Stability and Existence of Periodic Solutions for Cellular Neural Networks with State Dependent Delays on Time Scales

Abstract

We study delayed cellular neural networks on time scales. Without assuming the boundedness of the activation functions, we establish the exponential stability and existence of periodic solutions. The results in this paper are completely new even in case of the time scale 𝕋=ℝ or ℤ and improve some of the previously known results.

1. Introduction

Consider the following cellular neural networks with state-dependent delays on time scales: 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗(𝑥(𝑡))+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡),𝑡∈𝕋, (1.1) where 𝑖=1,2,…,𝑛, 𝕋 is an 𝜔-periodic time scale which has the subspace topology inherited from the standard topology on ℝ, 𝑥^𝜎_𝑖(𝑡)=𝑥_𝑖(𝜎(𝑡)), 𝜎(𝑡) will be defined in the next section, 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)), 𝑛 corresponds to the number of units in the neural network, 𝑥_𝑖(𝑡) corresponds to the state of the 𝑖th unit at time 𝑡, 𝑓_𝑗(𝑥_𝑗(𝑡)) denotes the output of the 𝑗th unit on 𝑖th unit at time 𝑡, 𝑏_𝑖𝑗 denotes the strength of the 𝑗th unit on the 𝑖th unit at time 𝜉_𝑖𝑗(𝑡,𝑥(𝑡)), 𝐼_𝑖 denotes the external bias on the 𝑖th unit at time 𝑡, 𝑡−𝜉_𝑖𝑗(𝑡,𝑥(𝑡)) corresponds to the transmission delay along the axon of the 𝑗th unit, 𝑐_𝑖 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.

It is well known that the cellular neural networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories, especially in image processing and solving nonlinear algebraic equations. They have been widely studied both in theory and applications [1–3]. Many results for the existence of their periodic solutions and the exponential convergence properties for cellular neural networks have been reported in the literatures. See, for instance, [4–17] and references cited therein.

In fact, continuous and discrete systems are very important in implementation and applications. It is well known that the theory of time scales has received a lot of attention which was introduced by Stefan Hilger in order to unify continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems see [18–28].

When 𝕋=ℝ, 𝜉_𝑖𝑗(𝑡,⋅)≡𝑡−𝜏_𝑖𝑗(𝑡), (1.1) reduces to 𝑥^′_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (1.2) where 𝑖=1,2,…,𝑛. By using Mawhin’s continuation theorem and Liapunov functions, the authors [6, 14] obtained the existence and stability of periodic solutions of (1.2), respectively.

Furthermore, (1.1) also covers discrete system (for when 𝕋=ℤ,𝜉_𝑖𝑗(𝑛,⋅)≡𝑛−𝜏_𝑖𝑗(𝑛); see [15]) 𝑥_𝑖(𝑛+1)=𝑥_𝑖(𝑛)𝑒^{−𝑐𝑖(𝑛)ℎ}+𝜃_𝑖(ℎ)^𝑚_𝑗=1𝑎_𝑖𝑗(𝑛)𝑓_𝑗𝑥_𝑗(𝑛)+𝜃_𝑖(ℎ)^𝑚_𝑗=1𝑏_𝑖𝑗(𝑛)𝑓_𝑗𝑥_𝑗𝑛−𝜏_𝑖𝑗(𝑛)+𝜃_𝑖(ℎ)𝐼_𝑖(𝑛), (1.3) where 𝜃_𝑖(ℎ)=(1−𝑒^{−𝑐𝑖(𝑛)ℎ})/𝑐_𝑖(𝑛), 𝑖=1,2,…,𝑚. In [15], the author firstly obtained the discrete-time analogue of (1.3) by the semidiscretization technique [29, 30], and then some sufficient conditions for the existence and global asymptotical stability of periodic solutions of (1.3) were established by using Mawhin’s continuation theorem and Liapunov functions.

However, in [5, 13–15], the activation functions 𝑓_𝑗, 𝑗=1,2,…,𝑛 are assumed to be bounded. Our main purpose of this paper is to establish the stability and existence of periodic solutions of (1.1) without assuming the boundedness of the activation functions.

