝑙𝑒_𝛼(𝑡,𝑡−𝑣_𝑖𝑗𝑙(𝑡))]<0,𝑖=1,2,…,𝑛. (4.4) We define a Lyapunov function 𝐻=(ℎ₁,ℎ₂,…,ℎ_𝑛)^𝑇 by ℎ_𝑖=𝑒_𝛼(𝑡,𝛿)|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|,𝛿∈[−𝜃,0]⋂𝕋,𝑖=1,2,…,𝑛. In view of (4.2), weobtain [ℎ^Δ_𝑖(𝑡)]⁺=𝛼𝑒_𝛼(𝑡,𝛿)|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|+𝑒_𝛼(𝜎(𝑡),𝛿)sign(𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡))×{−𝑐_𝑖(𝑡)(𝑥_𝑖(𝑡)−𝑥^∗_𝑖+(𝑡))_𝑛𝑗=1𝑎_𝑖𝑗(𝑡)(𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡)))−𝑓_𝑗(𝑥^∗_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡))))_{𝑛𝑛𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)(𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))))−𝑔_𝑗(𝑥^∗_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥^∗_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))}≤𝛼𝑒_𝛼(𝑡,𝛿)|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|+𝑒_𝛼(𝜎(𝑡),𝛿)×{−𝑐_𝑖(𝑡)|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|+^𝑛_𝑗=1|𝑎_𝑖𝑗(𝑡)||𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡))−𝑥^∗_𝑗(𝑡−𝛾_𝑖𝑗(𝑡))|𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1|𝑏_𝑖𝑗𝑙(𝑡)|[𝐻_𝑗𝑁_𝑙|𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡))−𝑥^∗_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡))|+𝐻_𝑙𝑁_𝑗|𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))−𝑥^∗_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))|]}≤𝑒_𝛼(𝜎(𝑡),𝛿)×{(−𝑐_𝑖(𝑡)+𝛼)|𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|+^𝑛_𝑗=1|𝑎_𝑖𝑗(𝑡)||𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡))−𝑥^∗_𝑗(𝑡−𝛾_𝑖𝑗(𝑡))|𝑀_𝑗+^𝑛𝑛_𝑗=1𝑙=1|𝑏_𝑖𝑗𝑙(𝑡)|[𝐻_𝑗𝑁_𝑙|𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡))−𝑥^∗_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡))|+𝐻_𝑙𝑁_𝑗|𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))−𝑥^∗_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))|]}≤[1+𝜇(𝑡)𝛼]{(−𝑐⁻_𝑖+𝛼)ℎ_𝑖(𝑡)+^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗𝑒_𝛼(𝑡,𝑡−𝛾_𝑖𝑗(𝑡))ℎ_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡))_{𝑛𝑛𝑗=1𝑙=1}𝑏⁺_𝑖𝑗𝑙×[𝐻_𝑗𝑁_𝑙𝑒_𝛼(𝑡,𝑡−𝜎_𝑖𝑗𝑙(𝑡))ℎ_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡))+𝐻_𝑙𝑁_𝑗𝑒_𝛼(𝑡,𝑡−𝑣_𝑖𝑗𝑙(𝑡))ℎ_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡))]},𝑖=1,2…,𝑛. (4.5) Defining