Advances in Difference Equations
Volume 2009 (2009), Article ID 798685, 14 pages
doi:10.1155/2009/798685
Research Article

Global Stability Analysis for Periodic Solution in Discontinuous Neural Networks with Nonlinear Growth Activations

Yingwei Li1 and Huaiqin Wu2

1College of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China
2Department of Applied Mathematics, Yanshan University, Qinhuangdao 066004, China

Received 30 December 2008; Accepted 18 March 2009

Academic Editor: Toka Diagana

Copyright © 2009 Yingwei Li and Huaiqin Wu. 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

This paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.

1. Introduction

The stability of neural networks, which includes the stability of periodic solution and the stability of equilibrium point, has been extensively studied by many authors so far; see, for example, [1–15]. In [1–4], the authors investigated the stability of periodic solutions of neural networks with or without time delays, where the assumptions on neuron activation functions include Lipschitz conditions, bounded and/or monotonic increasing property. Recently, in [13–15], the authors discussed global stability of the equilibrium points for the neural networks with discontinuous neuron activations. Particularly, in [14], Forti conjectures that all solutions of neural networks with discontinuous neuron activations converge to an asymptotically stable limit cycle whenever the neuron inputs are periodic functions. As far as we know, there are only works of Wu in [5, 7] and Papini and Taddei in [9] dealing with this conjecture. However, the activation functions are required to be monotonic in [5, 7, 9] and to be bounded in [5, 7].

In this paper, without assumptions of the boundedness and the monotonicity of the activation functions, by the Leray-Schauder alternative theorem in differential inclusion theory and some new analysis techniques, we study the existence of periodic solution for discontinuous neural networks with nonlinear growth activations. By constructing suitable Lyapunov functions we give a general condition on the global asymptotical stability of periodic solution. The results obtained in this paper show that Forti's conjecture in [14] for discontinuous neural networks with nonlinear growth activations is true.

For later discussion, we introduce the following notations.

Let 𝑥 = ( 𝑥 1 , … , 𝑥 𝑛 ) , 𝑦 = ( 𝑦 1 , … , 𝑦 𝑛 ) , where the prime means the transpose. By 𝑥 > 0 (resp., 𝑥 ≥ 0 ) we mean that 𝑥 𝑖 > 0 (resp., 𝑥 𝑖 ≥ 0 ) for all 𝑖 = 1 , … , 𝑛 . ∑ ‖ 𝑥 ‖ = ( 𝑛 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝑥 2 𝑖 ) 1 / 2 denotes the Euclidean norm of 𝑥 . ∑ ⟨ 𝑥 , 𝑦 ⟩ = 𝑛 𝑖 = 1 0 𝑥 0 2 0 0 𝑑 𝑥 𝑖 𝑦 𝑖 ⟨ ⋅ , ⋅ ⟩ denotes the inner product. ‖ 𝐵 ‖ denotes 2-norm of matrix 𝐵 ∈ 𝑅 𝑛 × 𝑛 , that is, √ ‖ 𝐵 ‖ = 𝜎 ( 𝐵 𝐵 ) , where 𝜎 ( 𝐵 𝐵 ) denotes the spectral radius of 𝐵 𝐵 .

Given a set 𝐶 ⊂ 𝑅 𝑛 , by 𝐾 [ 𝐶 ] we denote the closure of the convex hull of 𝐶 , and 𝑃 𝑘 𝑐 ( 𝐶 ) denotes the collection of all nonempty, closed, and convex subsets of 𝐶 . Let 𝑋 be a Banach space, and ‖ 𝑥 ‖ 𝑋 denotes the norm of 𝑥 , ∀ 𝑥 ∈ 𝑋 . By 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) we denote the Banach space of the Lebesgue integrable functions 𝑥 ( ⋅ ) : [ 0 , 𝜔 ] → 𝑅 𝑛 equipped with the norm ∫ 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 . Let 𝑉 ∶ 𝑅 𝑛 → 𝑅 be a locally Lipschitz continuous function. Clarke’s generalized gradient [16] of 𝑉 at 𝑥 is defined by 𝜕 𝑉 ( 𝑥 ) = 𝐾 l i m 𝑖 → ∞ 𝑥 ∇ 𝑉 𝑖 ∶ l i m 𝑖 → ∞ 𝑥 𝑖 = 𝑥 , 𝑥 𝑖 ∈ 𝑅 𝑛 ⧵ Ω 𝑉 ∪ 𝒩 , ( 1 . 1 ) where Ω 𝑉 ⊂ 𝑅 𝑛 is the set of Lebesgue measure zero where ∇ 𝑉 does not exist, and 𝒩 ⊂ 𝑅 𝑛 is an arbitrary set with measure zero.

The rest of this paper is organized as follows. Section 2 develops a discontinuous neural network model with nonlinear growth activations, and some preliminaries also are given. Section 3 presents the proof on the existence of periodic solution. Section 4 discusses global asymptotical stability of the neural network. Illustrative examples are provided to show the effectiveness of the obtained results in Section 5.