For the sake of convenience, we denote 𝑐_𝑖=max_{[]𝑡∈0,𝜔𝕋}𝑐_𝑖(𝑡),𝑐_𝑖=min_{[]𝑡∈0,𝜔𝕋}𝑐_𝑖(𝑡),𝑎_𝑖𝑗=max_{[]𝑡∈0,𝜔𝕋}𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗=max_{[]𝑡∈0,𝜔𝕋}𝑏_𝑖𝑗(𝑡),𝐼_𝑖=max_{[]𝑡∈0,𝜔𝕋}𝐼_𝑖(𝑡),𝑖,𝑗=1,2,…,𝑛. (1.4)

Throughout this paper, we assume that(H1)𝜉_𝑖𝑗(⋅,⋅)∈𝐶(𝕋×ℝ^𝑛,[0,∞)_𝕋) is 𝜔-periodic with respect to its first argument and satisfies 𝜉_𝑖𝑗(𝑡,𝑥)≤𝜏 for all (𝑡,𝑥)∈𝕋×ℝ^𝑛, 𝑐_𝑖∈𝐶(𝕋,(0,+∞)), 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, 𝐼_𝑖∈𝐶(𝕋,ℝ), 𝑖, 𝑗=1,2,…,𝑛 are 𝜔-periodic functions;(H2)𝑓_𝑗∈𝐶(ℝ,ℝ), and there exists a positive number 𝑀_𝑗 such that |𝑓_𝑗(𝑥)−𝑓_𝑗(𝑦)|⩽𝑀_𝑗|𝑥−𝑦| for all 𝑥, 𝑦∈ℝ, 𝑗=1,2,…,𝑛.

The initial conditions of system (1.1) are of the following form: 𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑖=1,2,…,𝑛,𝑠∈[−𝜏,0]_𝕋,𝜏=max_{1⩽𝑖⩽𝑛,1⩽𝑗⩽𝑛}max_{[]𝑡∈0,𝜔𝕋}𝜏_𝑖𝑗(𝑡), (1.5) where 𝜙_𝑖∈𝐶([−𝜏,0]_𝕋,ℝ), 𝑖=1,2,…,𝑛.

The organization of this paper is as follows. In Section 2, we introduce some lemmas and definitions and state some preliminary results needed in later sections, which will be used in latter sections. In Section 3, we will study the existence of periodic solutions of system (1.1) by using the method of coincidence degree. In Section 4, we will derive sufficient conditions to ensure that the periodic solutions of (1.1) are globally exponentially stable. In Section 5, an example is also provided to illustrate the effectiveness of the main results in Sections 3 and 4. The conclusions are drawn in Section 6.

2. Preliminaries

In this section, we will introduce some notations and definitions and state some preliminary results.

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. The set of continuous functions 𝑓∶𝕋→ℝ will be denoted by 𝐶(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|∀𝑠∈𝑈. (2.2)

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 𝑌^Δ(𝑡)=𝑦(𝑡), then we define the delta integral by ∫^𝑡_𝑎𝑦(𝑠)Δ𝑠=𝑌(𝑡)−𝑌(𝑎).

Definition 2.1 (see [31]). We say that a time scale 𝕋 is periodic if there exists 𝑝>0 such that if 𝑡∈𝕋, then 𝑡±𝑝∈𝕋. For 𝕋≠ℝ, the smallest positive 𝑝 is called the period of the time scale.

Definition 2.2 (see [31]). Let 𝕋≠ℝ be a periodic time scale with period 𝑝. We say that the function 𝑓∶𝕋→ℝ is periodic with period 𝜔 if there exists a natural number 𝑛 such that 𝜔=𝑛𝑝,𝑓(𝑡+𝜔)=𝑓(𝑡) for all 𝑡∈𝕋, and 𝜔 is the smallest number such that 𝑓(𝑡+𝜔)=𝑓(𝑡). If 𝕋=ℝ, we say that 𝑓 is periodic with period 𝜔>0 if 𝜔 is the smallest positive number such that 𝑓(𝑡+𝜔)=𝑓(𝑡) for all 𝑡∈𝕋.

If 𝕋 is 𝜔 periodic, then 𝜎(𝑡+𝜔)=𝜎(𝑡)+𝜔 and 𝜇(𝑡) is an 𝜔-periodic function.

