Abstract and Applied Analysis
Volume 2010 (2010), Article ID 915451, 20 pages
doi:10.1155/2010/915451
Research Article

Existence and Global Exponential Stability of Almost Periodic Solutions for SICNNs with Nonlinear Behaved Functions and Mixed Delays

Xinsong Yang,1 Jinde Cao,2 Chuangxia Huang,3 and Yao Long1

1Department of Mathematics, Honghe University, Mengzi Yunnan 661100, China
2Department of Mathematics, Southeast University, Nanjing 210096, China
3Department of Mathematics, College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, China

Received 9 July 2009; Accepted 2 February 2010

Academic Editor: Allan C. Peterson

Copyright © 2010 Xinsong Yang 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 using the Leray-Schauder fixed point theorem and differential inequality techniques, several new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with discrete and distributed delays. The model in this paper possesses two characters: nonlinear behaved functions and all coefficients are time varying. Hence, our model is general and applicable to many known models. Moreover, our main results are also general and can be easily deduced to many simple cases, including some existing results. An example and its simulation are employed to illustrate our feasible results.

1. Introduction

Consider the following shunting inhibitory cellular neural networks (SICNNs) with discrete and distributed delays (mixed delays):

𝑥 𝑖 𝑗 ( 𝑡 ) = − 𝑎 𝑖 𝑗 𝑡 , 𝑥 𝑖 𝑗 ( + 𝑡 ) 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑖 𝑗 𝑥 𝑘 𝑙 𝑡 − 𝜏 𝑘 𝑙 ( 𝑥 𝑡 ) 𝑖 𝑗 ( + 𝑡 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝑘 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑔 𝑖 𝑗 𝑥 𝑘 𝑙 ( 𝑠 ) d 𝑠 𝑥 𝑖 𝑗 ( 𝑡 ) + 𝐼 𝑖 𝑗 ( 𝑡 ) , ( 1 . 1 ) where 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 .    𝐶 𝑖 𝑗 ( 𝑡 ) denotes the cell at the ( 𝑖 , 𝑗 ) position of the lattice at time 𝑡 , the 𝑟 -neighborhood 𝑁 𝑟 ( 𝑖 , 𝑗 ) of 𝐶 𝑖 𝑗 ( 𝑡 ) is

𝑁 𝑟 𝐶 ( 𝑖 , 𝑗 ) = 𝑘 𝑙 | | | | , | | | | ( 𝑡 ) ∶ m a x 𝑘 − 𝑖 𝑙 − 𝑗 ≤ 𝑟 , 1 ≤ 𝑘 ≤ 𝑚 , 1 ≤ 𝑙 ≤ 𝑛 . ( 1 . 2 ) 𝑥 𝑖 𝑗 ( 𝑡 ) is the activity of the cell 𝐶 𝑖 𝑗 ( 𝑡 ) , 𝐼 𝑖 𝑗 ( 𝑡 ) is the external input to 𝐶 𝑖 𝑗 ( 𝑡 ) , 𝑎 𝑖 𝑗 ( 𝑡 , 𝑥 𝑖 𝑗 ( 𝑡 ) ) represents an appropriately behaved function of the cell 𝐶 𝑖 𝑗 ( 𝑡 ) at time 𝑡 ; 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) and 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell 𝐶 𝑖 𝑗 ( 𝑡 ) depending upon discrete delays and distributed delays, respectively; the activation functions 𝑓 𝑖 𝑗 ( ⋅ ) and 𝑔 𝑖 𝑗 ( ⋅ ) are continuous representing the output or firing rate of the cell 𝐶 𝑘 𝑙 ( 𝑡 ) ; 𝜏 𝑖 𝑗 ( 𝑡 ) represent axonal signal transmission delays; 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) , 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) , 𝑓 𝑖 𝑗 ( ⋅ ) , 𝑔 𝑖 𝑗 ( ⋅ ) , 𝐼 𝑖 𝑗 ( 𝑡 ) , 𝜏 𝑖 𝑗 ( 𝑡 ) are all continuous almost periodic functions.

