Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 415786, 14 pages
doi:10.1155/2009/415786
Research Article

Periodic Solutions and Exponential Stability of a Class of Neural Networks with Time-Varying Delays

Yingxin Guo1 and Mingzhi Xue2

1Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China
2Department of Mathematics, Shangqiu Normal University, Shangqiu 476000, Henan, China

Received 8 March 2009; Accepted 17 June 2009

Academic Editor: Guang Zhang

Copyright © 2009 Yingxin Guo and Mingzhi Xue. 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.

Abstract

Employing fixed point theorem, we make a further investigation of a class of neural networks with delays in this paper. A family of sufficient conditions is given for checking global exponential stability. These results have important leading significance in the design and applications of globally stable neural networks with delays. Our results extend and improve some earlier publications.

1. Introduction

The stability of dynamical neural networks with time delay which have been used in many applications such as optimization, control, and image processing has received much attention recently (see, e.g., [1–15]). Particularly, the authors [3, 8, 9, 14, 16] have studied the stability of neural networks with time-varying delays.

As pointed out in [8], Global dissipativity is also an important concept in dynamical neural networks. The concept of global dissipativity in dynamical systems is a more general concept, and it has found applications in areas such as stability theory, chaos and synchronization theory, system norm estimation, and robust control [8]. Global dissipativity of several classes of neural networks was discussed, and some sufficient conditions for the global dissipativity of neural networks with constant delays are derived in [8].

In this paper, without assuming the boundedness, monotonicity, and differentiability of activation functions, we consider the following delay differential equations: 𝑥 𝑖 ( 𝑡 ) = − 𝑑 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝐻 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑠 ) 𝑑 𝑠 + 𝐽 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 1 ) where 𝑛 denotes the number of the neurons in the network, 𝑥 𝑖 ( 𝑡 ) is the state of the 𝑖 th neuron at time 𝑡 , 𝑥 ( 𝑡 ) = [ 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ] 𝑇 ∈ 𝑅 𝑛 , 𝑓 ( 𝑥 ( 𝑡 ) ) = [ 𝑓 1 ( 𝑥 1 ( 𝑡 ) ) , 𝑓 2 ( 𝑥 2 ( 𝑡 ) ) , … , 𝑓 𝑛 ( 𝑥 𝑛 ( 𝑡 ) ) ] 𝑇 ∈ 𝑅 𝑛 denote the activation functions of the 𝑗 th neuron at time 𝑡 , and the kernels 𝐻 𝑖 𝑗 ∶ [ 0 , + ∞ ) → [ 0 , + ∞ ) are piece continuous functions with ∫ 0 + ∞ 𝐻 𝑖 𝑗 ( 𝑠 ) 𝑑 𝑠 = ℎ 𝑖 𝑗 < ∞ for 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 . Moreover, we consider model (1.1) with 𝜏 𝑖 𝑗 ( 𝑡 ) , 𝑑 𝑖 ( 𝑡 ) , 𝑎 𝑖 𝑗 ( 𝑡 ) , 𝑏 𝑖 𝑗 ( 𝑡 ) , 𝑐 𝑖 𝑗 ( 𝑡 ) , and 𝐽 𝑖 ( 𝑡 ) satisfying the following assumptions:

(A1) the time delays 𝜏 𝑖 𝑗 ( 𝑡 ) ∈ 𝐶 ( 𝑅 , [ 0 , ∞ ) ) are periodic functions with a common period 𝜔 ( > 0 ) for 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 ;(A2) 𝑐 𝑖 𝑗 ( 𝑡 ) ∈ 𝐶 ( 𝑅 , [ 0 , ∞ ) ) , 𝑎 𝑖 𝑗 ( 𝑡 ) , 𝑏 𝑖 𝑗 ( 𝑡 ) , 𝑐 𝑖 𝑗 ( 𝑡 ) , 𝐽 𝑖 ( 𝑡 ) ∈ 𝐶 ( 𝑅 , 𝑅 ) are periodic functions with a common period 𝜔 ( > 0 ) and 𝑓 𝑖 ∈ 𝐶 ( 𝑅 , 𝑅 ) , 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 .