2. Model Description and Preliminaries

The model we consider in the present paper is the neural networks modeled by the differential equation ̇ 𝑥 ( 𝑡 ) = − 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝑔 ( 𝑥 ( 𝑡 ) ) + 𝐼 ( 𝑡 ) , ( 2 . 1 ) where 𝑥 ( 𝑡 ) = ( 𝑥 1 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ) is the vector of neuron states at time 𝑡 ; 𝐴 is an 𝑛 × 𝑛 matrix representing the neuron inhibition; 𝐵 is an 𝑛 × 𝑛 neuron interconnection matrix; 𝑔 ∶ 𝑅 𝑛 → 𝑅 𝑛 , 𝑔 𝑖 , 𝑖 = 1 , … , 𝑛 , represents the neuron input-output activation and 𝐼 ( 𝑡 ) = ( 𝐼 1 ( 𝑡 ) , … , 𝐼 𝑛 ( 𝑡 ) ) is the continuous 𝜔 -periodic vector function denoting neuron inputs.

Throughout the paper, we assume that

𝐻 1 : ( 1 ) 𝑔 𝑖 has only a finite number of discontinuity points in every compact set of 𝑅 . Moreover, there exist finite right limit 𝑔 𝑖 ( 𝜈 + 𝑘 ) and left limit 𝑔 𝑖 ( 𝜈 − 𝑘 ) at discontinuity point 𝜈 𝑘 .

( 2 ) 𝑔 has the nonlinear growth property, that is, for all 𝑥 ∈ 𝑅 𝑛 [ 𝑔 ] ‖ 𝐾 ( 𝑥 ) ‖ = s u p 𝛾 ∈ 𝐾 [ 𝑔 ( 𝑥 ) ] ‖ 𝛾 ‖ ≤ 𝑐 1 + ‖ 𝑥 ‖ 𝛼 , ( 2 . 2 ) where 𝑐 > 0 , 𝛼 ∈ ( 0 , 1 ) are constants, and 𝐾 [ 𝑔 ( 𝑥 ) ] = ( 𝐾 [ 𝑔 1 ( 𝑥 1 ) ] , … , 𝐾 [ 𝑔 𝑛 ( 𝑥 𝑛 ) ] ) .

𝐻 2 : ⟨ 𝑥 , 𝐴 𝑥 ⟩ ≥ 𝜌 ‖ 𝑥 ‖ 2 for all 𝑥 ∈ 𝑅 𝑛 , where 𝜌 > 0 is a constant.

Under the assumption 𝐻 1 , 𝑔 is undefined at the points where 𝑔 is discontinuous. Equation (2.1) is a differential equation with a discontinuous right-hand side. For (2.1), we adopt the following definition of the solution in the sense of Filippov [17] in this paper.

Definition 2.1. Under the assumption 𝐻 1 , a solution of (2.1) on an interval [ 0 , 𝑏 ) with the initial value 𝑥 ( 0 ) = 𝑥 0 is an absolutely continuous function satisfying [ ] [ ̇ 𝑥 ( 𝑡 ) ∈ − 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝐾 𝑔 ( 𝑥 ( 𝑡 ) ) + 𝐼 ( 𝑡 ) , f o r a . e . 𝑡 ∈ 0 , 𝑏 ) . ( 2 . 3 )
It is easy to see that 𝜙 : ( 𝑥 , 𝑡 ) ↪ − 𝐴 𝑥 + 𝐵 𝐾 [ 𝑔 ( 𝑥 ) ] + 𝐼 ( 𝑡 ) is an upper semicontinuous set-valued map with nonempty compact convex values; hence, it is measurable [18]. By the measurable selection theorem [19], if 𝑥 ( ⋅ ) is a solution of (2.1), then there exists a measurable function 𝜂 ( 𝑡 ) ∈ 𝐾 [ 𝑔 ( 𝑥 ( 𝑡 ) ) ] such that [ ̇ 𝑥 ( 𝑡 ) = − 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝜂 ( 𝑡 ) + 𝐼 ( 𝑡 ) , f o r a . e . 𝑡 ∈ 0 , 𝑏 ) . ( 2 . 4 )
Consider the following differential inclusion problem [ ] [ ] , 𝑥 ̇ 𝑥 ( 𝑡 ) ∈ − 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝐾 𝑔 ( 𝑥 ( 𝑡 ) ) + 𝐼 ( 𝑡 ) , f o r a . e . 𝑡 ∈ 0 , 𝜔 ( 0 ) = 𝑥 ( 𝜔 ) . ( 2 . 5 )
It easily follows that if 𝑥 ( 𝑡 ) is a solution of (2.5), then 𝑥 ∗ ( 𝑡 ) defined by 𝑥 ∗ [ ] ( 𝑡 ) = 𝑥 ( 𝑡 − 𝑗 𝜔 ) , 𝑡 ∈ 𝑗 𝜔 , ( 𝑗 + 1 ) 𝜔 , 𝑗 ∈ 𝑁 ( 2 . 6 ) is an 𝜔 -periodic solution of (2.1). Hence, for the neural network (2.1), finding the periodic solutions is equivalent to finding solutions of (2.5).