Definition 2.3 (see [32]). A function 𝑝∶𝕋→ℝ is said to be regressive provided that 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋^𝑘, where 𝜇(𝑡)=𝜎(𝑡)−𝑡 is the graininess function. The set of all regressive rd-continuous functions 𝑓∶𝕋→ℝ is denoted by ℛ while the set ℛ⁺ is given by {𝑓∈ℛ∶1+𝜇(𝑡)𝑓(𝑡)>0} for all 𝑡∈𝕋. Let 𝑝∈ℛ. The exponential function is defined by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏, (2.3) where 𝜉_ℎ(𝑧) is the so-called cylinder transformation.

Let 𝑝,𝑞∶𝕋→ℝ be two regressive functions, and we define 𝑝𝑝⊕𝑞∶=𝑝+𝑞+𝜇𝑝𝑞,⊖𝑝∶=−.1+𝜇𝑝,𝑝⊖𝑞∶=𝑝⊕⊖𝑞 (2.4) Then the generalized exponential function has the following properties.

Lemma 2.4 (see [32]). Assume that 𝑝,𝑞∶𝕋→ℝ are two regressive functions, and then(1)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(2)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡); (3)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟).

Lemma 2.5 (see [32]). 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)). (2.5)

Lemma 2.6 (see [32]). 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+𝛼∫^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.

Lemma 2.7 (see [33]). Let 𝑡₁, 𝑡₂∈[0,𝜔]_𝕋. If 𝑥∶𝕋→ℝ is 𝜔 periodic, then 𝑥(𝑡)⩽𝑥𝑡₁+^𝜔₀𝑥^Δ(𝑠)Δ𝑠,𝑥(𝑡)⩾𝑥𝑡₂−^𝜔₀𝑥^Δ(𝑠)Δ𝑠. (2.6)

Definition 2.8. The periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇 of system (1.1) is said to be exponentially stable if there exists a positive constant 𝛼 with ⊖𝛼∈ℛ⁺ such that for every 𝛿∈[−𝜏,0]_𝕋, and there exists 𝑁=𝑁(𝛿)⩾1 such that the solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 of system (1.1) through (𝛿,𝑥(𝛿)) satisfies ^𝑛_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)⩽𝑁𝜙(𝛿)−𝑥^∗(𝛿)𝑒_⊖𝛼(𝑡,𝛿),𝑡∈𝕋⁺∶=[0,∞)_𝕋, (2.7) where ‖𝜙(𝛿)−𝑥^∗(𝛿)‖=∑^𝑛_𝑖=1max_{𝛿∈[−𝜏,0]𝕋}|𝜙_𝑖(𝛿)−𝑥^∗_𝑖(𝛿)|.

In order to show that there exists at least one 𝜔-periodic solution of system (1.1), we need the following concepts and result which are cited from [34].

Let 𝕏,𝕐 be Banach spaces, 𝐿∶Dom𝐿⊂𝕏→dim𝕐 be a linear mapping, and 𝑁∶𝕏→𝕐 be a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝕐. If 𝐿 is a Fredholm mapping of index zero and there exists continuous projector 𝑃∶𝕏→𝕏 and 𝑄∶𝕐→𝕐 such that Im𝑃=Ker𝐿,Ker𝑄=Im(𝐼−𝑄), it follows that mapping 𝐿|_{Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝕏→Im𝐿 is invertible. We denote the inverse of that mapping by 𝐾_𝑃. If Ω is an open bounded subset of 𝕏, the mapping 𝑁 will be called 𝐿 compact on Ω if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝕏 is compact.

Lemma 2.9 (see [34]). Let 𝕏,𝕐 be two Banach spaces and Ω⊂𝕏 be open bounded and symmetric with 0∈Ω. Suppose that 𝐿∶𝐷(𝐿)⊂𝕏→𝕐 is a linear Fredholm operator of index zero with 𝐷(𝐿)∩Ω≠∅, and 𝑁∶Ω→𝕐 is 𝐿 compact. Further, one also assumes that(H)𝐿𝑥−𝑁𝑥≠𝜆(−𝐿𝑥−𝑁(−𝑥)) for all 𝑥∈𝐷(𝐿)∩𝜕Ω, 𝜆∈(0,1].
Then equation 𝐿𝑥=𝑁𝑥 has at least one solution on 𝐷(𝐿)∩Ω.

