Abstract and Applied Analysis
Volume 2009 (2009), Article ID 957475, 15 pages
doi:10.1155/2009/957475
Research Article

Existence and Exponential Stability of Periodic Solution for a Class of Generalized Neural Networks with Arbitrary Delays

Yimin Zhang,1 Yongkun Li,2 and Kuohui Ye2

1Department of Mathematics, Zhaotong Teacher's College, Zhaotong, Yunnan 657000, China
2Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 4 August 2009; Accepted 1 December 2009

Academic Editor: Jean Mawhin

Copyright © 2009 Yimin Zhang 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.

Abstract

By the continuation theorem of coincidence degree and 𝑀 -matrix theory, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of generalized neural networks with arbitrary delays, which are milder and less restrictive than those of previous known criteria. Moreover our results generalize and improve many existing ones.

1. Introduction

Consider the following generalized neural networks with arbitrary delays:

𝑥 𝐵 ( 𝑡 ) = 𝐴 ( 𝑡 , 𝑥 ( 𝑡 ) ) ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝐹 𝑡 , 𝑥 𝑡 , ( 1 . 1 ) where 𝐴 ( 𝑡 , 𝑥 ( 𝑡 ) ) = d i a g ( 𝑎 1 ( 𝑡 , 𝑥 1 ( 𝑡 ) ) , 𝑎 2 ( 𝑡 , 𝑥 2 ( 𝑡 ) ) , … , 𝑎 𝑛 ( 𝑡 , 𝑥 𝑛 ( 𝑡 ) ) , 𝐵 ( 𝑡 , 𝑥 ( 𝑡 ) ) = 𝑏 1 ( 𝑡 , 𝑥 1 ( 𝑡 ) ) , 𝑏 2 ( 𝑡 , 𝑥 2 ( 𝑡 ) ) , … , 𝑏 𝑛 ( 𝑡 , 𝑥 𝑛 ( 𝑡 ) ) ) 𝑇 ,   𝐹 ( 𝑡 , 𝑥 𝑡 ) = ( 𝑓 1 ( 𝑡 , 𝑥 𝑡 ) , 𝑓 2 ( 𝑡 , 𝑥 𝑡 ) , … , 𝑓 𝑛 ( 𝑡 , 𝑥 𝑡 ) ) 𝑇 , 𝑓 𝑖 ( 𝑡 , 𝑥 𝑡 ) = 𝑓 𝑖 ( 𝑡 , 𝑥 1 𝑡 , 𝑥 2 𝑡 , … , 𝑥 𝑛 𝑡 ) ,    𝑥 𝑡 = ( 𝑥 1 𝑡 , 𝑥 2 𝑡 , … , 𝑥 𝑛 𝑡 ) 𝑇 is defined by 𝑥 𝑡 ( 𝜃 ) = 𝑥 ( 𝑡 + 𝜃 ) = ( 𝑥 1 ( 𝑡 + 𝜃 ) , 𝑥 2 ( 𝑡 + 𝜃 ) , … , 𝑥 𝑛 ( 𝑡 + 𝜃 ) ) 𝑇 , 𝜃 ∈ 𝐸 , and 𝐸 is a subset of 𝑅 − = ( − ∞ , 0 ] .

System (1.1) contains many neural networks, for examples, the higher-order Cohen-Grossberg type neural networks with delays (see [1])

𝑥 𝑖 ( 𝑡 ) = − 𝑎 𝑖 𝑥 𝑖 𝑏 ( 𝑡 ) 𝑖 𝑥 𝑖 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑔 𝑗 𝑥 𝑗 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑔 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑗 − ( 𝑡 ) 𝑛 𝑛 𝑗 = 1 𝑙 = 1 𝑏 𝑖 𝑗 𝑙 ( 𝑡 ) 𝑔 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑗 𝑔 ( 𝑡 ) 𝑙 𝑥 𝑙 𝑡 − 𝜏 𝑙 ( 𝑡 ) + 𝐼 𝑖 , ( 𝑡 ) 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 2 ) the Cohen-Grossberg neural networks with bounded and unbounded delays (see [2])

𝑥 𝑖 ( 𝑡 ) = − 𝑎 𝑖 𝑡 , 𝑥 𝑖 𝑏 ( 𝑡 ) 𝑖 𝑡 , 𝑥 𝑖 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑑 𝑖 𝑗 ( 𝑡 ) 𝑔 𝑗 ∞ 0 𝐾 𝑖 𝑗 ( 𝑢 ) 𝑥 𝑗 ( 𝑡 − 𝑢 ) d 𝑢 + 𝐼 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 3 ) the Cohen-Grossberg neural networks with time-varying delays (see [3])