The organization of this paper is as follows. In Section 2, problem formulation and preliminaries are given. In Section 3, some new results are given to ascertain the global robust dissipativity of the neural networks with time-varying delays. Section 4 gives an example to illustrate the effectiveness of our results.

2. Preliminaries and Lemmas

For the sake of convenience, two of the standing assumptions are formulated below as follows.

(A3) | 𝑓 𝑗 ( 𝑢 ) | ≤ 𝑝 𝑗 | 𝑢 | + 𝑞 𝑗 for all 𝑢 ∈ 𝑅 , 𝑗 = 1 , 2 , … , 𝑛 , where 𝑝 𝑗 , 𝑞 𝑗 are nonnegative constants.(A4) There exist nonnegative constants 𝑝 𝑗 , 𝑗 = 1 , 2 , … , 𝑛 , such that | 𝑓 𝑗 ( 𝑢 ) − 𝑓 𝑗 ( 𝑣 ) | ≤ 𝑝 𝑗 | 𝑢 − 𝑣 | for any 𝑢 , 𝑣 ∈ 𝑅 .

Let 𝜏 = m a x 1 ≤ 𝑖 , 𝑗 ≤ 𝑛 s u p 𝑡 ≥ 0 𝜏 𝑖 𝑗 ( 𝑡 ) . ( 2 . 1 )

The initial conditions associated with system (1.1) are of the form 𝑥 𝑖 ( 𝑠 ) = 𝜙 𝑖 [ ] ( 𝑠 ) , 𝑠 ∈ − 𝜏 , 0 , 𝑖 = 1 , 2 , … , 𝑛 , ( 2 . 2 ) in which 𝜙 𝑖 ( 𝑠 ) is continuous for 𝑠 ∈ [ − 𝜏 , 0 ] .

For continuous functions 𝜙 𝑖 defined on [ − 𝜏 , 0 ] , 𝑖 = 1 , 2 , … , 𝑛 , we set 𝜙 = ( 𝜙 1 , 𝜙 2 , … , 𝜙 𝑛 ) 𝑇 . If 𝑥 0 = ( 𝑥 0 1 , 𝑥 0 1 , … , 𝑥 0 𝑛 ) 𝑇 is an equilibrium of system (1.1), then we denote ‖ ‖ 𝜙 − 𝑥 0 ‖ ‖ = 𝑛 𝑖 = 1 s u p − 𝜏 ≤ 𝑡 ≤ 0 | | 𝜙 𝑖 ( 𝑡 ) − 𝑥 0 𝑖 | | . ( 2 . 3 )

Definition 2.1. The equilibrium 𝑥 0 = ( 𝑥 0 1 , 𝑥 0 1 , … , 𝑥 0 𝑛 ) 𝑇 is said to be globally exponentially stable, if there exist constants 𝜆 > 0 and 𝑚 ≥ 1 such that for any solution 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) 𝑇 of (1.1), we have | | 𝑥 𝑖 ( 𝑡 ) − 𝑥 0 𝑖 | | ‖ ‖ ≤ 𝑚 𝜙 − 𝑥 0 ‖ ‖ 𝑒 − 𝜆 𝑡 ( 2 . 4 ) for 𝑡 ≥ 0 , where 𝜆 is called to be globally exponentially convergent rate.

Lemma 2.2 ([17]). If 𝜌 ( 𝐾 ) < 1 for matrix 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 ≥ 0 , then ( 𝐸 − 𝐾 ) − 1 ≥ 0 , where 𝐸 denotes the identity matrix of size 𝑛 .

3. Periodic Solutions and Exponential Stability

We will use the coincidence degree theory to obtain the existence of a 𝜔 -periodic solution to systems (1.1). For the sake of convenience, we briefly summarize the theory as follows.