Definition 2.10 (see [35]). A real matrix 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛 is said to be a nonsingular 𝑀 matrix if 𝑝_𝑖𝑗≤0, 𝑖, 𝑗=1,2,…,𝑛, 𝑖≠𝑗, and all successive principal minors of 𝑃 are positive.

For 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑚×𝑛∈ℝ^𝑚𝑛, 𝐴≥𝐵(𝐴>𝐵) means that each pair of corresponding elements of 𝐴 and 𝐵 such that 𝑎_𝑖𝑗≥𝑏_𝑖𝑗(𝑎_𝑖𝑗>𝑏_𝑖𝑗).

Lemma 2.11 (see [35]). Assume that 𝑃 is a nonsingular 𝑀 matrix and 𝑃𝑦≤ℎ, then 𝑦≤𝑃⁻¹ℎ.

3. Existence of Periodic Solutions

In this section, by Lemma 2.9, we will study the existence of at least one periodic solution of system (1.1).

Theorem 3.1. Suppose that (H1)-(H2) holds, 𝐸=(𝑒_𝑖𝑗)_𝑛×𝑛 is a nonsingular 𝑀 matrix, where 𝑒_𝑖𝑗𝑟=𝑧^𝑖+𝑑_𝑖𝑖𝑑𝑖=𝑗,_𝑖𝑗𝑖≠𝑗, (3.1) and 𝑟_𝑖=𝑐_𝑖(1−𝜔𝑐_𝑖)/(1+𝑐_𝑖𝜔), 𝑑_𝑖𝑗=−(𝑎_𝑖𝑗+𝑏_𝑖𝑗)𝑀_𝑗, 𝑖,𝑗=1,2,…,𝑛, then system (1.1) has at least one 𝜔-periodic solution.