Since Bouzerdoum and Pinter described SICNNs as a new cellular neural networks [1–3], SICNNs have been extensively studied and found many important applications in different areas such as psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. There have been some results on the existence of periodic and almost solutions for SICNNs with discrete or distributed delays (distributed delay was first introduced in [4]) [5–21]. We find that all the behaved functions in the models in [5–21] are linear. Actually, 𝑎 𝑖 𝑗 ( 𝑡 , 𝑥 𝑖 𝑗 ( 𝑡 ) ) may be nonlinear. Moreover, we find that the models in [5–20] are special cases of (1.1). For example, let 𝑎 𝑖 𝑗 ( 𝑡 , 𝑥 𝑖 𝑗 ( 𝑡 ) ) = 𝑎 𝑖 𝑗 ( 𝑡 ) 𝑥 𝑖 𝑗 ( 𝑡 ) , then 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) ≡ 0 in [5], 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) ≡ 0 in [7], 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) ≡ 0 in [8], 𝑎 𝑖 𝑗 ( 𝑡 ) , 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) are constants and 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) ≡ 0 in [9], 𝑎 𝑖 𝑗 ( 𝑡 ) , 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) , 𝜏 𝑘 𝑙 ( 𝑡 ) are constants, and 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) ≡ 0 in [10, 11]. To the best of our knowledge, few authors have considered the existence and global exponential stability of almost periodic solutions for SICNNs with nonlinear behaved functions, periodic coefficients and mixed delays. Obviously, (1.1) is general and is worth to continue to investigate its dynamical properties such as existence and global exponential stability of almost periodic solutions.

The main purpose of this paper is to get sufficient conditions on the existence and global exponential stability of almost periodic solutions for SICNNs (1.1) by using the Leray-Schauder fixed point theorem and differential inequality techniques. Our results are general and possess infinitely adjustable real parameters and can be deduced to many simple results, including some existing results as special cases. Therefore, our results provide a wider application criteria for neural networks.

The remaining part of this paper is organized as follows. We first state some useful definitions and lemmas in Section 2. In Section 3, we study the existence of almost periodic solutions of system (1.1) by using the Leray-Schauder's fixed point theorem. In Section 4, by using Lemma 2.7, we will derive sufficient conditions for the global exponential stability of the almost periodic solution of system (1.1). A useful corollary is also obtained. An illustrative example and its simulation are given in Section 5.

2. Preliminaries

For convenience, we denote

𝑥 𝑥 = 𝑖 𝑗 = 𝑥 ( 𝑡 ) 1 1 ( 𝑡 ) , 𝑥 1 2 ( 𝑡 ) , … , 𝑥 1 𝑚 ( 𝑡 ) , … , 𝑥 𝑛 1 ( 𝑡 ) , 𝑥 𝑛 2 ( 𝑡 ) , … , 𝑥 𝑛 𝑚 ( 𝑡 ) 𝑇 . ( 2 . 1 )

Definition 2.1. The continuous function 𝑥 𝑖 𝑗 ( 𝑡 ) ∶ ℝ → ℝ is called almost periodic on ℝ , if for any 𝜀 > 0 , it is possible to find a real number 𝑙 = 𝑙 ( 𝜀 ) > 0 such that, for any interval with length 𝑙 , there exists a number 𝜏 = 𝜏 ( 𝜀 ) in this interval such that | 𝑥 𝑖 𝑗 ( 𝑡 + 𝜏 ) − 𝑥 𝑖 𝑗 ( 𝑡 ) | < 𝜀 , for any 𝑡 ∈ ℝ .

Definition 2.2 (see [22, page 21]). Let 𝔼 be a Banach space, 𝔻 an open subset in 𝔼 and 𝑓 ( 𝑡 , 𝑥 ) ∈ 𝐶 ( ℝ × 𝔻 , 𝔼 ) . For ∀ 𝑥 ∈ 𝔻 , 𝑓 ( 𝑡 , 𝑥 ) is called uniformly almost periodic about 𝑡 , if for any 𝜀 > 0 and any compact subset 𝕊 ⊂ 𝔻 , there exists a real number 𝑙 = 𝑙 ( 𝜀 , 𝕊 ) > 0 such that, for any interval with length 𝑙 , there exists a number 𝜏 = 𝜏 ( 𝜀 , 𝕊 ) in this interval such that ‖ 𝑓 ( 𝑡 + 𝜏 , 𝑥 ) − 𝑓 ( 𝑡 , 𝑥 ) ‖ < 𝜀 , for any ( 𝑡 , 𝑥 ) ∈ ℝ × 𝕊 .

The initial condition 𝜑 = { 𝜑 𝑖 𝑗 ( 𝑠 ) } of (1.1) is of the form

𝑥 𝑖 𝑗 ( 𝑠 ) = 𝜑 𝑖 𝑗 ] , ( 𝑠 ) , 𝑠 ∈ ( − ∞ , 0 ( 2 . 2 ) where 𝜑 𝑖 𝑗 ( 𝑠 ) , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , … , 𝑚 , are continuous almost periodic solutions.

Definition 2.3. Let 𝑥 ∗ ( 𝑡 ) be an almost periodic solution of (1.1) with initial value 𝜑 ∗ . If there exist constants 𝛼 > 0 and 𝑃 > 1 such that for every solution 𝑥 ( 𝑡 ) of (1.1) with initial value 𝜑 | | 𝑥 𝑖 𝑗 ( 𝑡 ) − 𝑥 ∗ 𝑖 𝑗 ( | | 𝑡 ) ≤ 𝑃 ‖ 𝜑 − 𝜑 ∗ ‖ 𝑒 − 𝛼 𝑡 , ∀ 𝑡 > 0 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , ( 2 . 3 ) where ‖ 𝜑 − 𝜑 ∗ ‖ = m a x ( 𝑖 , 𝑗 ) s u p 𝑠 ≤ 0 { | 𝜑 𝑖 𝑗 ( 𝑠 ) − 𝜑 ∗ 𝑖 𝑗 ( 𝑠 ) | } . Then 𝑥 ∗ ( 𝑡 ) is said to be globally exponentially stable.