𝑥 𝑖 ( 𝑡 ) = − 𝑎 𝑖 𝑡 , 𝑥 𝑖 𝑏 ( 𝑡 ) 𝑖 𝑡 , 𝑥 𝑖 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 − ( 𝑡 ) 𝑛 𝑗 = 1 𝑑 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 ( 𝑡 ) + 𝐼 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 4 ) the celluar neural networks (see [4, Page 193]):

𝑥 𝑖 ( 𝑡 ) = − 𝑟 𝑖 ( 𝑡 ) 𝑥 𝑖 ( 𝑡 ) + 𝑛 𝑗 = 1 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑔 𝑗 𝑥 𝑗 + ( 𝑡 ) 𝑛 𝑗 = 1 𝑏 𝑖 𝑗 ( 𝑡 ) 𝑔 𝑗 𝑥 𝑗 𝑡 − 𝜏 𝑖 𝑗 ( 𝑡 ) + 𝐼 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 5 ) and so on.

Since the model of Cohen-Grossberg neural networks was first introduced by Cohen and Grossberg in [5], the dynamical characteristics (including stable, unstable, and periodic oscillatory) of Cohen-Grossberg neural networks have been widely investigated for the sake of theoretical interest as well as application considerations. Many good results have already been obtained by some authors in [6–15] and the references cited therein. Moreover, the existing results are based on the assumption that demanding either the activation functions, the behaved functions, or delays is bounded in the above-mentioned literature. However, to the best of our knowledge, few authors have discussed the existence and exponential stability of periodic solutions of (1.1). In this paper, by using the continuation theorem of coincidence degree and 𝑀 -matrix theory, we study model (1.1), and get some sufficient conditions for the existence and exponential stability of the periodic solution of system (1.1); our results generalize and improve many existing ones.

Let 𝐴 = ( 𝑎 𝑖 𝑗 ) 𝑛 × 𝑛 , 𝐵 = ( 𝑏 𝑖 𝑗 ) 𝑛 × 𝑛 ∈ 𝑅 𝑛 × 𝑛 be two matrices, 𝑢 = ( 𝑢 1 , 𝑢 2 , … , 𝑢 𝑛 ) 𝑇 ∈ 𝑅 𝑛 , 𝑣 = ( 𝑣 1 , 𝑣 2 , … , 𝑣 𝑛 ) 𝑇 ∈ 𝑅 𝑛 be two vectors. For convenience, we introduce the following notations.

(i) 𝐴 ≥ 0 ( 𝐴 > 0 ) means that each element 𝑎 𝑖 𝑗 is nonnegative (positive) respectively,(ii) 𝐴 ≥ 𝐵 ( > 𝐵 ) means 𝐴 − 𝐵 ≥ 0 ( > 0 ) ,(iii) 𝑢 ≥ 0 ( 𝑢 > 0 ) means each element 𝑢 𝑖 ≥ 0 ( 𝑢 𝑖 > 0 ) ,(iv) 𝑢 ≤ 𝑣 ( 𝑢 < 𝑣 ) means 𝑣 − 𝑢 ≥ 0 ( 𝑣 − 𝑢 > 0 ) ,(v) | 𝑢 | = ( | 𝑢 1 | , | 𝑢 2 | , … , | 𝑢 𝑛 | ) 𝑇 .

For continuous 𝜔 -periodic function 𝑔 ∶ 𝑅 → 𝑅 , we denote | 𝑔 | = m a x 0 ≤ 𝑡 ≤ 𝜔 | 𝑔 ( 𝑡 ) | , 𝐶 𝐸 = 𝐶 [ 𝐸 , 𝑅 𝑛 ] is the family of continuous functions 𝜙 = ( 𝜙 1 , 𝜙 2 , … , 𝜙 𝑛 ) 𝑇 from 𝐸 ⊂ ( − ∞ , 0 ] to 𝑅 𝑛 . Clearly, it is a Banach space with the norm ‖ 𝜙 ‖ = m a x 0 ≤ 𝑖 ≤ 𝑛 | 𝜙 𝑖 | , where | 𝜙 𝑖 | = s u p 𝜃 ∈ 𝐸 | 𝜙 𝑖 ( 𝜃 ) | . The initial conditions of system (1.1) are of the form

𝑥 0 = 𝜙 , t h a t i s , 𝑥 𝑖 ( 𝜃 ) = 𝜙 𝑖 ( 𝜃 ) , 𝜃 ∈ 𝐸 , 𝑖 = 1 , 2 , … , 𝑛 , ( 1 . 6 ) where 𝜙 = ( 𝜙 1 , 𝜙 2 , … , 𝜙 𝑛 ) 𝑇 ∈ 𝐶 𝐸 . For 𝑉 ( 𝑡 ) ∈ 𝐶 ( ( 𝑎 , + ∞ ) , 𝑅 ) , let