Proof. Let 𝕏=𝕐=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛^𝑇∈𝐶(𝕋,ℝ)∶𝑥(𝑡+𝜔)=𝑥(𝑡),𝑡∈𝕋 (3.2) with the norm defined by ‖𝑥‖=max_{1≤𝑖≤𝑛}|𝑥_𝑖|₀, where |𝑥_𝑖|₀=max_{𝑡∈[0,𝜔]𝕋}|𝑥_𝑖(𝑡)|, then 𝕏 and 𝕐 are Banach spaces.
Set 𝐿𝑥=𝑥^Δ(𝑡)1,𝑃𝑥=𝑄𝑥=𝜔^𝜔₀𝑥(𝑡)Δ𝑡,𝑥∈𝕏, (3.3) and ℕ∶𝕏→𝕐𝑁𝑥_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡), (3.4) where 𝑖=1,2,…,𝑛. Obviously, Ker𝐿=ℝ^𝑛, Im𝐿={𝑥∈𝕏∶∫^𝜔₀𝑥(𝑠)Δ𝑠=0} is closed in 𝕐 and dimKer𝐿=𝑛=codimIm𝐿. (3.5) Hence, 𝐿 is a Fredholm mapping of index zero. Furthermore, similar to the proof of Theorem  3.4 in [21], one can easily show that 𝑁 is 𝐿 compact on Ω with any open bounded set Ω⊂𝑋. Corresponding to the operator equation 𝐿𝑥−𝑁𝑥=𝜆(−𝐿𝑥−𝑁(−𝑥)), 𝜆∈(0,1], we have 𝑥^Δ_𝑖(𝑡)=11+𝜆−𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡)−𝜆𝑐1+𝜆^𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗−𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗−𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡), (3.6) where 𝑖=1,2,…,𝑛.
Suppose that 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 is a solution of system (3.6) for some 𝜆∈(0,1]. In view of (3.6) and (H2), we have ^𝜔₀𝑥^Δ_𝑖(𝑡)Δ𝑡≤^𝜔₀𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡+^𝜔₀𝐼_𝑖(𝑡+1)Δ𝑡1+𝜆^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗((𝑡+𝜆𝑡,𝑥))Δ𝑡1+𝜆^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗−𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗−𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))Δ𝑡⩽^𝜔₀𝑐_𝑖𝑥^𝜎_𝑖(𝑡)Δ𝑡+^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗𝑥_𝑗(𝑡)+𝑓_𝑗(0+_𝑛𝑗=1)𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝑓_𝑗(0)+𝐼_𝑖⩽Δ𝑡𝑐_𝑖𝜔𝑥_𝑖0+𝜔^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝜔^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑓_𝑗(0)+𝐼_𝑖⩽𝑐_𝑖𝜔𝑥_𝑖0+𝜔^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝜔𝛼_𝑖,𝑖=1,2,…,𝑛, (3.7) where 𝛼_𝑖=∑^𝑛_𝑗=1(𝑎_𝑖𝑗+𝑏_𝑖𝑗)|𝑓_𝑗(0)|+𝐼_𝑖.
Integrating both sides of (3.6) from 0 to 𝜔, we obtain that ^𝜔₀𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)1Δ𝑡=1+𝜆^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡)−𝜆Δ𝑡1+𝜆^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗−𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗−𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡)Δ𝑡, (3.8) where 𝑖=1,2,…,𝑛. Then we have from (3.7) that ^𝜔₀𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡⩽^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))+𝐼_𝑖(𝑡)Δ𝑡⩽𝜔^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝜔𝛼_𝑖,𝑖=1,2,…,𝑛. (3.9) From Lemma 2.7, for any 𝑡^𝑖₁, 𝑡^𝑖₂∈[0,𝜔]_𝕋, 𝑖=1,2,…,𝑛, we have ^𝜔₀𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡⩽^𝜔₀𝑐_𝑖(𝑡)𝑥_𝑖𝑡^𝑖₁Δ𝑡+^𝜔₀𝑐_𝑖(𝑡)^𝜔₀𝑥^Δ_𝑖(𝑡)Δ𝑡Δ𝑡,_𝜔0𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡⩾^𝜔₀𝑐_𝑖(𝑡)𝑥_𝑖𝑡^𝑖₂Δ𝑡−^𝜔₀𝑐_𝑖(𝑡)^𝜔₀𝑥^Δ_𝑖(𝑡)Δ𝑡Δ𝑡,𝑖=1,2,…,𝑛. (3.10) Dividing by ∫^𝜔₀𝑐_𝑖(𝑡)Δ𝑡 on the two sides of the inequalities above, we obtain that −𝑥_𝑖𝑡^𝑖₁1⩽−∫^𝜔₀𝑐_𝑖(𝑡)Δ𝑡^𝜔0𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡+^𝜔₀𝑥^Δ_𝑖(𝑡𝑥)Δ𝑡,_𝑖𝑡^𝑖₂⩽1∫^𝜔₀𝑐_𝑖(𝑡)Δ𝑡^𝜔0𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡+^𝜔₀𝑥^Δ_𝑖(𝑡)Δ𝑡,𝑖=1,2,…,𝑛. (3.11)(3.12)
Let |𝑥_𝑖(𝑡^𝑖₀)|=max_{𝑡∈[0,𝜔]𝕋}|𝑥_𝑖(𝑡)|, 𝑖=1,2,…,𝑛. If for some 𝑖=1,2,…,𝑛, 𝑥_𝑖(𝑡^𝑖₀)⩾0, we choose 𝑡^𝑖₂=𝑡^𝑖₀. Hence max_{𝑡∈[0,𝜔]𝕋}|𝑥_𝑖(𝑡)|=𝑥_𝑖(𝑡^𝑖₂). From (3.12), we have 𝑐_𝑖𝑥_𝑖0⩽𝑐_𝑖1∫^𝜔₀𝑐_𝑖(𝑡)Δ𝑡^𝜔0𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡+^𝜔₀𝑥^Δ_𝑖(𝑡⩽1)Δ𝑡𝜔^𝜔₀𝑐_𝑖(𝑡)𝑥^𝜎_𝑖(𝑡)Δ𝑡+𝑐_𝑖^𝜔₀𝑥^Δ_𝑖(𝑡)Δ𝑡. (3.13) If for some 𝑖=1,2,…,𝑛, 𝑥_𝑖(𝑡^𝑖₀)⩽0, we choose 𝑡^𝑖₁=𝑡^𝑖₀. Hence max_{𝑡∈[0,𝜔]𝕋}|𝑥_𝑖(𝑡)|=−𝑥_𝑖(𝑡^𝑖₁). From (3.11), we also have (3.13).
By using (3.7) and (3.9) and (3.13), for 𝑖=1,2,…,𝑛, we obtain that 𝑐_𝑖𝑥_𝑖0⩽^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝛼_𝑖+𝑐_𝑖𝜔𝑥_𝑖0+𝑐_𝑖𝜔^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝑐_𝑖𝜔𝛼_𝑖, (3.14) so, 𝑐_𝑖1−𝜔𝑐_𝑖1+𝑐_𝑖𝜔𝑥_𝑖0⩽^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑀_𝑗𝑥_𝑗0+𝛼_𝑖, (3.15) namely, 𝑟_𝑖𝑥_𝑖0−^𝑛_𝑗=1𝑑_𝑖𝑗𝑥_𝑗0⩽𝛼_𝑖,𝑖=1,2,…,𝑛. (3.16) Denote the following: |𝑥|₀=𝑥₁₀,𝑥₂₀,…,𝑥_𝑛0^𝑇,𝐶=𝛼₁,𝛼₂,…,𝛼_𝑛^𝑇. (3.17) Thus (3.16) is rewritten in the matrix form 𝐸|𝑥|₀⩽𝐶. (3.18) From the conditions of Theorem 3.1, 𝐸 is a nonsingular 𝑀 matrix, therefore |𝑥|₀⩽𝐸⁻¹𝐶∶=𝐵₁,𝐵₂,…,𝐵_𝑛^𝑇, (3.19) that is, |𝑥_𝑖|₀⩽𝐵_𝑖, 𝑖=1,2,…,𝑛.
Take 𝐴=max_{1≤𝑖≤𝑛}𝐵_𝑖+1 and Ω={𝑥∈𝕏∶‖𝑥‖<𝐴}. (3.20) It is clear that Ω satisfies all the requirements in Lemma 2.9, and condition (H) is satisfied. In view of all the discussions above, we conclude from Lemma 2.9 that system (1.1) has at least one 𝜔-periodic solution. This completes the proof.