Let 𝑋 and 𝑍 be normed spaces, and let 𝐿 ∶ D o m 𝐿 ⊂ 𝑋 ↦ 𝑍 be a linear mapping and be a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if d i m K e r 𝐿 = c o d i m I m 𝐿 < ∞ and I m 𝐿 is closed in 𝑍 . If 𝐿 is a Fredholm mapping of index zero, then there exist continuous projectors 𝑃 ∶ 𝑋 ↦ 𝑋 and 𝑄 ∶ 𝑍 ↦ 𝑍 such that I m 𝑃 = K e r 𝐿 and I m 𝐿 = K e r 𝑄 = I m ( 𝐼 − 𝑄 ) . It follows that 𝐿 ∣ D o m 𝐿 ∩ K e r 𝑃 ∶ ( 𝐼 − 𝑃 ) 𝑋 ↦ I m 𝐿 is invertible. We denote the inverse of this map by 𝐾 𝑝 . If Ω is a bounded open subset of 𝑋 , the mapping 𝑁 is called 𝐿 -compact on Ω if 𝑄 𝑁 ( Ω ) is bounded and 𝐾 𝑝 ( 𝐼 − 𝑄 ) 𝑁 ∶ Ω ↦ 𝑋 is compact. Because I m 𝑄 is isomorphic to K e r 𝐿 , there exists an isomorphism 𝐽 ∶ I m 𝑄 ↦ K e r 𝐿 .

Let Ω ⊂ 𝑅 𝑛 be open and bounded, 𝑓 ∈ 𝐶 1 ( Ω , 𝑅 𝑛 ) ∩ 𝐶 ( Ω , 𝑅 𝑛 ) and 𝑦 ∈ 𝑅 𝑛 ⧵ 𝑓 ( 𝜕 Ω ∪ 𝑆 𝑓 ) , that is, 𝑦 is a regular value of 𝑓 . Here, 𝑆 𝑓 = { 𝑥 ∈ Ω ∶ 𝐽 𝑓 ( 𝑥 ) = 0 } , the critical set of 𝑓 , and 𝐽 𝑓 is the Jacobian of 𝑓 at 𝑥 . Then the degree d e g { 𝑓 , Ω , 𝑦 } is defined by d e g { 𝑓 , Ω , 𝑦 } = 𝑥 ∈ 𝑓 − 1 ( 𝑦 ) s g n 𝐽 𝑓 ( 𝑥 ) ( 3 . 1 ) with the agreement that the above sum is zero if 𝑓 − 1 ( 𝑦 ) = ∅ . For more details about the degree theory, we refer to the book of Deimling [18].

Lemma 3.1 (continuation theorem [19, page 40]). Let 𝐿 be a Fredholm mapping of index zero, and let 𝑁 be 𝐿 -compact on Ω . Suppose that (a)for each 𝜆 ∈ ( 0 , 1 ) , every solution 𝑥 of 𝐿 𝑥 = 𝜆 𝑁 𝑥 is such that 𝑥 ∈ 𝜕 Ω ;(b) 𝑄 𝑁 𝑥 ≠ 0 for each 𝑥 ∈ 𝜕 Ω ∩ K e r 𝐿 and d e g { 𝐽 𝑄 𝑁 , Ω ∩ K e r 𝐿 , 0 } ≠ 0 . ( 3 . 2 ) Then the equation 𝐿 𝑥 = 𝑁 𝑥 has at least one solution lying in D o m 𝐿 ∩ Ω .

For the simplicity of presentation, in the remaining part of this paper, for a continuous function 𝑔 ∶ [ 0 , 𝜔 ] ↦ 𝑅 , we denote 𝑔 ∗ = m a x [ ] 𝑡 ∈ 0 , 𝜔 𝑔 ( 𝑡 ) , 𝑔 ∗ = m i n [ ] 𝑡 ∈ 0 , 𝜔 𝑔 ( 𝑡 ) , 1 𝑔 = 𝜔 𝜔 0 𝑔 ( 𝑡 ) 𝑑 𝑡 . ( 3 . 3 )

Theorem 3.2. Let (A1)–(A3) hold, 𝑘 𝑖 𝑗 = ( 1 / 𝑑 𝑖 + 𝜔 ) ( | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | ) 𝑝 𝑗 and 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 . If 𝜌 ( 𝐾 ) < 1 , then system (1.1) has at least a 𝜔 -periodic solution.