𝐷 − 𝑉 ( 𝑡 ) = l i m s u p ℎ → 0 − 𝐷 ( 𝑡 + ℎ ) − 𝐷 ( 𝑡 ) ℎ , 𝐷 − 𝑉 ( 𝑡 ) = l i m i n f ℎ → 0 − 𝐷 ( 𝑡 + ℎ ) − 𝐷 ( 𝑡 ) ℎ , 𝑡 ∈ ( 𝑎 , + ∞ ) . ( 1 . 7 )

Throughout this paper, we assume the following:

( 𝐻 1 ) For 𝑖 = 1 , 2 , … , 𝑛 , 𝑎 𝑖 , 𝑏 𝑖 ∈ 𝐶 [ 𝑅 2 , 𝑅 ] , 𝑓 𝑖 ∈ 𝐶 [ 𝑅 × 𝐶 𝐸 , 𝑅 ] and are 𝜔 -periodic for their first arguments, respectively, that is, 𝑎 𝑖 ( 𝑡 + 𝜔 , 𝑢 ) = 𝑎 𝑖 ( 𝑡 , 𝑢 ) , 𝑏 𝑖 ( 𝑡 + 𝜔 , 𝑢 ) = 𝑏 𝑖 ( 𝑡 , 𝑢 ) , 𝑓 𝑖 ( 𝑡 + 𝜔 , 𝜙 ) = 𝑓 𝑖 ( 𝑡 , 𝜙 ) so 𝐴 ( 𝑡 + 𝜔 , 𝑢 ) = 𝐴 ( 𝑡 , 𝑢 ) , 𝐵 ( 𝑡 + 𝜔 , 𝑢 ) = 𝐵 ( 𝑡 , 𝑢 ) , 𝐹 ( 𝑡 + 𝜔 , 𝜙 ) = 𝐹 ( 𝑡 , 𝜙 ) , for all 𝑡 ∈ 𝑅 , 𝑢 ∈ 𝑅 𝑛 , 𝜙 ∈ 𝐶 𝐸 .( 𝐻 2 ) There exists a positive diagonal matrix 𝐴 = d i a g ( 𝑎 1 , 𝑎 2 , … , 𝑎 𝑛 ) such that 𝐴 ( 𝑡 , 𝑢 ) ≥ 𝐴 , for all ( 𝑡 , 𝑢 ) ∈ 𝑅 𝑛 + 1 .( 𝐻 3 ) There is a positive diagonal matrix 𝐵 = d i a g ( 𝑏 1 , 𝑏 2 , … , 𝑏 𝑛 ) such that | 𝐵 ( 𝑡 , 𝑢 ) | ≥ 𝐵 | 𝑢 | , and 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 > 0 or 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 < 0 for all ( 𝑡 , 𝑢 ) ∈ 𝑅 𝑛 + 1 , 𝑖 = 1 , 2 , … , 𝑛 . ( 𝐻 4 ) There exist a nonnegative matrix 𝐶 = ( 𝑐 𝑖 𝑗 ) 𝑛 × 𝑛 ∈ 𝑅 𝑛 × 𝑛 and a nonnegative vector 𝐷 = ( 𝐷 1 , 𝐷 2 , … , 𝐷 𝑛 ) 𝑇 such that | 𝐹 ( 𝑡 , 𝜙 ) | ≤ 𝐶 | 𝜙 | + 𝐷 , for all ( 𝑡 , 𝜙 ) ∈ 𝑅 × 𝐶 𝐸 , where 𝜙 = ( 𝜙 1 , 𝜙 2 , … , 𝜙 𝑛 ) 𝑇 ∈ 𝐶 𝐸 , | 𝜙 | = ( | 𝜙 1 | , | 𝜙 2 | , … , | 𝜙 𝑛 | ) 𝑇 .

2. Preliminaries

In this section, we first introduce some definitions and lemmas which play an important role in the proof of our main results in this paper.

Definition 2.1. Let ̃ 𝑥 ( 𝑡 ) = ( ̃ 𝑥 1 ( 𝑡 ) , ̃ 𝑥 2 ( 𝑡 ) , … , ̃ 𝑥 𝑛 ( 𝑡 ) ) 𝑇 be an 𝜔 -periodic solution of system (1.1) with initial value 𝜙 ∈ 𝐶 𝐸 , if there exist two constants 𝛼 > 0 and 𝑀 > 1 such that for every solution 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ) 𝑇 of system (1.1) with initial value (1.6), | | 𝑥 𝑖 ( 𝑡 ) − ̃ 𝑥 𝑖 | | ‖ ‖ 𝜙 ‖ ‖ 𝑒 ( 𝑡 ) ≤ 𝑀 𝜙 − − 𝛼 𝑡 , ∀ 𝑡 > 0 , 𝑖 = 1 , 2 , … , 𝑛 . ( 2 . 1 ) Then ̃ 𝑥 ( 𝑡 ) is said to be globally exponential stable.