Lemma 2.4 (see [22, page 136]). Suppose that 𝑓 ( 𝑡 , 𝑥 ) is uniformly continuous on ℝ × 𝕊 , where 𝕊 is any compact set on ℝ 𝑛 , and that there exists a nonsingular matrix 𝑃 ( 𝑡 ) ∈ 𝐶 1 , such that (a)there exists a constant 𝜌 satisfies ‖ 𝑃 ( 𝑡 ) ‖ < 𝜌 , for all 𝑡 ∈ ℝ , (b)the eigenvalues 𝜆 𝑖 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , of 𝑃 ( 𝑡 ) satisfy | 𝜆 𝑖 ( 𝑡 ) | ≥ 𝜗 > 0 , where 𝜗 is a constant, and there are 𝑘 negative eigenvalues, 𝑛 − 𝑘 nonnegative eigenvalues, (c)all the eigenvalues 𝜆 𝑖 ( 𝑡 , 𝑥 ) , 𝑖 = 1 , 2 , … , 𝑛 , of the following symmetric matrix: 𝑀 ( 𝑡 , 𝑥 ) = 𝑃 ( 𝑡 ) 𝑓 ( 𝑡 , 𝑥 ) + 𝑓 𝑇 ̇ ( 𝑡 , 𝑥 ) 𝑃 ( 𝑡 ) + 𝑃 ( 𝑡 ) ( 2 . 4 ) satisfy 𝜆 𝑖 ( 𝑡 , 𝑥 ) ≤ − 𝛿 < 0 , ∀ ‖ 𝑥 ‖ ≤ 𝑅 0 , ( 2 . 5 ) where 𝛿 , 𝑅 0 are constants.Then, for any fixed 𝜙 ( 𝑡 ) ∈ 𝐶 ( ℝ , ℝ 𝑛 ) and any 𝑡 ∈ ℝ satisfying ‖ 𝜙 ( 𝑡 ) ‖ < 𝑅 0 , the linear differential equation ̇ 𝑥 = 𝑓 ( 𝑡 , 𝜙 ( 𝑡 ) ) 𝑥 ( 2 . 6 ) admits an exponential dichotomy on ℝ : | | 𝑋 𝜙 ( 𝑡 ) 𝑄 𝜙 𝑋 𝜙 − 1 | | ( 𝑠 ) ≤ 𝑀 𝜌 𝜗 𝑒 − ( 𝛿 / 2 𝜌 ) ( 𝑡 − 𝑠 ) | | 𝑋 , 𝑡 ≥ 𝑠 , 𝜙 ( 𝑡 ) 𝐼 − 𝑄 𝜙 𝑋 𝜙 − 1 | | ( 𝑠 ) ≤ 𝑀 𝜌 𝜗 𝑒 − ( 𝛿 / 2 𝜌 ) ( 𝑡 − 𝑠 ) , 𝑠 ≥ 𝑡 , ( 2 . 7 ) where 𝑋 𝜙 ( 𝑡 ) is fundamental solution matrix of (2.6) satisfying 𝑋 𝜙 ( 0 ) = 𝐼 𝑛 × 𝑛 , 𝐼 𝑛 × 𝑛 is the identity matrix, 𝑄 𝜙 is a constant projection, the constant 𝑀 has no relationship with 𝜙 ( 𝑡 ) .

Lemma 2.5 (see [22, page 139]). Suppose that the 𝑛 × 𝑛 matrix function 𝑓 ( 𝑡 , 𝑥 ) and the 𝑛 -dimensional vector function 𝑔 ( 𝑡 , 𝑥 ) are uniformly almost periodic on ℝ × ℝ 𝑛 , and that there exists real symmetric nonsingular matrix 𝑃 ( 𝑡 ) satisfying the conditions (a)–(c) in Lemma 2.4, then, ̇ 𝑥 = 𝑓 ( 𝑡 , 𝜙 ) 𝑥 + 𝑔 ( 𝑡 , 𝜙 ) ( 2 . 8 ) has a unique almost periodic solution 𝑥 ( 𝑡 ) , where 𝜙 ∈ 𝐶 ( ℝ , ℝ 𝑛 ) is almost periodic function and 𝑥 ( 𝑡 ) = 𝑡 − ∞ 𝑋 𝜙 ( 𝑠 ) 𝑄 𝜙 𝑋 𝜙 − 1 ( 𝑠 ) 𝑔 ( 𝑠 , 𝜙 ( 𝑠 ) ) d 𝑠 − 𝑡 + ∞ 𝑋 𝜙 ( 𝑠 ) 𝐼 − 𝑄 𝜙 𝑋 𝜙 − 1 ( 𝑠 ) 𝑔 ( 𝑠 , 𝜙 ( 𝑠 ) ) d 𝑠 . ( 2 . 9 )