Proof. Take 𝑋 = 𝑍 = { 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) 𝑇 ∈ 𝐶 ( 𝑅 , 𝑅 𝑛 ) ∶ 𝑥 ( 𝑡 ) = 𝑥 ( 𝑡 + 𝜔 ) , f o r a l l 𝑡 ∈ 𝑅 } , and denote | | 𝑥 𝑖 | | = m a x 𝑡 ∈ [ 0 , 𝜔 ] | | 𝑥 𝑖 | | ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , ‖ 𝑥 ‖ = m a x 1 ≤ 𝑖 ≤ 𝑛 | | 𝑥 𝑖 | | . ( 3 . 4 ) Equipped with the norms ‖ ⋅ ‖ , both 𝑋 and 𝑍 are Banach spaces. Denote Δ 𝑥 𝑖 , 𝑡 ∶ = − 𝑑 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝐻 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑠 ) 𝑑 𝑠 + 𝐽 𝑖 ( 𝑡 ) . ( 3 . 5 ) Since 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝐻 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑠 ) 𝑑 𝑠 = 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) ∞ 0 𝐻 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑡 − 𝑠 ) 𝑑 𝑠 , ( 3 . 6 ) then, for any 𝑥 ( 𝑡 ) ∈ 𝑋 , because of the periodicity, it is easy to check that Δ 𝑥 𝑖 , 𝑡 = − 𝑑 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) ∞ 0 𝐻 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑡 − 𝑠 ) 𝑑 𝑠 + 𝐽 𝑖 ( 𝑡 ) ∈ 𝑍 . ( 3 . 7 ) Let 𝐿 ∶ D o m 𝐿 = 𝑥 ∈ 𝑋 ∶ 𝑥 ∈ 𝐶 1 ( 𝑅 , 𝑅 𝑛 ) ∋ 𝑥 ⟼ 𝑥 1 ( ⋅ ) ∈ 𝑍 , 𝑃 ∶ 𝑋 ∋ 𝑥 ⟼ 𝜔 𝜔 0 𝑥 1 ( 𝑡 ) 𝑑 𝑡 ∈ 𝑋 , 𝑄 ∶ 𝑍 ∋ 𝑧 ⟼ 𝜔 𝜔 0 𝑥 𝑧 ( 𝑡 ) 𝑑 𝑡 ∈ 𝑍 , 𝑁 ∶ 𝑋 ∋ 𝑥 ⟼ Δ 𝑖 , 𝑡 ∈ 𝑍 . ( 3 . 8 ) Here, for any 𝑊 = ( 𝑤 1 , 𝑤 2 , … , 𝑤 𝑛 ) 𝑇 ∈ 𝑅 𝑛 , we identify it as the constant function in 𝑋 or 𝑍 with the value vector 𝑊 = ( 𝑤 1 , 𝑤 2 , … , 𝑤 𝑛 ) 𝑇 . Then system (1.1) can be reduced to the operator equation 𝐿 𝑥 = 𝑁 𝑥 . It is easy to see that K e r 𝐿 = 𝑅 𝑛 , 1 I m 𝐿 = 𝑧 ∈ 𝑍 ∶ 𝜔 𝜔 0 𝑧 ( 𝑡 ) 𝑑 𝑡 = 0 , w h i c h i s c l o s e d i n 𝑍 , d i m K e r 𝐿 = c o d i m I m 𝐿 = 𝑛 < ∞ , ( 3 . 9 ) and 𝑃 , 𝑄 are continuous projectors such that I m 𝑃 = k e r 𝐿 , K e r 𝑄 = I m 𝐿 = I m ( 𝐼 − 𝑄 ) . ( 3 . 1 0 ) It follows that 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿 ) 𝐾 𝑝 ∶ I m 𝐿 ↦ K e r 𝑃 ∩ D o m 𝐿 is given by ( 𝐾 𝑝 ( 𝑧 ) ) 𝑖 ( 𝑡 ) = 𝑡 0 𝑧 𝑖 ( 1 𝑠 ) 𝑑 𝑠 − 𝜔 𝜔 0 𝑠 0 𝑧 𝑖 ( 𝑣 ) 𝑑 𝑣 𝑑 𝑠 . ( 3 . 1 1 ) Then, ( 𝑄 𝑁 𝑥 ) 𝑖 1 ( 𝑡 ) = 𝜔 𝜔 0 Δ 𝑥 𝑖 , 𝑠 𝑑 𝑠 , ( 𝐾 𝑝 ( 𝐼 − 𝑄 ) 𝑁 𝑥 ) 𝑖 ( 𝑡 ) = 𝑡 0 Δ 𝑥 𝑖 1 , 𝑠 𝑑 𝑠 − 𝜔 𝜔 0 𝑡 0 Δ 𝑥 𝑖 1 , 𝑠 𝑑 𝑠 𝑑 𝑡 + 2 − 𝑡 𝜔 𝜔 0 Δ 𝑥 𝑖 , 𝑠 𝑑 𝑠 . ( 3 . 