Definition 2.2. A real matrix 𝑊 = ( 𝑤 𝑖 𝑗 ) 𝑛 × 𝑛 ∈ 𝑅 𝑛 × 𝑛 is said to be an 𝑀 -matrix if 𝑤 𝑖 𝑗 ≤ 0 , 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 , 𝑖 ≠ 𝑗 , and 𝑊 − 1 ≥ 0 .

Lemma 2.3 (see [15, 16]). Assume that 𝐴 is an 𝑀 -matrix and 𝐴 𝑢 ≤ 𝑑 , 𝑢 , 𝑑 ∈ 𝑅 𝑛 , then 𝑢 ≤ 𝐴 − 1 𝑑 .

Lemma 2.4 (see [15, 16]). Let 𝑊 = ( 𝑤 𝑖 𝑗 ) 𝑛 × 𝑛 with 𝑤 𝑖 𝑗 ≤ 0 , 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 , 𝑖 ≠ 𝑗 , then the following statements are equivalent. (i) 𝑊 is an 𝑀 -matrix.(ii)There exists a positive vector 𝜂 = ( 𝜂 1 , 𝜂 2 , … , 𝜂 𝑛 ) > 0 such that 𝜂 𝑊 > 0 .(iii)There exists a positive vector 𝜉 = ( 𝜉 1 , 𝜉 2 , … , 𝜉 𝑛 ) 𝑇 > 0 such that 𝑊 𝜉 > 0 .

Lemma 2.5 (see [15, 16]). Let 𝐴 ≥ 0 be an 𝑛 × 𝑛 matrix and 𝜌 ( 𝐴 ) < 1 , then ( 𝐸 𝑛 − 𝐴 ) − 1 ≥ 0 , where 𝐸 𝑛 denotes the identity matrix of size 𝑛 , so 𝐸 𝑛 − 𝐴 is an 𝑀 -matrix.

Now we introduce Mawhin's continuation theorem which will be fundamental in this paper.

Lemma 2.6 (see [17]). Let 𝑋 and 𝑌 be two Banach spaces and 𝐿 a Fredholm mapping of index zero. Assume that Ω ⊂ 𝑋 is an open bounded set and 𝑁 ∶ 𝑋 → 𝑍 is a continuous operator which is 𝐿 -compact on Ω . Then 𝐿 𝑥 = 𝑁 𝑥 has at least one solution in D o m 𝐿 ∩ Ω , if the following conditions are satisfied: (1) 𝐿 𝑥 ≠ 𝜆 𝑁 𝑥 , for all ( 𝑥 , 𝜆 ) ∈ ( D o m 𝐿 ∩ 𝜕 Ω ) × ( 0 , 1 ) ,(2) 𝑄 𝑁 𝑥 ≠ 0 , for all 𝑥 ∈ 𝜕 Ω ∩ K e r 𝐿 ,(3) d e g { 𝐽 𝑄 𝑁 | K e r 𝐿 ∩ Ω , Ω ∩ K e r 𝐿 , 0 } ≠ 0 .

Let

𝑥 𝑋 = 𝑌 = 𝑥 = 1 , 𝑥 2 , … , 𝑥 𝑛 𝑇 ∈ 𝐶 ( 𝑅 , 𝑅 𝑛 ) ∶ 𝑥 ( 𝑡 + 𝜔 ) = 𝑥 ( 𝑡 ) , 𝑡 ∈ 𝑅 ( 2 . 2 ) with the norm defined by ‖ 𝑥 ‖ = m a x 1 ≤ 𝑖 ≤ 𝑛 | 𝑥 𝑖 | , where | 𝑥 𝑖 | = m a x 𝑡 ∈ [ 0 , 𝜔 ] | 𝑥 𝑖 ( 𝑡 ) | . Clearly, 𝑋 and 𝑌 are two Banach spaces. Let 𝑥 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝑇 ∈ 𝑋 = 𝑌 , we define the linear operator 𝐿 ∶ D o m 𝐿 ⊂ 𝑋 → 𝑌 as

( 𝐿 𝑥 ) ( 𝑡 ) = 𝑥 𝑥 ( 𝑡 ) = 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) 𝑇 , D o m 𝐿 = 𝑥 ∈ 𝑋 ∶ 𝑥 , ∈ 𝑌 ( 2 . 3 ) and the operators 𝑁 ∶ 𝑋 → 𝑋 , 𝑃 ∶ 𝑋 → 𝑋 , 𝑄 ∶ 𝑌 → 𝑌 as