From Theorem 3.1, when 𝕋=ℝ,𝜉_𝑖𝑗(𝑡,⋅)≡𝑡−𝜏_𝑖𝑗(𝑡), we have the following.

Corollary 3.2. Under assumptions of Theorem 3.1, system (1.2) has at least one 𝜔-periodic solution.

When 𝕋=ℤ, 𝜉_𝑖𝑗(𝑛,⋅)≡𝑛−𝜏_𝑖𝑗(𝑛), from Theorem 3.1, we have the following.

Corollary 3.3. Under assumptions of Theorem 3.1, system (1.3) has at least one 𝜔-periodic solution.

4. Global Exponential Stability of the Periodic Solution

In this section, we will establish some results for the global exponential stability of the periodic solution of (1.1). The following lemma is essential.

Lemma 4.1 (see [36]). Let 𝑝∶𝕋→ℝ be rd-continuous and regressive. Suppose that 𝑓∶𝕋→ℝ is rd-continuous, 𝛿∈𝕋, and 𝑦₀∈ℝ. Then 𝑦 is the unique solution of the initial value problem 𝑦^Δ(𝑡)+𝑝(𝑡)𝑦^𝜎(𝑡)=ℎ(𝑡),𝑦(𝛿)=𝑦₀, (4.1) if and only if 𝑦(𝑡)=𝑒_⊖𝑝(𝑡,𝛿)𝑦₀+^𝑡_𝛿𝑒_⊖𝑝(𝑡,𝑠)ℎ(𝑠)Δ𝑠. (4.2)

Theorem 4.2. Assume (H₁), (H₂), and (H₄) hold. Furthermore, suppose that(H₅) for 𝑡∈[0,𝜔]_𝕋, 𝑖=1,2,…,𝑛, 𝑐_𝑖(𝑡)−𝑀>0, where 𝑀=max_{1≤𝑖≤𝑛}{∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗+∑^𝑛_𝑗=1𝑏_𝑖𝑗𝑀_𝑗}
holds. Then the 𝜔-periodic solution of system (1.1) is globally exponentially stable.