the curve 𝜌={𝜔(𝑙)∶𝜔_𝑖=𝜖_𝑖𝑙,𝑙>0,𝑖=1,2,…,𝑛} and the set Ω(𝜔)={𝑢∶0≤𝑢≤𝜔}, 𝑆_𝑖(𝜔)={𝑢∈Ω(𝜔)∶𝑢_𝑖=𝜔},𝑖=1,2…,𝑛. It is obvious that if 𝑙<𝑙, then Ω(𝜔(𝑙))⊂Ω(𝜔(𝑙)). We will prove that the zero solution of (4.2) is exponential stable, namely, there exists a constant 𝛽>0 such that ‖𝑥(𝑡)−𝑥^∗(𝑡)‖≤𝑁(𝛿)𝑒_−𝛽(𝑡,𝛿)‖𝜑(𝑡)−𝑥^∗(𝑡)‖,𝑡≥0. (4.6) Let 𝜖^𝑀=max_{1≤𝑖≤𝑛}{𝜖_𝑖},𝜖^𝑚=min_{1≤𝑖≤𝑛}{𝜀_𝑖},𝑙₀=(1−𝛿)|𝜑_𝑖−𝑥^∗_𝑖|₀/𝜖^𝑚, where −𝛿≥0 is a constant, |𝜑_𝑖−𝑥^∗_𝑖|₀=max_{1≤𝑖≤𝑛}{max_{⋂𝕋𝛿∈[−𝜃,0]}|𝜑_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)|}. Then, {|𝐻|∶|𝐻|=𝑒_𝛼(𝑡,𝛿)|𝜑(𝛿)−𝑥^∗(𝛿)|,−𝜃≤𝑡≤𝛿≤0}⊂Ω(𝜔(𝑙₀)), namely, |ℎ_𝑖(𝛿)|=𝑒_𝛼(𝑡,𝛿)|𝜑_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)|<𝜖_𝑖𝑙₀,−𝜃≤𝛿≤0,𝑖=1,2,…,𝑛. (4.7) We can claim that |ℎ_𝑖(𝑡)|<𝜖_𝑖𝑙₀ for 𝑡>0,𝑖=1,2,…,𝑛. If it is not true, then there exist some 𝑖 and 𝑡₁(𝑡_𝑖>0) such that |ℎ_𝑖(𝑡₁)|=𝜖_𝑖𝑙₀,[ℎ^Δ_𝑖(𝑡₁)]⁺≥0 and |ℎ_𝑖(𝑡)|≤𝜖_𝑖𝑙₀ for −𝜃≤𝑡≤𝑡₁,𝑖=1,2,…,𝑛. However, from (4.4) and (4.5), we get [ℎ^Δ_𝑖(𝑡₁)]⁺≤[1+𝜇(𝑡)𝛼][(−𝑐⁻_𝑖+𝛼)𝜖_𝑖+^𝑛_𝑗=1𝑎⁺_𝑖𝑗𝑀_𝑗𝑒_𝛼(𝑡,𝑡−𝛾_𝑖𝑗(𝑡))𝜖_𝑗+^𝑛𝑛_𝑗=1𝑙=1𝑏⁺_𝑖𝑗𝑙[𝐻_𝑗𝑁_𝑙𝑒_𝛼(𝑡,𝑡−𝜎_𝑖𝑗𝑙(𝑡))𝜖_𝑗+𝐻_𝑙𝑁_𝑗𝑒_𝛼(𝑡,𝑡−𝑣_𝑖𝑗𝑙(𝑡))𝜖_𝑙]]𝑙₀<0 (4.8) for 𝑡>0,𝑖=1,2,…,𝑛, this is a contradiction. So |ℎ_𝑖(𝑡)|<𝜖_𝑖𝑙₀, for 𝑡≥0. Also |𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|<𝑒_Θ𝛼(𝑡,𝛿)𝜖_𝑖𝑙₀≤1𝜖^𝑚𝑒_Θ𝛼(𝑡,𝛿)𝜖_𝑖(1−𝛿)|𝜑_𝑖−𝑥^∗_𝑖|₀,𝑖=1,2,…,𝑛,𝑡≥0, (4.9) which means that ‖𝑥(𝑡)−𝑥^∗1(𝑡)‖≤𝜖^𝑚𝑒_Θ𝛼(𝑡,𝛿)𝜖^𝑀(1−𝛿)‖𝜑(𝛿)−𝑥^∗(𝛿)‖. (4.10) Denote −𝛽=Θ𝛼=−𝛼/(1+𝜇𝛼)∈ℜ⁺,𝑁=𝑁(𝛿)=(𝜖^𝑀/𝜖^𝑚)(1−𝛿)>1, in view of (4.10), we have ‖𝑥(𝑡)−𝑥^∗(𝑡)‖≤𝑁𝑒_−𝛽(𝑡,𝛿)‖𝜑(𝑠)−𝑥^∗(𝑠)‖. (4.11) From Definition 4.2, the periodic solution of system (1.2) is globally exponentially stable. The proof is complete.