𝐵 ( 𝑁 𝑥 ) ( 𝑡 ) = 𝐴 ( 𝑡 , 𝑥 ( 𝑡 ) ) ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝐹 𝑡 , 𝑥 𝑡 ∶ = Δ 𝑡 , 𝑥 𝑡 , 1 𝑃 𝑥 = 𝑄 𝑥 = 𝜔 𝜔 0 = 1 𝑥 ( 𝑡 ) d 𝑡 𝜔 𝜔 0 𝑥 1 1 ( 𝑡 ) d 𝑡 , 𝜔 𝜔 0 𝑥 2 1 ( 𝑡 ) d 𝑡 , … , 𝜔 𝜔 0 𝑥 𝑛 ( 𝑡 ) d 𝑡 𝑇 . ( 2 . 4 ) It is not difficult to show that 𝑃 and 𝑄 are continuous projectors and the following conditions are satisfied:

K e r 𝐿 = 𝑅 𝑛 = I m 𝑃 = I m 𝑄 , I m 𝐿 = 𝑦 ∈ 𝑌 = 𝑋 ∶ 𝜔 0 𝑦 ( 𝑡 ) d 𝑡 = 0 = K e r 𝑄 = I m ( 𝐼 − 𝑄 ) , I m 𝐿 i s c l o s e d i n 𝑌 , d i m K e r 𝐿 = 𝑛 = c o d i m I m 𝐿 . ( 2 . 5 ) Thus, the mapping 𝐿 is a Fredholm mapping of index zero and the isomorphism 𝐽 ∶ I m 𝑄 → K e r 𝐿 is the identity operator; the generalized inverse (of 𝐿 | D o m 𝐿 ∩ K e r 𝑃 ) 𝐾 𝑃 ∶ I m 𝐿 → K e r 𝑃 ∩ D o m 𝐿 exists, which has the form

𝐾 𝑃 𝑥 ( 𝑡 ) = 𝑡 0 1 𝑥 ( 𝑠 ) d 𝑠 − 𝜔 𝜔 0 𝑡 0 𝑥 ( 𝑠 ) d 𝑠 d 𝑡 , ∀ 𝑥 ∈ I m 𝐿 . ( 2 . 6 ) Therefore

1 ( 𝑄 𝑁 𝑥 ) ( 𝑡 ) = 𝜔 𝜔 0 Δ 𝑡 , 𝑥 𝑡 𝐾 d 𝑡 , 𝑃 ( 𝐼 − 𝑄 ) 𝑁 𝑥 ( 𝑡 ) = 𝑡 0 Δ 𝑠 , 𝑥 𝑠 1 d 𝑠 − 𝜔 𝜔 0 𝑡 0 Δ 𝑠 , 𝑥 𝑠 1 d 𝑠 d 𝑡 + 2 − 𝑡 𝜔 𝜔 0 Δ 𝑠 , 𝑥 𝑠 d 𝑠 . ( 2 . 7 )

3. Existence of Periodic Solutions

In this section, we shall use Lemma 2.6 to study the existence of at least one periodic solution of system (1.1).

Theorem 3.1. Let ( 𝐻 1 )–( 𝐻 4 ) hold. Moveover, suppose that ( 𝐻 5 ) 𝐸 − 𝐾 is a 𝑀 -matrix, where the matrix 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 = 𝐵 − 1 𝐶 .then (i)system (1.1) has at least one 𝜔 -periodic solution;(ii)there exists a nonnegative constant 𝛿 such that for all 𝜔 -periodic solution 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) 𝑇 of system (1.1), | 𝑥 𝑖 ( 𝑡 ) | ≤ 𝛿 , 𝑖 = 1 , 2 , … , 𝑛 .