Lemma 2.6 (Leray-Schauder). Let 𝔼 be a Banach space, and let the operator Φ ∶ 𝔼 → 𝔼 be completely continuous. If the set { ‖ 𝑥 ‖ ∣ 𝑥 ∈ 𝔼 , 𝑥 = 𝜆 Φ 𝑥 , 0 < 𝜆 < 1 } is bounded, then Φ has a fixed point in 𝕋 , where 𝕋 = { 𝑥 ∣ 𝑥 ∈ 𝐸 , ‖ 𝑥 ‖ ≤ 𝑅 } , 𝑅 = s u p { ‖ 𝑥 ‖ ∣ 𝑥 = 𝜆 Φ 𝑥 , 0 < 𝜆 < 1 } . ( 2 . 1 0 )

Lemma 2.7 (see [23]). Let 𝑎 ≥ 0 , 𝑏 𝑘 ≥ 0 ( 𝑘 = 1 , 2 , … , 𝑚 ) , the following inequality holds 𝑎 𝑚 𝑘 = 1 𝑏 𝑞 𝑘 𝑘 ≤ 1 𝑟 𝑚 𝑘 = 1 𝑞 𝑘 𝑏 𝑟 𝑘 + 1 𝑟 𝑎 𝑟 , ( 2 . 1 1 ) where 𝑞 𝑘 > 0 , ( 𝑘 = 1 , 2 , … , 𝑚 ) is some constants, ∑ 𝑚 𝑘 = 1 𝑞 𝑘 = 𝑟 − 1 , and 𝑟 > 1 .

Obviously, inequality (2.11) also holds for 𝑎 ≥ 0 , 𝑏 𝑘 ≥ 0 , 𝑟 = 1 , and 𝑞 𝑘 = 0 , 𝑘 = 1 , 2 , … , 𝑚 . Hence, we always assume that 𝑎 ≥ 0 , 𝑏 𝑘 ≥ 0 , 𝑟 ≥ 1 , and 𝑞 𝑘 ≥ 0 , 𝑘 = 1 , 2 , … , 𝑚 in (2.11) in the later sections of this paper.

Furthermore, throughout this paper, we assume that

(H1) 𝑎 𝑖 𝑗 ( 𝑡 , 𝑢 ) ∈ 𝐶 ( ℝ 2 , ℝ ) is continuous almost periodic about the first argument and, there exists a positive continuous almost periodic function 𝜇 𝑖 𝑗 ( 𝑡 ) such that 𝜕 𝑎 𝑖 𝑗 ( 𝑡 , 𝑢 ) / 𝜕 𝑢 ≥ 𝜇 𝑖 𝑗 ( 𝑡 ) , 𝑢 ∈ ℝ , and 𝑎 𝑖 𝑗 ( 𝑡 , 0 ) = 0 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , (H2)there exist nonnegative constants 𝑀 𝑖 𝑗 and 𝑁 𝑖 𝑗 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , such that | | 𝑓 𝑖 𝑗 | | ( 𝑢 ) ≤ 𝑀 𝑖 𝑗 , | | 𝑔 𝑖 𝑗 | | ( 𝑢 ) ≤ 𝑁 𝑖 𝑗 , ∀ 𝑢 ∈ ℝ ; ( 2 . 1 2 ) (H3)the delay kernels 𝑘 𝑖 𝑗 ∶ [ 0 , + ∞ ) → ℝ are continuous, integrable and there are positive constants 𝑘 𝑖 𝑗 such that 0 + ∞ | | 𝑘 𝑖 𝑗 | | ( 𝑠 ) d 𝑠 ≤ 𝑘 𝑖 𝑗 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 ; ( 2 . 1 3 ) (H4)there exists a constant 𝛼 0 > 0 such that 0 + ∞ | | 𝑘 𝑖 𝑗 | | 𝑒 ( 𝑠 ) 𝛼 0 𝑠 d 𝑠 < + ∞ , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 . ( 2 . 1 4 ) (H5)the following inequality holds: m a x ( 𝑖 , 𝑗 ) s u p 𝑡 ∈ ℝ ∑ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) | | 𝐵 𝑘 𝑙 𝑖 𝑗 | | 𝑀 ( 𝑡 ) 𝑖 𝑗 + ∑ 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) | | 𝐶 𝑘 𝑙 𝑖 𝑗 | | ( 𝑡 ) 𝑘 𝑖 𝑗 𝑁 𝑖 𝑗 𝜇 𝑖 𝑗 ( 𝑡 ) = 𝜂 < 1 ; ( 2 . 1 5 ) (H6) there are nonnegative constants 𝛼 𝑖 𝑗 , 𝛽 𝑖 𝑗 such that 𝛼 𝑖 𝑗 = s u p 𝑢 ≠ 𝑣 | | | | 𝑓 𝑖 𝑗 ( 𝑢 ) − 𝑓 𝑖 𝑗 ( 𝑣 ) | | | | 𝑢 − 𝑣 , 𝛽 𝑖 𝑗 = s u p 𝑢 ≠ 𝑣 | | | | 𝑔 𝑖 𝑗 ( 𝑢 ) − 𝑔 𝑖 𝑗 ( 𝑣 ) | | | | 𝑢 − 𝑣 ( 2 . 1 6 ) for all 𝑢 , 𝑣 ∈ ℝ , 𝑢 ≠ 𝑣 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 .