Definition 2.2. The periodic solution 𝑥 ∗ ( 𝑡 ) with initial value 𝑥 ∗ ( 0 ) = 𝑥 ∗ 0 of the neural network (2.1) is said to be globally asymptotically stable if 𝑥 ∗ is stable and for any solution 𝑥 ( 𝑡 ) , whose existence interval is [ 0 , + ∞ ) , we have l i m 𝑡 → + ∞ ‖ 𝑥 ( 𝑡 ) − 𝑥 ∗ ( 𝑡 ) ‖ = 0 .

Lemma 2.3. If 𝑋 is a Banach space, 𝐶 ⊆ 𝑋 is nonempty closed convex with 0 ∈ 𝐶 and 𝐺 ∶ 𝐶 → 𝑃 𝑘 𝑐 ( 𝐶 ) is an upper semicontinuous set-valued map which maps bounded sets into relatively compact sets, then one of the following statements is true: (a)the set Γ = { 𝑥 ∈ 𝐶 ∶ 𝑥 ∈ 𝜆 𝐺 ( 𝑥 ) , 𝜆 ∈ ( 0 , 1 ) } is unbounded; (b)the 𝐺 ( ⋅ ) has a fixed point in 𝐶 , that is, there exists 𝑥 ∈ 𝐶 , such that 𝑥 ∈ 𝐺 ( 𝑥 ) .

Lemma 2.3 is said to be the Leray-Schauder alternative theorem, whose proof can be found in [20]. Define the following: 𝑊 1 , 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 [ ] 𝑊 ) = { 𝑥 ( ⋅ ) ∶ 𝑥 ( ⋅ ) i s a b s o l u t e c o n t i n u o u s o n 0 , 𝜔 } , 𝑝 1 , 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 ) = 𝑥 ( ⋅ ) ∈ 𝑊 1 , 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 , ) ∣ 𝑥 ( 0 ) = 𝑥 ( 𝜔 ) ‖ 𝑥 ‖ 𝑊 1 , 1 = 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 + 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ ̇ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 , ∀ 𝑥 ∈ 𝑊 1 , 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 ) , ( 2 . 7 ) then ‖ ⋅ ‖ 𝑊 1 , 1 is a class of norms of 𝑊 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , 𝑊 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , and 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) ⊂ 𝑊 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) are Banach space under the norm ‖ 𝑥 ‖ 𝑊 1 , 1 .

If 𝑉 ∶ 𝑅 𝑛 → 𝑅 is (i) regular in 𝑅 𝑛 [16]; (ii) positive definite, that is, 𝑉 ( 𝑥 ) > 0 for 𝑥 ≠ 0 , and 𝑉 ( 0 ) = 0 ; (iii) radially unbounded, that is, 𝑉 ( 𝑥 ) → + ∞ as ‖ 𝑥 ‖ → + ∞ , then 𝑉 ( 𝑥 ) is said to be C-regular.

Lemma 2.4 (Chain Rule [15]). If 𝑉 ( 𝑥 ) is C-regular and 𝑥 ( ⋅ ) ∶ [ 0 , + ∞ ) → 𝑅 𝑛 is absolutely continuous on any compact interval of [ 0 , + ∞ ) , then 𝑥 ( 𝑡 ) and 𝑉 ( 𝑥 ( 𝑡 ) ) ∶ [ 0 , + ∞ ) → 𝑅 are differential for a.e. 𝑡 ∈ [ 0 , + ∞ ) , and one has 𝑑 𝑑 𝑡 𝑉 ( 𝑥 ( 𝑡 ) ) = 𝜁 , 𝑑 𝑥 ( 𝑡 ) 𝑑 𝑡 , ∀ 𝜁 ∈ 𝜕 𝑉 ( 𝑥 ) . ( 2 . 8 )

3. Existence of Periodic Solution

Theorem 3.1. If the assumptions 𝐻 1 and 𝐻 2 hold, then for any 𝑥 0 ∈ 𝑅 𝑛 , (2.1) has at least a solution defined on [ 0 , + ∞ ) with the initial value 𝑥 ( 0 ) = 𝑥 0 .