5. An Example

Let 𝕋=ℝ, in this case, 𝑥^Δ(𝑡)=𝑑𝑥(𝑡)/𝑑𝑡. Consider the following equation 𝑑𝑥_𝑖(𝑡)𝑑𝑡=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+²_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥_𝑗(𝑡−𝛾_𝑖𝑗+(𝑡)))_{22𝑗=1𝑙=1}𝑏_𝑖𝑗𝑙(𝑡)𝑔_𝑗(𝑥_𝑗(𝑡−𝜎_𝑖𝑗𝑙(𝑡)))𝑔_𝑙(𝑥_𝑙(𝑡−𝑣_𝑖𝑗𝑙(𝑡)))+𝐼_𝑖(𝑡),𝑖=1,2, (5.1) where 𝑓₁(𝑥₁1)=sin(2^1/2𝑥₁),𝑓₂(𝑥₂1)=sin(2^3/2𝑥₂𝑔),₁(𝑥₁1)=arctan(2^1/2𝑥₁),𝑔₂(𝑥₂1)=arctan(2^3/2𝑥₂). (5.2) Obviously, 𝑓_𝑖(𝑥_𝑖),𝑔_𝑖(𝑥_𝑖)(𝑖=1,2) satisfy (𝐻₁) and (𝐻₂), and 𝐻₁=𝐻₂=𝑀₁=𝑀₂=1,𝑁₁=𝑁₂=𝜋2. (5.3) Take 𝑎₁₁(𝑡)=1+cos(2𝜋𝑡),𝑎₁₂(𝑡)=2+cos(2𝜋𝑡),𝑎₂₁(𝑡)=2+cos(2𝜋𝑡),𝑎₂₂𝑐(𝑡)=3+cos(2𝜋𝑡),₁(𝑡)=20+5sin(2𝜋𝑡),𝑐₂(𝑡)=33+16sin(2𝜋𝑡),𝐼₁(𝑡)=1+sin(2𝜋𝑡),𝐼₂𝑏(𝑡)=1+cos(2𝜋𝑡),₁₁₁(𝑡)=𝑏₂₂₂1(𝑡)=4+14sin(2𝜋𝑡),𝑏₁₁₂(𝑡)=𝑏₂₁₂1(𝑡)=3+13𝑏cos(2𝜋𝑡),₁₂₁(𝑡)=𝑏₂₂₁1(𝑡)=5+15cos(2𝜋𝑡),𝑏₁₂₂(𝑡)=𝑏₂₁₁1(𝑡)=6+16sin(2𝜋𝑡). (5.4) One can verifies that (𝐻₁) is satisfied, and 𝜔=1,𝑐⁻₁=15,𝑐⁻₂=17,𝑎⁺₁₁=2,𝑎⁺₁₂=3,𝑎⁺₂₁=3,𝑎⁺₂₂=4,𝑏⁺₁₁₁=𝑏⁺₂₂₂=1/2,𝑏⁺₁₁₂=𝑏⁺₂₁₂=2/3,𝑏⁺₁₂₁=𝑏⁺₂₂₁=2/5,𝑏⁺₁₂₂=𝑏⁺₂₁₁=1/3, so, if we take 𝜖₁=𝜖₂=1, we can obtain −𝜖₁𝑐⁻₁+²_𝑗=1𝑎⁺_1𝑗𝑀_𝑗𝜖_𝑗+²²_𝑗=1𝑙=1𝑏⁺_1𝑗𝑙(𝐻_𝑗𝑁_𝑙𝜖_𝑗+𝐻_𝑙𝑁_𝑗𝜖_𝑙)=−15+5+1910𝜋<0,−𝜖₂𝑐⁻₂+²_𝑗=1𝑎⁺_2𝑗𝑀_𝑗𝜖_𝑗+²²_𝑗=1𝑙=1𝑏⁺_2𝑗𝑙(𝐻_𝑗𝑁_𝑙𝜖_𝑗+𝐻_𝑙𝑁_𝑗𝜖_𝑙)=−17+7+1910𝜋<0. (5.5) Condition (4.2) is satisfied. From Theorem 3.1 and 4.3, we know that (5.1) has at least one 1-periodic solution and this solution is exponential stability.

6. Conclusion

Sufficient conditions are derived to guarantee the stability and existence of periodic solutions for a class of delayed high-order Hopfield neural networks on time scales. To the best of our knowledge, the results presented here have been not appeared in the related literature. In fact, both continuous and discrete systems, are very important in implementing and applications. But it is troublesome to study the existence and stability of periodic solutions for continuous and discrete systems respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations. Also, our methods used in this paper may be applied to some other systems.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 1097118.