3. Existence of Almost Periodic Solutions

Let 𝜉 𝑖 𝑗 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 be constants. Make the following transformation:

𝑥 𝑖 𝑗 = 𝜉 𝑖 𝑗 𝑦 𝑖 𝑗 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , ( 3 . 1 ) then (1.1) can be reformulated as

𝑦 𝑖 𝑗 ( 𝑡 ) = − 𝜉 − 1 𝑖 𝑗 𝑎 𝑖 𝑗 𝑡 , 𝜉 𝑖 𝑗 𝑦 𝑖 𝑗 ( + 𝑡 ) 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝑦 𝑘 𝑙 𝑡 − 𝜏 𝑘 𝑙 ( 𝑦 𝑡 ) 𝑖 𝑗 ( + 𝑡 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝑘 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝑦 𝑘 𝑙 ( 𝑠 ) d 𝑠 𝑦 𝑖 𝑗 ( 𝑡 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ( 𝑡 ) . ( 3 . 2 )

System (3.2) can be rewritten as

𝑦 𝑖 𝑗 ( 𝑡 ) = − 𝑑 𝑖 𝑗 𝑡 , 𝑦 𝑖 𝑗 ( 𝑦 𝑡 ) 𝑖 𝑗 ( 𝑡 ) + 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝑦 𝑘 𝑙 𝑡 − 𝜏 𝑘 𝑙 ( 𝑦 𝑡 ) 𝑖 𝑗 ( + 𝑡 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝑘 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝑦 𝑘 𝑙 ( 𝑠 ) d 𝑠 𝑦 𝑖 𝑗 ( 𝑡 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ( 𝑡 ) , ( 3 . 3 ) where 𝑑 𝑖 𝑗 ( 𝑡 , 𝑦 𝑖 𝑗 ( 𝑡 ) ) ≜ ( 𝜕 𝑎 𝑖 𝑗 ( 𝑡 , 𝑧 ) / 𝜕 𝑧 ) | 𝑧 = 𝑒 𝑖 𝑗 , 𝑒 𝑖 𝑗 is between 0 and 𝜉 𝑖 𝑗 𝑦 𝑖 𝑗 ( 𝑡 ) , 𝑒 𝑖 𝑗 ∈ ℝ . By ( H 1 ) , we know that 𝑎 𝑖 𝑗 ( 𝑡 , 𝜉 𝑖 𝑗 𝑦 𝑖 𝑗 ) is strictly monotone increasing about 𝑦 𝑖 𝑗 . Hence, 𝑑 𝑖 𝑗 ( 𝑡 , 𝑦 𝑖 𝑗 ( 𝑡 ) ) is unique for any 𝑦 𝑖 𝑗 ( 𝑡 ) . Obviously, 𝑑 𝑖 𝑗 ( 𝑡 , 𝑦 𝑖 𝑗 ( 𝑡 ) ) is continuous almost periodic about the first argument and 𝑑 𝑖 𝑗 ( 𝑡 , 𝑦 𝑖 𝑗 ( 𝑡 ) ) ≥ 𝜇 𝑖 𝑗 ( 𝑡 ) .