Proof. By the assumption 𝐻 1 , it is easy to get that 𝜙 : ( 𝑥 , 𝑡 ) ↪ − 𝐴 𝑥 + 𝐵 𝐾 [ 𝑔 ( 𝑥 ) ] + 𝐼 ( 𝑡 ) is an upper semicontinuous set-valued map with nonempty, compact, and convex values. Hence, by Definition 2.1, the local existence of a solution 𝑥 ( ⋅ ) for (2.1) on [ 0 , 𝑡 0 ] , 𝑡 0 > 0 , with 𝑥 ( 0 ) = 𝑥 0 , is obvious [17].
Set 𝜓 ( 𝑡 , 𝑥 ) = 𝐵 𝐾 [ 𝑔 ( 𝑥 ) ] + 𝐼 ( 𝑡 ) . Since 𝐼 ( ⋅ ) is a continuous 𝜔 -periodic vector function, 𝐼 ( ⋅ ) is bounded, that is, there exists a constant ℐ > 0 such that ‖ 𝐼 ( 𝑡 ) ‖ ≤ ℐ , 𝑡 ∈ [ 0 , 𝜔 ] . By the assumption 𝐻 1 , we have s u p 𝑥 ∈ 𝑅 𝑛 , 𝑡 ∈ [ 0 , + ∞ ] [ 𝑔 ] ‖ 𝜓 ( 𝑡 , 𝑥 ) ‖ ≤ ‖ 𝐵 ‖ ‖ 𝐾 ( 𝑥 ) ‖ + ‖ 𝐼 ( 𝑡 ) ‖ ≤ 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ‖ 𝛼 + ℐ . ( 3 . 1 ) By l i m ‖ 𝑥 ‖ → + ∞ ( 𝑐 ‖ 𝐵 ‖ ( 1 + ‖ 𝑥 ‖ 𝛼 ) + ℐ ) / ‖ 𝑥 ‖ = 0 , we can choose a constant 𝑅 > 0 , such that when 𝑅 ‖ 𝑥 ‖ > , 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ‖ 𝛼 + ℐ < 𝜌 ‖ 𝑥 ‖ 2 . ( 3 . 2 ) By (2.4), (3.1), (3.2), and the Cauchy inequality, when 𝑅 ‖ 𝑥 ( 𝑡 ) ‖ > , 1 2 𝑑 𝑑 𝑡 ‖ 𝑥 ( 𝑡 ) ‖ 2 = ⟨ 𝑥 ( 𝑡 ) , ̇ 𝑥 ( 𝑡 ) ⟩ = ⟨ 𝑥 ( 𝑡 ) , − 𝐴 𝑥 ( 𝑡 ) + 𝐵 𝜂 ( 𝑡 ) + 𝐼 ( 𝑡 ) ⟩ = − ⟨ 𝑥 ( 𝑡 ) , 𝐴 𝑥 ( 𝑡 ) ⟩ + ⟨ 𝑥 ( 𝑡 ) , 𝐵 𝜂 ( 𝑡 ) + 𝐼 ( 𝑡 ) ⟩ ≤ − 𝜌 ‖ 𝑥 ( 𝑡 ) ‖ 2 + ‖ 𝑥 ( 𝑡 ) ‖ 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ( 𝑡 ) ‖ 𝛼 = + ℐ − 𝜌 + 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ( 𝑡 ) ‖ 𝛼 + ℐ ‖ ‖ 𝑥 ( 𝑡 ) ‖ 𝑥 ( 𝑡 ) ‖ 2 𝜌 < − 2 ‖ 𝑥 ( 𝑡 ) ‖ 2 < 0 . ( 3 . 3 ) Therefore, let 𝑅 = m a x { ‖ 𝑥 0 ‖ , 𝑅 } , then, by (3.3), it follows that ‖ 𝑥 ( 𝑡 ) ‖ ≤ 𝑅 on [ 0 , 𝑡 0 ] . This means that the local solution 𝑥 ( ⋅ ) is bounded. Thus, (2.1) has at least a solution with the initial value 𝑥 ( 0 ) = 𝑥 0 on [ 0 , + ∞ ) . This completes the proof.

Theorem 3.1 shows the existence of solutions of (2.1). In the following, we will prove that (2.1) has an 𝜔 -periodic solution.

Let 𝐿 𝑥 = ̇ 𝑥 + 𝐴 𝑥 for all 𝑥 ∈ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , then 𝐿 ∶ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) → 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) is a linear operator.

Proposition 3.2. 𝐿 ∶ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) → 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) is bounded, one to one and surjective.