Proof. Clearly, 𝑄 𝑁 and 𝐾 𝑃 ( 𝐼 − 𝑄 ) 𝑁 are continuous functions and for every bounded subset Ω ⊂ 𝑋 , 𝑄 𝑁 ( Ω ) , 𝐾 𝑃 ( 𝐼 − 𝑄 ) 𝑁 ( Ω ) , and ( 𝐾 𝑃 ( 𝐼 − 𝑄 ) 𝑁 𝑥 ) , 𝑥 ∈ Ω are bounded. By using the Arzela-Ascoli theorem, 𝑄 𝑁 ( Ω ) and 𝐾 𝑃 ( 𝐼 − 𝑄 ) 𝑁 ( Ω ) are compact, therefore 𝑁 is 𝐿 -compact on Ω . Consider the following operator equation: 𝐿 𝑥 = 𝜆 𝑁 𝑥 , 𝜆 ∈ ( 0 , 1 ) . ( 3 . 1 ) That is, 𝑥 𝐵 ( 𝑡 ) = 𝜆 𝐴 ( 𝑡 , 𝑥 ( 𝑡 ) ) ( 𝑡 , 𝑥 ( 𝑡 ) ) + 𝐹 𝑡 , 𝑥 𝑡 , ( 3 . 2 ) or 𝑥 𝑖 ( 𝑡 ) = 𝜆 𝑎 𝑖 𝑡 , 𝑥 𝑖 𝑏 ( 𝑡 ) 𝑖 𝑡 , 𝑥 𝑖 ( 𝑡 ) + 𝑓 𝑖 𝑡 , 𝑥 𝑡 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 3 ) Assume that 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) 𝑇 ∈ 𝑋 is a solution of (3.3) for some 𝜆 ∈ ( 0 , 1 ) . Then, for any 𝑖 = 1 , 2 , … , 𝑛 , 𝑥 𝑖 ( 𝑡 ) are all continuous 𝜔 -periodic functions, and there exist 𝑡 𝑖 ∈ [ 0 , 𝜔 ] , such that | | 𝑥 𝑖 𝑡 𝑖 | | = m a x 𝑡 ∈ [ 0 , 𝜔 ] | | 𝑥 𝑖 | | = ( 𝑡 ) | | 𝑥 𝑖 | | , 𝑥 𝑖 𝑡 𝑖 = 0 , 𝑖 = 1 , 2 , … , 𝑛 , ( 3 . 4 ) from ( 𝐻 2 ), we have 𝑏 𝑖 𝑡 𝑖 , 𝑥 𝑖 𝑡 𝑖 + 𝑓 𝑖 𝑡 𝑖 , 𝑥 𝑡 𝑖 = 0 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 5 ) It follows from ( 𝐻 3 ) that 𝑏 𝑖 | | 𝑥 𝑖 | | ≤ | | 𝑏 𝑖 𝑡 𝑖 , 𝑥 𝑖 𝑡 𝑖 | | = | | 𝑓 𝑖 𝑡 𝑖 , 𝑥 𝑡 𝑖 | | ≤ 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 | | 𝑥 𝑗 𝑡 𝑖 | | + 𝐷 𝑖 ≤ 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 | | 𝑥 𝑗 | | + 𝐷 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 6 ) Thus | | 𝑥 𝑖 | | ≤ 𝑏 𝑖 𝑛 − 1 𝑗 = 1 𝑐 𝑖 𝑗 | | 𝑥 𝑗 | | + 𝑏 𝑖 − 1 𝐷 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 , ( 3 . 7 ) we denote the vector 𝑑 = ( 𝑑 1 , 𝑑 2 , … , 𝑑 𝑛 ) 𝑇 , | 𝑥 | = ( | 𝑥 1 | , | 𝑥 2 | , … , | 𝑥 𝑛 | ) 𝑇 , where 𝑑 𝑖 = 𝑏 𝑖 − 1 ( 𝐷 𝑖 + 1 ) > 0 , 𝑖 = 1 , 2 , … , 𝑛 . It follows from (3.7) that ( 𝐸 − 𝐾 ) | 𝑥 | < 𝑑 . ( 3 . 8 ) Since ( 𝐻 5 ) , and application of Lemma 2.3 yields | 𝑥 | < ( 𝐸 − 𝐾 ) − 1 ℎ 𝑑 = 1 , ℎ 2 , … , ℎ 𝑛 𝑇 = ℎ , ( 3 . 9 ) where ℎ satisfies the equation ℎ = 𝐾 ℎ + 𝑑 , that is, ℎ 𝑖 = ∑ 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 ℎ 𝑗 + 𝑑 𝑖 > 0 .
Take Ω = 𝑥 ∈ 𝑋 ∶ | | 𝑥 𝑖 | | < ℎ 𝑖 , 𝑖 = 1 , 2 , … , 𝑛 . ( 3 . 1 0 ) It is easy to see that Ω satisfies condition ( 1 ) in Lemma 2.6.
For all 𝑥 = ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑛 ) 𝑇 ∈ 𝜕 Ω ∩ K e r 𝐿 , 𝑥 is a constant vector in 𝑅 𝑛 and there exists some 𝑖 ∈ { 1 , 2 , … , 𝑛 } such that | 𝑥 𝑖 | = | 𝑥 𝑖 | = ℎ 𝑖 , we claim that | | ( 𝑄 𝑁 𝑥 ) 𝑖 | | > 0 , s o t h a t 𝑄 𝑁 𝑥 ≠ 0 . ( 3 . 1 1 ) We firstly claim that (1)if 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 > 0 , then 𝑥 𝑖 ( 𝑄 𝑁 𝑥 ) 𝑖 > 0 ,(2)if 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 < 0 , then 𝑥 𝑖 ( 𝑄 𝑁 𝑥 ) 𝑖 < 0 .