Proof. From Theorem 3.1, we see that system (1.1) has at least one 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 be an arbitrary solution of system (1.1). Let 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡), and then we obtain that 𝑦^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑦^𝜎_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓𝑥_𝑗(𝑡)−𝑓𝑥^∗_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))−𝑓_𝑗𝑥^∗_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡)),𝑖=1,2,…,𝑛 (4.3) with initial values given by 𝑦_𝑖(𝑠)=𝜑_𝑖(𝑠)−𝑥^∗_𝑖(𝑠),𝑠∈[−𝜏,0]_𝕋,𝑖=1,2,…,𝑛, (4.4) where each 𝜑_𝑖∈𝐶[(−𝜏,0]_𝕋,ℝ] is bounded.
Let ℎ(𝑡)=(ℎ₁(𝑡),ℎ₂(𝑡),…,ℎ_𝑛(𝑡))^𝑇 and ℎ_𝑖(𝑡)=^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓𝑥_𝑗(𝑡)−𝑓𝑥^∗_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡))−𝑓_𝑗𝑥^∗_𝑗𝜉_𝑖𝑗(𝑡,𝑥(𝑡)),𝑖=1,2,…,𝑛. (4.5) From Lemma 4.1, then 𝑦_𝑖(𝑡)=𝑒_⊖𝑐𝑖(𝑡,𝛿)𝑦₀+^𝑡_𝛿𝑒_⊖𝑐𝑖(𝑡,𝑠)ℎ_𝑖(𝑠)Δ𝑠,𝑖=1,2,…,𝑛,𝛿∈(−∞,0]_𝕋,𝑡≥𝛿 (4.6) is the unique solution of (4.3) with initial value 𝑦(𝛿)=𝑦₀. From (4.6), we have 𝑦_𝑖(𝑡)𝑒_⊖𝑐𝑖()≤𝑦𝑡,𝛿⁰+^𝑡_𝛿1𝑒_⊖𝑐𝑖()ℎ𝑠,𝛿^𝑖(𝑠)Δ𝑠,𝑖=1,2,…,𝑛,𝑡≥𝛿 (4.7) and, according to (4.5), we can obtain that ‖ℎ‖≤^𝑛_𝑗=1𝑎_𝑖𝑗𝑀_𝑗+^𝑛_𝑗=1𝑏_𝑖𝑗𝑀_𝑗𝑦≤𝑀𝑦,𝑖=1,2,…,𝑛,𝑡≥𝛿. (4.8) Thus, by (4.7), we have 𝑦𝑒_⊖𝑐𝑖()≤𝑦𝑡,𝛿⁰+^𝑡_𝛿𝑀𝑦𝑒_⊖𝑐𝑖(𝑠,𝛿)Δ𝑠,𝑖=1,2,…,𝑛,𝑡≥𝛿. (4.9) By virtue of Gronwall’s inequality ([32], Corollary  6.7), we obtain 𝑦𝑒_⊖𝑐𝑖()≤𝑦𝑡,𝛿⁰𝑒𝑀(𝑡,𝛿),𝑖=1,2,…,𝑛,𝑡≥𝛿. (4.10) Hence we have 𝑦≤𝑦₀𝑒𝑀(𝑡,𝛿)𝑒_⊖𝑐𝑖()≤𝑦𝑡,𝛿₀𝑒_{⊖𝑐𝑖⊕}𝑀()=𝑦𝑡,𝛿₀𝑒_{−(𝑐𝑖−}𝑀_{)/(1+𝜇𝑐𝑖)}()≤𝑦𝑡,𝛿₀𝑒_−𝛾(𝑡,𝛿),𝑡≥𝛿, (4.11) where 𝛾=min_{1≤𝑖≤𝑛}{(𝑐_𝑖−𝑀)/(1+𝜇𝑐_𝑖)}, 𝑖=1,2,…,𝑛. We can rewrite (4.10) as follows ^𝑛_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)≤𝜙(𝛿)−𝑥^∗(𝛿)𝑒_−𝛾(𝑡,𝛿),𝑡≥𝛿. (4.12) Therefore, the 𝜔-periodic solution of system (1.1) is globally exponentially stable. This completes the proof.

From Theorem 4.2, when 𝕋=ℝ,𝜉_𝑖𝑗(𝑡,⋅)≡𝑡−𝜏_𝑖𝑗(𝑡), we have the following.