Proof. For any 𝑥 ∈ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , we have ‖ 𝐿 𝑥 ‖ 𝐿 1 = 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ ̇ 𝑥 ( 𝑡 ) + 𝐴 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 ≤ 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ ̇ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 + 𝜔 0 ≤ 0 𝑥 0 2 0 0 𝑑 ‖ 𝐴 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ ̇ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 + ‖ 𝐴 ‖ 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ 𝑥 ( 𝑡 ) ‖ 𝑑 𝑡 ≤ m a x { 1 , ‖ 𝐴 ‖ } ‖ 𝑥 ‖ 𝑊 1 , 1 ( 3 . 4 ) this implies that 𝐿 is bounded.
Let 𝑥 1 , 𝑥 2 ∈ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . If 𝐿 𝑥 1 = 𝐿 𝑥 2 , then ̇ 𝑥 1 ( 𝑡 ) − ̇ 𝑥 2 𝑥 ( 𝑡 ) = − 𝐴 1 ( 𝑡 ) − 𝑥 2 ( 𝑡 ) . ( 3 . 5 ) By the assumption 𝐻 2 , 𝑑 − ‖ ‖ 𝑥 𝑑 𝑡 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ ( 𝑡 ) 2 = − 2 ⟨ 𝑥 1 ( 𝑡 ) − 𝑥 2 ( 𝑡 ) , ̇ 𝑥 1 ( 𝑡 ) − ̇ 𝑥 2 𝑥 ( 𝑡 ) ⟩ = 2 1 ( 𝑡 ) − 𝑥 2 𝑥 ( 𝑡 ) , 𝐴 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ 𝑥 ( 𝑡 ) ≥ 2 𝜌 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ 2 . ( 3 . 6 ) Noting 𝑥 1 ( 0 ) = 𝑥 1 ( 𝜔 ) , 𝑥 2 ( 0 ) = 𝑥 2 ( 𝜔 ) , we have 𝜔 0 𝑑 0 𝑥 0 2 0 0 𝑑 − ‖ ‖ 𝑥 𝑑 𝑡 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ ( 𝑡 ) 2 ‖ ‖ 𝑥 𝑑 𝑡 = 1 ( 0 ) − 𝑥 2 ‖ ‖ ( 0 ) 2 − ‖ ‖ 𝑥 1 ( 𝜔 ) − 𝑥 2 ‖ ‖ ( 𝜔 ) 2 = 0 , ( 3 . 7 ) By (3.6), 0 ≤ 𝜔 0 ‖ ‖ 𝑥 0 𝑥 0 2 0 0 𝑑 2 𝜌 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ ( 𝑡 ) 2 𝑑 𝑡 ≤ 𝜔 0 𝑑 0 𝑥 0 2 0 0 𝑑 − ‖ ‖ 𝑥 𝑑 𝑡 1 ( 𝑡 ) − 𝑥 2 ‖ ‖ ( 𝑡 ) 2 𝑑 𝑡 = 0 . ( 3 . 8 ) Hence ‖ 𝑥 1 − 𝑥 2 ‖ = 0 . It follows 𝑥 1 = 𝑥 2 . This shows that 𝐿 is one to one.
Let 𝑓 ( ⋅ ) ∈ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . In order to verify that 𝐿 is surjective, in the following, we will prove that there exists 𝑥 ( ⋅ ) ∈ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) such that 𝐿 𝑥 = 𝑓 , ( 3 . 9 ) that is, we will prove that there exists a solution for the differential equation 𝑥 ̇ 𝑥 ( 𝑡 ) = − 𝐴 𝑥 ( 𝑡 ) + 𝑓 ( 𝑡 ) , ( 0 ) = 𝑥 ( 𝜔 ) . ( 3 . 1 0 )
Consider Cauchy problem 𝑥 ̇ 𝑥 ( 𝑡 ) = − 𝐴 𝑥 ( 𝑡 ) + 𝑓 ( 𝑡 ) , ( 0 ) = 𝜉 . ( 3 . 1 1 )
It is easily checked that 𝑥 ( 𝑡 ) = 𝑒 − 𝐴 𝑡 𝜉 + 𝑡 0 0 𝑥 0 2 0 0 𝑑 𝑓 ( 𝑠 ) 𝑒 𝐴 𝑠 𝑑 𝑠 ( 3 . 1 2 ) is the solution of (3.11). By (3.12), we want 𝜉 = 𝑥 ( 𝜔 ) , then 𝑒 − 𝐴 𝜔 𝜉 + 𝜔 0 0 𝑥 0 2 0 0 𝑑 𝑓 ( 𝑠 ) 𝑒 𝐴 𝑠 𝑑 𝑠 = 𝜉 , ( 3 . 1 3 ) that is, 𝐼 − 𝑒 − 𝐴 𝜔 𝜉 = 𝑒 − 𝐴 𝜔 𝜔 0 0 𝑥 0 2 0 0 𝑑 𝑓 ( 𝑠 ) 𝑒 𝐴 𝑠 𝑑 𝑠 . ( 3 . 1 4 ) By the assumption 𝐻 2 , 𝐼 − 𝑒 − 𝐴 𝜔 is a nonsingular matrix, where 𝐼 is a unit matrix. Thus, by (3.14), if we take 𝜉 as 𝜉 = 𝐼 − 𝑒 − 𝐴 𝜔 − 1 𝑒 − 𝐴 𝜔 𝜔 0 0 𝑥 0 2 0 0 𝑑 𝑓 ( 𝑠 ) 𝑒 𝐴 𝑠 𝑑 𝑠 ( 3 . 1 5 ) in (3.12), then (3.12) is the solution of (3.10). This shows that 𝐿 is surjective. This completes the proof.