We only prove (1), since the proof of (2) is similar. If 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 > 0 , we have 𝑥 𝑖 𝑏 𝑖 𝑡 , 𝑥 𝑖 ( 𝑡 ) + 𝑓 𝑖 𝑡 , 𝑥 𝑡 ≥ 𝑏 𝑖 𝑥 2 𝑖 − | | 𝑥 𝑖 | | 𝑛 𝑗 = 1 𝑐 𝑖 𝑗 | | 𝑥 𝑗 | | + 𝐷 𝑖 > 𝑏 𝑖 ℎ 𝑖 ℎ 𝑖 − 𝑛 𝑗 = 1 𝑏 𝑖 − 1 𝑐 𝑖 𝑗 ℎ 𝑗 + 𝑑 𝑖 = 𝑏 𝑖 ℎ 𝑖 ℎ 𝑖 − 𝑛 𝑗 = 1 𝑘 𝑖 𝑗 ℎ 𝑗 + 𝑑 𝑖 = 0 . ( 3 . 1 2 ) Therefore 𝑥 𝑖 ( 𝑄 𝑁 𝑥 ) 𝑖 = 𝜔 − 1 𝑥 𝑖 𝜔 0 𝑎 𝑖 𝑡 , 𝑥 𝑖 𝑏 ( 𝑡 ) 𝑖 𝑡 , 𝑥 𝑖 ( 𝑡 ) + 𝑓 𝑖 𝑡 , 𝑥 𝑡 d 𝑡 > 0 . ( 3 . 1 3 ) Thus (3.11) is valid.
Next, we define continuous functions 𝐻 𝑖 ∶ ( Ω ∩ K e r 𝐿 ) × [ 0 , 1 ] → Ω ∩ K e r 𝐿 , 𝑖 = 1 , 2 , by 𝐻 1 [ ] , 𝐻 ( 𝑥 , 𝑡 ) = 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑄 𝑁 𝑥 , ∀ ( 𝑥 , 𝑡 ) ∈ ( Ω ∩ K e r 𝐿 ) × 0 , 1 2 [ ] , ( 𝑥 , 𝑡 ) = − 𝑡 𝑥 + ( 1 − 𝑡 ) 𝑄 𝑁 𝑥 , ∀ ( 𝑥 , 𝑡 ) ∈ ( Ω ∩ K e r 𝐿 ) × 0 , 1 ( 3 . 1 4 ) respectively. If 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 > 0 , from (i) we have 𝐻 1 [ ] ( 𝑥 , 𝑡 ) ≠ 0 , ∀ ( 𝑥 , 𝑡 ) ∈ K e r 𝐿 𝜕 Ω × 0 , 1 , ( 3 . 1 5 ) If 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 < 0 , from ( 2 ) we can get 𝐻 2 [ ] ( 𝑥 , 𝑡 ) ≠ 0 , ∀ ( 𝑥 , 𝑡 ) ∈ K e r 𝐿 𝜕 Ω × 0 , 1 . ( 3 . 1 6 ) Using the homotopy invariance theorem, we obtain if 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 > 0 , d e g 𝐽 𝑄 𝑁 | Ω ∩ K e r 𝐿 𝐻 , Ω ∩ K e r 𝐿 , 0 = d e g 1 𝐻 ( ⋅ , 0 ) , Ω ∩ K e r 𝐿 , 0 = d e g 1 ( ⋅ , 1 ) , Ω ∩ K e r 𝐿 , 0 = d e g { 𝑥 , Ω ∩ K e r 𝐿 , 0 } = 1 , ( 3 . 1 7 ) or if 𝑏 𝑖 ( 𝑡 , 𝑢 𝑖 ) 𝑢 𝑖 < 0 , d e g 𝐽 𝑄 𝑁 | Ω ∩ K e r 𝐿 𝐻 , Ω ∩ K e r 𝐿 , 0 = d e g 2 𝐻 ( ⋅ , 0 ) , Ω ∩ K e r 𝐿 , 0 = d e g 2 ( ⋅ , 1 ) , Ω ∩ K e r 𝐿 , 0 = d e g { − 𝑥 , Ω ∩ K e r 𝐿 , 0 } = ( − 1 ) 𝑛 . ( 3 . 1 8 ) To summarize, Ω satisfies all the conditions of Lemma 2.6. This completes the proof of (i).
For all 𝜔 -periodic solution 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) 𝑇 of system (1.1), from (3.3)–(3.7) we have | | 𝑥 𝑖 | | ≤ 𝑏 𝑖 𝑛 − 1 𝑗 = 1 𝑐 𝑖 𝑗 | | 𝑥 𝑗 | | + 𝑏 𝑖 − 1 𝐷 𝑖 , | 𝑥 | ≤ ( 𝐸 − 𝐾 ) − 1 𝜈 𝜐 = 𝜈 = 1 , 𝜈 2 , … , 𝜈 𝑛 𝑇 , ( 3 . 1 9 ) where 𝜐 = ( 𝜐 1 , 𝜐 2 , … , 𝜐 𝑛 ) 𝑇 , | 𝑥 | = ( | 𝑥 1 | , | 𝑥 2 | , … , | 𝑥 𝑛 | ) 𝑇 , 𝜐 𝑖 = 𝑏 𝑖 − 1 𝐷 𝑖 ≥ 0 , 𝑖 = 1 , 2 , … , 𝑛 . Notes 𝛿 = m a x 1 ≤ 𝑖 ≤ 𝑛 𝜈 𝑖 ≥ 0 , thus | 𝑥 𝑖 ( 𝑡 ) | ≤ 𝛿 , for all 𝑖 = 1 , 2 , … , 𝑛 . This completes the proof of (ii).