Corollary 4.3. Under assumptions of Theorem 4.2, all the 𝜔-periodic solutions of system (1.2) are globally exponentially stable.

When 𝕋=ℤ, 𝜉_𝑖𝑗(𝑛,⋅)≡𝑛−𝜏_𝑖𝑗(𝑛), we have the following.

Corollary 4.4. Under assumptions of Theorem 4.2, all the 𝜔-periodic solutions of system (1.3) are globally exponentially stable.

Remark 4.5. Corollaries 3.2 and 4.3 improve the corresponding results obtained in [4–6, 13, 14] and are different from those in [8, 11, 12, 16, 17] and the references cited therein. Therefore, the results of this paper are new, and they complement previously known results. Corollaries 3.3 and 4.4 improve the results obtained in [15].

5. An Example

Consider the following cellular neural networks with delays: 𝑥^Δ_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+²_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+²_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐼_𝑖(𝑡), (5.1) where 𝑡∈𝕋, 𝑖=1, 2, 𝕋 is a 1-periodic time scale, 𝑐₁(𝑡)=1/4+(1/8)sin(2𝜋𝑡), 𝑐₂(𝑡)=1/3+(1/6)cos(2𝜋𝑡), 𝑎₁₁(𝑡)=sin²(𝜋𝑡), 𝑎₁₂(𝑡)=𝑎₂₁(𝑡)=(1/4)sin⁴(𝜋𝑡), 𝑎₂₂(𝑡)=|cos(𝜋𝑡)|, 𝑏₁₁(𝑡)=2cos²(𝜋𝑡), 𝑏₁₂(𝑡)=𝑏₂₁(𝑡)=(1/8)cos⁶(𝜋𝑡), 𝑏₂₂(𝑡)=(1/2)|sin(𝜋𝑡)|, 𝑓₁(𝑥₁)=(1/112)𝑥 + (1/56)sin((1/2)𝑥₁)+3, 𝑓₂(𝑥₂)=(1/126)𝑥+(1/42)cos((2/3)𝑥₂)−11, 𝐼₁(𝑡)=sin⁶(𝜋𝑡), 𝐼₂(𝑡)=3cos(2𝜋𝑡), 𝜏₁₁(𝑡)=𝑒^{cos(2𝜋𝑡)}, 𝜏₁₂(𝑡)=3^{sin2(𝜋𝑡)}, 𝜏₂₁(𝑡)=ln(sin⁴(𝜋𝑡)), 𝜏₂₂(𝑡)=3/7+5^{cos6(𝜋𝑡)}.

By calculating, we have 𝑐₁=3/8, 𝑐₁=1/8, 𝑐₂=1/2, 𝑐₂=1/6, 𝑀₁=1/56, 𝑀₂=1/42, 𝑎₁₁=1, 𝑎₁₂=𝑎₂₁=1/4, 𝑎₂₂=1, 𝑏₁₁=2, 𝑏₁₂=𝑏₂₁=1/8, 𝑏₂₂=1/2. It is not difficult to verify that (H1)-(H2) are satisfied.

Also by calculating, we have that 𝐸=𝑒_𝑖𝑗2×2=1−128−11215122828 (5.2) is a nonsingular 𝑀 matrix. If we take 𝜉₁=𝜉₂=1, we can obtain that −𝜉₁𝑐₁+²_𝑗=1𝜉_𝑗𝑎_1𝑗+𝑏_1𝑗𝑀_𝑗7=−112<0,−𝜉₂𝑐₂+²_𝑗=1𝜉_𝑗𝑎_2𝑗+𝑏_2𝑗𝑀_𝑗=−1671344<0. (5.3) The condition (H3) is satisfied. By Theorems 3.1 and 4.2, we know that system (5.1) has at least one 1-periodic solution, and this solution is globally exponentially stable.

6. Conclusion

In this paper, without assuming the boundedness of the activation functions, we establish the stability and existence of periodic solutions of cellular neural networks with delays on time scales. Our results obtained in this paper are completely new even in case of the time scale 𝕋 = ℝ or ℤ. Besides, our method used in this paper may be used to study other type neural networks such as BAM neural networks, high-order Hopfield neural networks, and Cohen-Grossberg neural networks.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183, and this work was also supported by IRTSTYN.