By the Banach inverse operator theorem, 𝐿 − 1 ∶ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) → 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) ⊆ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) is a bounded linear operator.

For any 𝑥 ( ⋅ ) ∈ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , define the set-valued map 𝒩 as 𝒩 ( 𝑥 ) = 𝑣 ( ⋅ ) ∈ 𝐿 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 [ ] ) ∣ 𝑣 ( 𝑡 ) ∈ 𝜓 ( 𝑡 , 𝑥 ( 𝑡 ) ) , f o r a . e . 𝑡 ∈ 0 , 𝜔 . ( 3 . 1 6 ) Then 𝒩 has the following properties.

Proposition 3.3. 𝒩 ( ⋅ ) ∶ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) ↪ 2 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) 𝑤 has nonempty closed convex values in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) and is also upper semicontinuous from 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) into 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) endowed with the weak topology.

Proof. The closedness and convexity of values of 𝒩 ( ⋅ ) are clear. Next, we verify the nonemptiness. In fact, for any 𝑥 ∈ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , there exists a sequence of step functions { 𝑠 𝑛 } 𝑛 ≥ 1 such that 𝑠 𝑛 ( 𝑡 ) → 𝑥 ( 𝑡 ) and ‖ 𝑠 𝑛 ( 𝑡 ) ‖ ≤ ‖ 𝑥 ( 𝑡 ) ‖ a.e. on [ 0 , 𝜔 ] . By the assumption 𝐻 1 (1) and the continuity of 𝐼 ( 𝑡 ) , we can get that ( 𝑡 , 𝑥 ) → 𝜓 ( 𝑡 , 𝑥 ) is graph measurable. Hence, for any 𝑛 ≥ 1 , 𝑡 → 𝜓 ( 𝑡 , 𝑠 𝑛 ( 𝑡 ) ) admits a measurable selector 𝑓 𝑛 ( 𝑡 ) . By the assumption 𝐻 1 (2), { 𝑓 𝑛 ( ⋅ ) } 𝑛 ≥ 1 is uniformly integrable. So by Dunford-Pettis theorem, there exists a subsequence { 𝑓 𝑛 𝑘 ( ⋅ ) } 𝑘 ≥ 1 ⊆ { 𝑓 𝑛 ( ⋅ ) } 𝑛 ≥ 1 such that 𝑓 𝑛 𝑘 → 𝑓 weakly in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Hence, from [21, Theorem  3.1], we have 𝑓 ( 𝑡 ) ∈ 𝐾 𝑤 − 𝑓 l i m 𝑛 𝑘 ( 𝑡 ) 𝑘 ≥ 1 ⊆ 𝐾 𝑤 − l i m 𝜓 𝑡 , 𝑠 𝑛 𝑘 [ ] ( 𝑡 ) , a . e . o n 0 , 𝜔 . ( 3 . 1 7 ) Noting that 𝜓 ( 𝑡 , ⋅ ) is an upper semicontinuous set-valued map with nonempty closed convex values on 𝑥 for a.e. 𝑡 ∈ [ 0 , 𝜔 ] , 𝐾 [ 𝑤 − l i m 𝜓 ( 𝑡 , 𝑠 𝑛 𝑘 ( 𝑡 ) ) ] ⊆ 𝜓 ( 𝑡 , 𝑥 ( 𝑡 ) ) . Therefore, 𝑓 ∈ 𝒩 ( 𝑥 ) . This shows that 𝒩 ( 𝑥 ( ⋅ ) ) is nonempty.
At last we will prove that 𝒩 ( ⋅ ) is upper semicontinuous from 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) into 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) 𝑤 . Let 𝐶 be a nonempty and weakly closed subset of 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , then we need only to prove that the set 𝒩 − 1 ( 𝐶 ) = 𝑥 ∈ 𝐿 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 ) ∶ 𝒩 ( 𝑥 ) 0 𝑥 0 2 0 0 𝑑 𝐶 ≠ ∅ ( 3 . 1 8 ) is closed. Let { 𝑥 𝑛 } 𝑛 ≥ 1 ⊆ 𝒩 − 1 ( 𝐶 ) and assume 𝑥 𝑛 → 𝑥 in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , then there exists a subsequence { 𝑥 𝑛 𝑘 } 𝑘 ≥ 1 ⊆ { 𝑥 𝑛 } 𝑛 ≥ 1 such that 𝑥 𝑛 𝑘 ( 𝑡 ) → 𝑥 ( 𝑡 ) a.e. on [ 0 , 𝜔 ] . Take 𝑓 𝑘 ∈ 𝒩 ( 𝑥 𝑛 𝑘 ) ⋂ 𝐶 , 𝑛 ≥ 1 , then By the assumption 𝐻 1 (2) and Dunford-Pettis theorem, there exists a subsequence { 𝑓 𝑘 𝑚 ( 𝑡 ) } 𝑚 ≥ 1 ⊆ { 𝑓 𝑘 ( 𝑡 ) } 𝑘 ≥ 1 such that 𝑓 𝑘 𝑚 → 𝑓 ∈ 𝐶 weakly in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . As before we have 𝑓 ( 𝑡 ) ∈ 𝐾 𝑤 − 𝑓 l i m 𝑘 𝑚 ( 𝑡 ) 𝑚 ≥ 1 ⊆ 𝐾 𝑤 − l i m 𝜓 𝑡 , 𝑥 𝑛 𝑘 𝑚 [ ] ( 𝑡 ) ⊆ 𝜓 ( 𝑡 , 𝑥 ( 𝑡 ) ) , a . e . o n 0 , 𝜔 . ( 3 . 1 9 ) This implies ⋂ 𝐶 𝑓 ∈ 𝒩 ( 𝑥 ) , that is, 𝒩 − 1 ( 𝐶 ) is closed in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . The proof is complete.