1 2 ) Clearly, 𝑄 𝑁 and 𝐾 𝑝 ( 𝐼 − 𝑄 ) 𝑁 are continuous. For any bounded open subset Ω ⊂ 𝑋 , 𝑄 𝑁 ( Ω ) is obviously bounded. Moreover, applying the ArzelaCAscoli theorem, one can easily show that 𝐾 𝑝 ( 𝐼 − 𝑄 ) 𝑁 ( Ω ) is compact. Therefore, 𝑁 is 𝐿 -compact on with any bounded open subset Ω ∈ 𝑋 . Since I m 𝑄 = K e r 𝐿 , we take the isomorphism 𝐽 of I m 𝑄 onto K e r 𝐿 to be the identity mapping.
Now, we reach the point to search for an appropriate open bounded set Ω for the application of the continuation theorem corresponding to the operator equation 𝐿 𝑥 = 𝜆 𝑁 𝑥 , 𝜆 ∈ ( 0 , 1 ) , and we have 𝑥 𝑖 𝑥 ( 𝑡 ) = 𝜆 Δ 𝑖 , 𝑡 f o r 1 = 1 , 2 , … , 𝑛 . ( 3 . 1 3 ) Assume that 𝑥 = 𝑥 ( 𝑡 ) ∈ 𝑋 is a solution of system (1.1) for some 𝜆 ∈ ( 0 , 1 ) . Integrating both sides of (3.13) over the interval [ 0 , 𝜔 ] , we obtain 0 = 𝜔 0 𝑥 𝑖 ( 𝑡 ) 𝑑 𝑡 = 𝜆 𝜔 0 Δ 𝑥 𝑖 , 𝑡 𝑑 𝑡 . ( 3 . 1 4 ) Then 𝜔 0 𝑑 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) 𝑑 𝑡 = 𝜔 0 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) ∞ 0 𝐻 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑡 − 𝑠 ) 𝑑 𝑠 + 𝐽 𝑖 ( 𝑡 ) 𝑑 𝑡 . ( 3 . 1 5 ) Noting that | | 𝑓 𝑗 | | ( 𝑢 ) ≤ 𝑝 𝑗 | 𝑢 | + 𝑞 𝑗 ∀ 𝑢 ∈ 𝑅 , 𝑗 = 1 , 2 , … , 𝑛 , ( 3 . 1 6 ) we get | | 𝑥 𝑖 | | ∗ 𝑑 𝑖 ≤ 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 | | 𝑥 𝑗 | | ∗ + 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | . ( 3 . 1 7 ) It follows that | | 𝑥 𝑖 | | ∗ ≤ 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 | | 𝑥 𝑗 | | ∗ + 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | . ( 3 . 1 8 ) Note that each 𝑥 𝑖 ( 𝑡 ) is continuously differentiable for 𝑖 = 1 , 2 , … , 𝑛 , and it is certain that there exists 𝑡 𝑖 ∈ [ 0 , 𝜔 ] such that | 𝑥 𝑖 ( 𝑡 𝑖 ) | = | 𝑥 𝑖 ( 𝑡 ) | ∗ . Set 𝐷 = ( 𝐷 1 , 𝐷 2 , … , 𝐷 𝑛 ) 𝑇 , 𝐷 𝑖 = 1 𝑑 𝑖 + 𝜔 𝑛 𝑗 = 1 | | 𝑎 𝑖 𝑗 | | + | | 𝑏 𝑖 𝑗 | | + | | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | | 𝑞 𝑗 + | | 𝐽 𝑖 | | . ( 3 . 