From the proof of Theorem 3.1, we can easily obtain the following corollary.

Corollary 3.2. Suppose that ( 𝐻 1 )–( 𝐻 5 ) hold, and 𝐷 = 0 in ( 𝐻 4 ) , then system (1.1) has only one 𝜔 -periodic solution 𝑥 ( 𝑡 ) = 0 .

Some special cases of Theorem 3.1 are in what follows.

Corollary 3.3. Equation (1.3) has at least one 𝜔 -periodic solution, if the following conditions are satisfied. ( 𝐴 1 ) For 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 , 𝑎 𝑖 , 𝑏 𝑖 , 𝑎 𝑖 𝑗 , 𝑏 𝑖 𝑗 , 𝜏 𝑗 , 𝐼 𝑖 ∶ 𝑅 → 𝑅 are continuous 𝜔 -periodic ( 𝜔 > 0 ) functions.( 𝐴 2 ) For 𝑖 = 1 , 2 , … , 𝑛 , 𝑎 𝑖 ( 𝑥 ) are positive, and there exist 𝑎 𝑖 > 0 such that 𝑎 𝑖 ( 𝑥 ) ≥ 𝑎 𝑖 > 0 .( 𝐴 3 ) For 𝑖 = 1 , 2 , … , 𝑛 , there exist 𝑏 𝑖 > 0 such that | | 𝑏 𝑖 | | ( 𝑥 ) ≥ 𝑏 𝑖 | 𝑥 | , 𝑏 𝑖 ( 𝑥 ) 𝑥 > 0 , o r 𝑏 𝑖 ( 𝑥 ) 𝑥 < 0 , ∀ 𝑥 ∈ 𝑅 . ( 3 . 2 0 ) ( 𝐴 4 ) For 𝑖 = 1 , 2 , … , 𝑛 , there exist 𝐺 𝑖 , 𝑝 𝑖 , 𝑞 𝑖 ≥ 0 such that | | 𝑔 𝑖 | | ( 𝑥 ) ≤ 𝐺 𝑖 , | | 𝑔 𝑖 | | ( 𝑥 ) ≤ 𝑝 𝑖 | 𝑥 | + 𝑞 𝑖 . ( 3 . 2 1 ) ( 𝐴 5 ) 𝜌 ( 𝐾 ) < 1 , 𝐾 = ( 𝑘 𝑖 𝑗 ) 𝑛 × 𝑛 , a n d 𝑘 𝑖 𝑗 = 𝑏 𝑖 − 1 ( | 𝑎 𝑖 𝑗 | + | 𝑏 𝑖 𝑗 ∑ | + 𝑛 𝑗 = 1 | 𝑏 𝑖 𝑗 𝑙 | 𝐺 𝑙 ) 𝑝 𝑗 , 𝑖 , 𝑗 = 1 , 2 , … , 𝑛 .

Proof. It is clear that 𝐴 𝑎 ( 𝑡 , 𝑥 ) = d i a g 1 𝑥 1 , 𝑎 2 𝑥 2 , … , 𝑎 𝑛 𝑥 𝑛 𝑎 ≥ d i a g 1 , 𝑎 2 , … , 𝑎 𝑛 | | | | = | | 𝑏 = 𝐴 , 𝐵 ( 𝑡 , 𝑥 ) 1 𝑥 1 | | , | | 𝑏 2 𝑥 2 | | | | 𝑏 , … , 𝑛 𝑥 𝑛 | | 𝑇 ≥ 𝑏 1 | | 𝑥 1 | | , 𝑏 2 | | 𝑥 2 | | , … , 𝑏 𝑛 | | 𝑥 𝑛 | | 𝑇