Take 𝕏 = { 𝜙 = { 𝜙 𝑖 𝑗 ( 𝑡 ) } ∣ 𝜙 𝑖 𝑗 ∶ ℝ → ℝ is an almost periodic function, 𝑖 = 1 , … , 𝑛 , 𝑗 = 1 , … , 𝑚 } . Then 𝕏 is a Banach space with the norm

‖ 𝜙 ‖ = m a x ( 𝑖 , 𝑗 ) | | 𝜙 𝑖 𝑗 | | 0 , | | 𝜙 𝑖 𝑗 | | 0 = s u p 𝑡 ∈ ℝ | | 𝜙 𝑖 𝑗 | | ( 𝑡 ) , 𝑖 = 1 , … , 𝑛 , 𝑗 = 1 , … , 𝑚 . ( 3 . 4 )

For for all 𝜙 ∈ 𝕏 , we consider the following auxiliary equation:

𝑦 𝑖 𝑗 ( 𝑡 ) = − 𝑑 𝑖 𝑗 𝑡 , 𝜙 𝑖 𝑗 ( 𝑦 𝑡 ) 𝑖 𝑗 ( 𝑡 ) + 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 𝑡 − 𝜏 𝑘 𝑙 ( 𝜙 𝑡 ) 𝑖 𝑗 ( + 𝑡 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝑘 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 ( 𝑠 ) d 𝑠 𝜙 𝑖 𝑗 ( 𝑡 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ( 𝑡 ) . ( 3 . 5 ) From ( H 1 ) , we know that

𝑑 𝑖 𝑗 𝑡 , 𝑦 𝑖 𝑗 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , ( 3 . 6 ) are uniformly almost periodic functions on ℝ × ℝ . Since 𝜇 𝑖 𝑗 ( 𝑡 ) , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , are positive continuous almost periodic functions, there exists a positive constant 𝛿 such that

𝑑 𝑖 𝑗 𝑡 , 𝜙 𝑖 𝑗 ( 𝑡 ) ≥ 𝜇 𝑖 𝑗 ( 𝑡 ) ≥ 𝛿 > 0 . ( 3 . 7 ) Hence, the conditions in Lemma 2.5 are satisfied (take 𝑃 ( 𝑡 ) = 𝐼 𝑛 𝑚 × 𝑛 𝑚 ).

According to the fact that 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) , 𝜙 𝑖 𝑗 ( 𝑡 ) , 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) , 𝐼 𝑖 𝑗 ( 𝑡 ) are almost periodic functions, in view of Lemma 2.5, we know that system (3.5) has a unique almost periodic solution

𝑦 𝜙 ( 𝑡 ) = 𝑡 − ∞ 𝑒 − ∫ 𝑡 𝑠 𝑑 𝑖 𝑗 ( 𝑢 , 𝜙 𝑖 𝑗 ( 𝑢 ) ) d 𝑢 × ⎡ ⎢ ⎢ ⎣ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 𝑠 − 𝜏 𝑘 𝑙 𝜙 ( 𝑠 ) 𝑖 𝑗 + ( 𝑠 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑠 − ∞ 𝑘 𝑖 𝑗 ( 𝑠 − 𝑢 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 ( 𝑢 ) d 𝑢 𝜙 𝑖 𝑗 ( 𝑠 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ⎤ ⎥ ⎥ ⎦ ⎫ ⎪ ⎬ ⎪ ⎭ . ( 𝑠 ) d 𝑠 ( 3 . 8 )

Set a mapping Φ ∶ 𝕏 → 𝕏 by setting

( Φ 𝜙 ) ( 𝑡 ) = 𝑦 𝜙 ( 𝑡 ) , ∀ 𝜙 ∈ 𝕏 . ( 3 . 9 )

Before using Lemma 2.6 to obtain conditions of the existence of almost periodic solution for (1.1), we have to prove the following lemma.

Lemma 3.1. Suppose that (H1)–( H5) hold. Then Φ ∶ 𝕏 → 𝕏 is completely continuous.