1 9 ) In view of 𝜌 ( 𝐾 ) < 1 and Lemma 2.2, we have ( 𝐸 − 𝐾 ) − 1 𝐷 = 𝑙 = ( 𝑙 1 , 𝑙 2 , … , 𝑙 𝑛 ) 𝑇 ≥ 0 , where 𝑙 𝑖 is given by 𝑙 𝑖 = 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 𝑙 𝑗 + 𝐷 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 2 0 ) Let Ω = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝑇 ∈ 𝑅 𝑛 ; | | 𝑥 𝑖 | | ≤ 𝑙 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 2 1 ) Then, for 𝑡 ∈ [ 𝑡 𝑖 , 𝑡 𝑖 + 𝜔 ] , we have | | 𝑥 𝑖 ( | | ≤ | | 𝑥 𝑡 ) 𝑖 𝑡 𝑖 | | + 𝑡 𝑡 𝑖 𝐷 + | | 𝑥 𝑖 ( | | ≤ | | 𝑥 𝑡 ) 𝑑 𝑡 𝑖 | | ( 𝑡 ) ∗ + 𝑡 𝑖 𝑡 + 𝜔 𝑖 𝐷 + | | 𝑥 𝑖 | | ≤ 1 ( 𝑡 ) 𝑑 𝑡 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 | | 𝑥 𝑗 | | ∗ + 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | + 𝑡 𝑖 𝑡 + 𝜔 𝑖 𝐷 + | | 𝑥 𝑖 | | ≤ 1 ( 𝑡 ) 𝑑 𝑡 𝑑 𝑖 + 𝜔 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 | | 𝑥 𝑗 | | ∗ + 1 𝑑 𝑖 + 𝜔 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | ≤ 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 𝑙 𝑗 + 𝐷 𝑖 = 𝑙 𝑖 , ( 3 . 2 2 ) where 𝐷 + denotes the right derivative. Clearly, 𝑙 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 , are independent of 𝜆 . Then there are no 𝜆 ∈ ( 0 , 1 ) and 𝑥 ∈ Ω such that 𝐿 𝑥 = 𝜆 𝑁 𝑥 . When 𝑢 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝑇 ∈ 𝜕 Ω ∩ K e r 𝐿 = 𝜕 Ω ∩ 𝑅 𝑛 , 𝑢 is a constant vector in 𝑅 𝑛 with | 𝑥 𝑖 | = 𝑙 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 . Note that 𝑄 𝑁 𝑢 = 𝐽 𝑄 𝑁 𝑢 ; when 𝑢 ∈ K e r 𝐿 , it must be ( 𝑄 𝑁 𝑢 ) 𝑖 = − 𝑑 𝑖 + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 + 𝑏 𝑖 𝑗 + 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 𝑓 𝑗 𝑥 𝑗 + 𝐽 𝑖 . ( 3 . 2 3 ) We claim that | | ( 𝑄 𝑁 𝑢 ) 𝑖 | | > 0 f o r 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 2 4 ) On the contrary, suppose that there exists some 𝑖 such that | ( 𝑄 𝑁 𝑢 ) 𝑖 | = 0 , that is, 𝑑 𝑖 𝑥 𝑖 = 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 + 𝑏 𝑖 𝑗 + 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 𝑓 𝑗 𝑥 𝑗 + 𝐽 𝑖 . ( 3 . 2 5 ) Then, we have 𝑙 𝑖 = | | 𝑥 𝑖 | | = 1 𝑑 𝑖 | | | | | 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 + 𝑏 𝑖 𝑗 + 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 𝑓 𝑗 𝑥 𝑗 + 𝐽 𝑖 | | | | | ≤ 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 𝑙 𝑗 + 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | ≤ 1 𝑑 𝑖 + 𝜔 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 𝑙 𝑗 + 1 𝑑 𝑖 + 𝜔 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | = 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 𝑙 𝑗 + 𝐷 𝑖 = 𝑙 𝑖 , ( 3 . 