Theorem 3.4. Under the assumptions 𝐻 1 and 𝐻 2 , there exists a solution for the boundary-value problem (2.5), that is, the neural network (2.1) has an 𝜔 -periodic solution.

Proof. Consider the set-valued map 𝐿 − 1 ∘ 𝒩 . Since 𝐿 − 1 is continuous and 𝒩 is upper semicontinuous, the set-valued map 𝐿 − 1 ∘ 𝒩 is upper semicontinuous. Let 𝐾 ⊂ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) be a bounded set, then 𝒩 ( 𝐾 ) = 𝑥 ∈ 𝐾 0 𝑥 0 2 0 0 𝑑 𝒩 ( 𝑥 ) ( 3 . 2 0 ) is a bounded subset of 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Since 𝐿 − 1 is a bounded linear operator, 𝐿 − 1 ( 𝒩 ( 𝐾 ) ) is a bounded subset of 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Noting that 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) is compactly embedded in 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , 𝐿 − 1 ( 𝒩 ( 𝐾 ) ) is a relatively compact subset of 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Hence by Proposition 3.3, 𝐿 − 1 ∘ 𝒩 ∶ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) → 𝑃 𝑘 𝑐 ( 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) ) is the upper semicontinuous set-valued map which maps bounded sets into relatively compact sets.
For any 𝑟 ( 𝑥 ) ∈ 𝐾 [ 𝑔 ( 𝑥 ) ] , when 𝑥 ≠ 0 , 𝜆 ∈ ( 0 , 1 ) , by (3.1) and the Cauchy inequality, ⟨ 𝑥 , − 𝐴 𝑥 + 𝜆 𝐵 𝑟 ( 𝑥 ) + 𝜆 𝐼 ( 𝑡 ) ⟩ = − ⟨ 𝑥 , 𝐴 𝑥 ⟩ + ⟨ 𝑥 , 𝜆 𝐵 𝑟 ( 𝑥 ) + 𝜆 𝐼 ( 𝑡 ) ⟩ ≤ − 𝜌 ‖ 𝑥 ‖ 2 + 𝜆 ‖ 𝜓 ( 𝑡 , 𝑥 ) ‖ ‖ 𝑥 ‖ ≤ − 𝜌 ‖ 𝑥 ‖ 2 + 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ‖ 𝛼 ‖ = + ℐ 𝑥 ‖ − 𝜌 + 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ‖ 𝛼 + ℐ ‖ ‖ 𝑥 ‖ 𝑥 ‖ 2 . ( 3 . 2 1 ) Arguing as (3.2), we can choose a constant 𝐾 0 > 0 , such that when ‖ 𝑥 ‖ > 𝐾 0 , 𝑐 ‖ 𝐵 ‖ 1 + ‖ 𝑥 ‖ 𝛼 + ℐ < 𝜌 ‖ 𝑥 ‖ 2 . ( 3 . 2 2 ) Therefore, when ‖ 𝑥 ‖ > 𝐾 0 , by (3.21), 𝜌 ⟨ 𝑥 , − 𝐴 𝑥 + 𝜆 𝐵 𝑟 ( 𝑥 ) + 𝜆 𝐼 ( 𝑡 ) ⟩ < − 2 ‖ 𝑥 ‖ 2 [ 𝑔 ] , ∀ 𝑟 ( 𝑥 ) ∈ 𝐾 ( 𝑥 ) . ( 3 . 