Proof. Under our assumptions, it is clear that the operator Φ is continuous. Next, we show that Φ is compact.
For any constant 𝐷 > 0 , let Ω = { 𝜙 ∣ 𝜙 ∈ 𝕏 , ‖ 𝜙 ‖ < 𝐷 } . Then, for any 𝜙 ∈ Ω , we have
‖ ‖ ( Φ 𝜙 ) = m a x ( 𝑖 , 𝑗 ) s u p 𝑡 ∈ ℝ | | | | 𝑡 − ∞ 𝑒 − ∫ 𝑡 𝑠 𝑑 𝑖 𝑗 ( 𝑢 , 𝜙 𝑖 𝑗 ( 𝑢 ) ) d 𝑢 × ⎡ ⎢ ⎢ ⎣ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 𝑠 − 𝜏 𝑘 𝑙 𝜙 ( 𝑠 ) 𝑖 𝑗 + ( 𝑠 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) C 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑠 − ∞ 𝑘 𝑖 𝑗 ( 𝑠 − 𝑢 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 ( 𝑢 ) d 𝑢 𝜙 𝑖 𝑗 ( 𝑠 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ⎤ ⎥ ⎥ ⎦ | | | | | ⎫ ⎪ ⎬ ⎪ ⎭ ( 𝑠 ) d 𝑠 ≤ m a x ( 𝑖 , 𝑗 ) s u p 𝑡 ∈ ℝ ⎧ ⎪ ⎨ ⎪ ⎩ 𝑡 − ∞ 𝑒 − ∫ 𝑡 𝑠 𝜇 𝑖 𝑗 ( 𝑢 ) d 𝑢 ⎡ ⎢ ⎢ ⎣ ⎛ ⎜ ⎜ ⎝ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) | | 𝐵 𝑘 𝑙 𝑖 𝑗 ( | | 𝑀 𝑠 ) 𝑖 𝑗 + 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) | | 𝐶 𝑘 𝑙 𝑖 𝑗 ( | | 𝑠 ) 𝑘 𝑖 𝑗 𝑁 𝑖 𝑗 ⎞ ⎟ ⎟ ⎠ × | | 𝜙 𝑖 𝑗 | | ( 𝑠 ) + 𝜉 − 1 𝑖 𝑗 | | 𝐼 𝑖 𝑗 | | ⎤ ⎥ ⎥ ⎦ ⎫ ⎪ ⎬ ⎪ ⎭ ( 𝑠 ) d 𝑠 ≤ m a x ( 𝑖 , 𝑗 ) s u p 𝑡 ∈ ℝ 𝑡 − ∞ 𝑒 − ∫ 𝑡 𝑠 𝜇 𝑖 𝑗 ( 𝑢 ) d 𝑢 𝜇 𝑖 𝑗 ( 𝑠 ) d 𝑠 𝜂 ‖ 𝜙 ‖ + m a x ( 𝑖 , 𝑗 ) 𝐼 𝑖 𝑗 𝜉 𝑖 𝑗 𝜇 𝑖 𝑗 < 𝜂 𝐷 + m a x ( 𝑖 , 𝑗 ) 𝐼 𝑖 𝑗 𝜉 𝑖 𝑗 𝜇 𝑖 𝑗 , ( 3 . 1 0 ) where 𝐼 𝑖 𝑗 = m a x 𝑡 ∈ 𝑅 | 𝐼 𝑖 𝑗 ( 𝑡 ) | , 𝜇 𝑖 𝑗 = i n f 𝑡 ∈ ℝ 𝜇 𝑖 𝑗 ( 𝑡 ) . Hence, Φ ( Ω ) is uniformly bounded.
By the definition of Φ , we get
( Φ 𝜙 ) 𝑖 𝑗 ( d 𝑡 ) = d 𝑡 𝑡 − ∞ 𝑒 − ∫ 𝑡 𝑠 𝑑 𝑖 𝑗 ( 𝑢 , 𝜙 𝑖 𝑗 ( 𝑢 ) ) d 𝑢 × ⎡ ⎢ ⎢ ⎣ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 𝑠 − 𝜏 𝑘 𝑙 𝜙 ( 𝑠 ) 𝑖 𝑗 + ( 𝑠 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑠 ) 𝑠 − ∞ 𝑘 𝑖 𝑗 ( 𝑠 − 𝑢 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 ( 𝑢 ) d 𝑢 𝜙 𝑖 𝑗 ( 𝑠 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎠ ( 𝑠 ) d 𝑠 = − 𝑑 𝑖 𝑗 𝑡 , 𝜙 𝑖 𝑗 ( 𝑡 ) Φ 𝜙 𝑖 𝑗 ( 𝑡 ) + 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑓 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 𝑡 − 𝜏 𝑘 𝑙 𝜙 ( 𝑡 ) 𝑖 𝑗 + ( 𝑡 ) 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) 𝑡 − ∞ 𝑘 𝑖 𝑗 ( 𝑡 − 𝑠 ) 𝑔 𝑖 𝑗 𝜉 𝑘 𝑙 𝜙 𝑘 𝑙 ( 𝑠 ) d 𝑠 𝜙 𝑖 𝑗 ( 𝑡 ) + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ( 𝑡 ) . ( 3 . 1 1 ) Since 𝑑 𝑖 𝑗 ( 𝑡 , 𝜙 𝑖 𝑗 ( 𝑡 ) ) , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , are uniformly almost periodic functions on ℝ × Ω , there exists a positive constant Θ such that | | 𝑑 𝑖 𝑗 𝑡 , 𝜙 𝑖 𝑗 | | ( 𝑡 ) ≤ Θ , f o r 𝑡 ∈ ℝ , 𝜙 𝑖 𝑗 ( 𝑡 ) ∈ Ω , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 . ( 3 . 1 2 ) Hence, ‖ ‖ ( Φ 𝜙 ) 𝑖 𝑗 ‖ ‖ ( 𝑡 ) ≤ Θ 𝜂 𝐷 + m a x ( 𝑖 , 𝑗 ) 𝐼 𝑖 𝑗 𝜉 𝑖 𝑗 𝜇 𝑖 𝑗 + m a x ( 𝑖 , 𝑗 ) ⎧ ⎪ ⎨ ⎪ ⎩ 𝐵 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐵 𝑘 𝑙 𝑖 𝑗 𝑀 𝑖 𝑗 𝐷 + 𝐶 𝑘 𝑙 ∈ 𝑁 𝑟 ( 𝑖 , 𝑗 ) 𝐶 𝑘 𝑙 𝑖 𝑗 𝑘 𝑖 𝑗 𝑁 𝑖 𝑗 𝐷 + 𝜉 − 1 𝑖 𝑗 𝐼 𝑖 𝑗 ⎫ ⎪ ⎬ ⎪ ⎭ , ( 3 . 1 3 ) where 𝐵 𝑘 𝑙 𝑖 𝑗 = s u p 𝑡 ∈ 𝑅 | 𝐵 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) | , 𝐶 𝑘 𝑙 𝑖 𝑗 = s u p 𝑡 ∈ ℝ | 𝐶 𝑘 𝑙 𝑖 𝑗 ( 𝑡 ) | . So, Φ ( Ω ) ⊆ 𝕏 is a family of uniformly bounded and equicontinuous subsets. By using the Arzela-Ascoli theorem, Φ ∶ 𝕏 → 𝕏 is compact. Therefore, Φ ∶ 𝕏 → 𝕏 is completely continuous. This completes the proof.