2 6 ) which is a contradiction. Therefore, 𝑄 𝑁 𝑢 ≠ 0 f o r a n y 𝑢 ∈ 𝜕 Ω ∩ K e r 𝐿 = 𝜕 Ω ∩ 𝑅 𝑛 . ( 3 . 2 7 ) Consider the homotopy 𝐹 ∶ ( Ω ∩ K e r 𝐿 ) × [ 0 , 1 ] ↦ Ω ∩ K e r 𝐿 defined by − 𝐹 ( 𝑢 , 𝜇 ) = 𝜇 d i a g 𝑑 1 , − 𝑑 2 , … , − 𝑑 𝑛 𝑢 + ( 1 − 𝜇 ) 𝑄 𝑁 𝑢 , ( 3 . 2 8 ) ( 𝑢 , 𝜇 ) ∈ ( Ω ∩ K e r 𝐿 ) × [ 0 , 1 ] . Note that 𝐹 ( ⋅ , 0 ) = 𝐽 𝑄 𝑁 ; if 𝐹 ( 𝑢 , 𝜇 ) = 0 , then, as before, we have | | 𝑥 𝑖 | | = 1 − 𝜇 𝑑 𝑖 | | | | | 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 + 𝑏 𝑖 𝑗 + 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 𝑓 𝑗 𝑥 𝑗 + 𝐽 𝑖 | | | | | ≤ 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 | | 𝑥 𝑗 | | + 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | ≤ 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑝 𝑗 𝑙 𝑗 + 1 𝑑 𝑖 𝑛 𝑗 = 1 | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | 𝑞 𝑗 + | 𝐽 𝑖 | < 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 𝑙 𝑗 + 𝐷 𝑖 = 𝑙 𝑖 , ( 3 . 2 9 ) Hence [ ] 𝐹 ( 𝑢 , 𝜇 ) ≠ 0 , f o r ( 𝑢 , 𝜇 ) ∈ ( 𝜕 Ω ∩ K e r 𝐿 ) × 0 , 1 . ( 3 . 3 0 ) It follows from the property of invariance under a homotopy that − d e g { 𝐽 𝑄 𝑁 , Ω ∩ K e r 𝐿 , 0 } = d e g { 𝐹 ( ⋅ , 0 ) , Ω ∩ K e r 𝐿 , 0 } = d e g { 𝐹 ( ⋅ , 1 ) , Ω ∩ K e r 𝐿 , 0 } = d e g d i a g 𝑑 1 , − 𝑑 2 , … , − 𝑑 𝑛 ≠ 0 . ( 3 . 3 1 ) Thus, we have shown that Ω satisfies all the assumptions of Lemma 3.1. Hence, 𝐿 𝑢 = 𝑁 𝑢 has at least one 𝜔 -periodic solution on D o m 𝐿 ∩ Ω . This completes the proof.

When 𝑐 𝑖 𝑗 = 0 , (1.1) turns into the following system: 𝑥 𝑖 ( 𝑡 ) = − 𝑑 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 ( 𝑡 − 𝜏 ( 𝑡 ) ) + 𝐽 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 . ( ( 𝑃 ) )

Corollary 3.3. Let (A1)–(A3) hold, 𝑘 𝑖 𝑗 = ( 1 / 𝑑 𝑖 + 𝜔 ) ( | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | ) 𝑝 𝑗 , and 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 . If 𝜌 ( 𝐾 ) < 1 , then system ( 𝑃 ) has at least a 𝜔 -periodic solution.

Theorem 3.4. Let (A1), (A2), and (A4) hold, 𝑘 𝑖 𝑗 = ( 1 / 𝑑 𝑖 + 𝜔 ) ( | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 | + | 𝑐 𝑖 𝑗 ℎ 𝑖 𝑗 | ) 𝑝 𝑗 , and 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 . If 𝜌 ( 𝐾 ) < 1 , and that 𝑑 𝑖 − 𝑛 𝑗 = 1