Theorem 3.2. Suppose that ( H 1 ) - ( H 5 ) hold. Let 𝜉 𝑖 𝑗 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , be constants. Then system (1.1) has an almost periodic solution 𝑥 ∗ ( 𝑡 ) with ‖ 𝑥 ∗ ‖ ≤ m a x ( 𝑖 , 𝑗 ) { 𝜉 𝑖 𝑗 } 𝑅 ≜ 𝑅 0 , where 𝑅 = m a x ( 𝑖 , 𝑗 ) 𝐼 𝑖 𝑗 / 𝜉 𝑖 𝑗 𝜇 𝑖 𝑗 . 1 − 𝜂 ( 3 . 1 4 )

Proof. Let 𝜙 ∈ 𝕏 . From Lemma 3.1, we get that Φ ∶ 𝕏 → 𝕏 is completely continuous. Consider the following operator equation: 𝜙 = 𝜆 Φ 𝜙 , 𝜆 ∈ ( 0 , 1 ) . ( 3 . 1 5 ) If 𝜙 is a solution of (3.15), we obtain ‖ 𝜙 ‖ ≤ ‖ Φ 𝜙 ‖ ≤ 𝜃 ‖ 𝜙 ‖ + m a x ( 𝑖 , 𝑗 ) 𝐼 𝑖 𝑗 𝜉 𝑖 𝑗 𝜇 𝑖 𝑗 . ( 3 . 1 6 ) This and ( H 5 ) imply that ‖ 𝜙 ‖ ≤ 𝑅 . ( 3 . 1 7 ) In view of Lemma 2.6, we obtain that Φ has a fixed point 𝜙 ∗ ( 𝑡 ) with ‖ 𝜙 ∗ 𝑅 ‖ ≤ . From (3.5) and (3.8), we know that 𝜙 ∗ satisfies (3.3) and (3.2). Hence, system (3.2) has an almost periodic solution 𝜙 ∗ ( 𝑡 ) = { 𝜙 ∗ 𝑖 𝑗 ( 𝑡 ) } with ‖ 𝜙 ∗ 𝑅 ‖ ≤ . It follows from (3.1) that 𝑥 ∗ ( 𝑡 ) = { 𝑥 ∗ 𝑖 𝑗 ( 𝑡 ) } = { 𝜉 𝑖 𝑗 𝜙 ∗ 𝑖 𝑗 ( 𝑡 ) } is one almost periodic solution of (1.1) with ‖ 𝑥 ∗ ‖ ≤ m a x ( 𝑖 , 𝑗 ) 𝜉 𝑖 𝑗 𝑅 ≜ 𝑅 0 . ( 3 . 1 8 ) This completes the proof.

4. Stability of Almost Periodic solution

In this section, we prove that, under suitable conditions, the almost periodic solution obtained in Theorem 3.2 is globally exponentially stable.

Theorem 4.1. Assume that ( H 1 ) - ( H 4 ) and ( H 6 ) hold and (H7)there are constants 𝑜 𝑠 , 𝜀 𝑠 , 𝑙 𝑠 , 𝛾 𝑠 , 𝛿 𝑠 , 𝑑 𝑠 ∈ ℝ , 𝑞 ≥ 1 , 𝑝 𝑠 ≥ 0 , 𝜎 > 0 , 𝜉 𝑖 𝑗 > 0 , 𝑠 = 1 , 2 , … , 𝑚 + 1 , 𝑖 = 1 , 2 , … , 𝑛 , 𝑗 = 1 , 2 , … , 𝑚 , such that Λ = m a x ( 𝑖 , 𝑗 ) s u p 𝑡 ∈ 𝑅 ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ − 𝑞 𝜇 𝑖 𝑗 (