2 3 ) Set Γ = 𝑥 ∈ 𝐿 1 ( [ ] 0 , 𝜔 , 𝑅 𝑛 ) ∶ 𝑥 ∈ 𝜆 𝐿 − 1 ∘ 𝒩 ( 𝑥 ) , 𝜆 ∈ ( 0 , 1 ) . ( 3 . 2 4 ) In the following, we will prove that Γ is a bounded subset of 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Let 𝑥 ∈ Γ , then 𝑥 ∈ 𝜆 𝐿 − 1 ∘ 𝒩 ( 𝑥 ) , that is, 𝐿 𝑥 ∈ 𝜆 𝒩 ( 𝑥 ) . By the definition of 𝒩 ( ⋅ ) , there exists a measurable selection 𝑣 ( 𝑡 ) ∈ 𝐾 [ 𝑔 ( 𝑥 ( 𝑡 ) ) ] , such that ̇ 𝑥 ( 𝑡 ) + 𝐴 𝑥 ( 𝑡 ) = 𝜆 𝐵 𝑣 ( 𝑡 ) + 𝜆 𝐼 ( 𝑡 ) . ( 3 . 2 5 ) By (3.23) and (3.25), m a x 𝑡 ∈ [ 0 , 𝜔 ] ‖ 𝑥 ( 𝑡 ) ‖ ≤ 𝐾 0 . Otherwise, m a x 𝑡 ∈ [ 0 , 𝜔 ] ‖ 𝑥 ( 𝑡 ) ‖ > 𝐾 0 . By 𝐿 − 1 ∶ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) → 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) , we have 𝑥 ( 0 ) = 𝑥 ( 𝜔 ) . Since 𝑥 ( ⋅ ) is continuous, we can choose 𝑡 0 ∈ ( 0 , 𝜔 ] , such that ‖ ‖ 𝑥 𝑡 0 ‖ ‖ = m a x 𝑡 ∈ [ 0 , 𝜔 ] ‖ 𝑥 ( 𝑡 ) ‖ > 𝐾 0 . ( 3 . 2 6 ) Thus, there exists a constant 𝛿 𝑡 0 > 0 , such that when 𝑡 ∈ ( 𝑡 0 − 𝛿 𝑡 0 , 𝑡 0 ] , ‖ 𝑥 ( 𝑡 ) ‖ > 𝐾 0 . By (3.23) and (3.25), 1 0 ≤ 2 ‖ ‖ 𝑥 ( 𝑡 0 ) ‖ ‖ 2 − 1 2 ( ‖ 𝑥 𝑡 ) ‖ 2 = 1 2 𝑡 0 𝑡 𝑑 0 𝑥 0 2 0 0 𝑑 ( 𝑑 𝑠 ‖ 𝑥 𝑠 ) ‖ 2 𝑑 𝑠 = 𝑡 0 𝑡 0 𝑥 0 2 0 0 𝑑 ⟨ 𝑥 ( 𝑠 ) , ̇ 𝑥 ( 𝑠 ) ⟩ 𝑑 𝑠 = 𝑡 0 𝑡 𝜌 0 𝑥 0 2 0 0 𝑑 ⟨ 𝑥 ( 𝑠 ) , − 𝐴 𝑥 ( 𝑠 ) + 𝜆 𝐵 𝑣 ( 𝑠 ) + 𝜆 𝐼 ( 𝑠 ) ⟩ 𝑑 𝑠 < − 2 𝑡 0 𝑡 ‖ 0 𝑥 0 2 0 0 𝑑 ‖ 𝑥 ( 𝑠 ) 2 𝑡 𝑑 𝑠 < 0 , f o r 𝑡 ∈ 0 − 𝛿 𝑡 0 , 𝑡 0 . ( 3 . 2 7 ) This is a contradiction. Thus, for any 𝑥 ∈ Γ , m a x 𝑡 ∈ [ 0 , 𝜔 ] ‖ 𝑥 ( 𝑡 ) ‖ ≤ 𝐾 0 . Furthermore, we have ‖ x ‖ 𝐿 1 = 𝜔 0 0 𝑥 0 2 0 0 𝑑 ‖ 𝑥 ( 𝑠 ) ‖ 𝑑 𝑠 ≤ 𝜔 𝐾 0 , ∀ 𝑥 ∈ Γ . ( 3 . 2 8 ) This shows that Γ is a bounded subset of 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) .
By Lemma 2.3, the set-valued map 𝐿 − 1 ∘ 𝒩 has a fixed point, that is, there exists 𝑥 ∗ ∈ 𝐿 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) such that 𝑥 ∗ ∈ 𝐿 − 1 ∘ 𝒩 ( 𝑥 ∗ ) , 𝐿 𝑥 ∗ ∈ 𝒩 ( 𝑥 ∗ ) . Hence there exists a measurable selection 𝜂 ∗ ( 𝑡 ) ∈ 𝐾 [ 𝑔 ( 𝑥 ∗ ( 𝑡 ) ) ] , such that ̇ 𝑥 ∗ ( 𝑡 ) + 𝐴 𝑥 ∗ ( 𝑡 ) = 𝐵 𝜂 ∗ ( 𝑡 ) + 𝐼 ( 𝑡 ) . ( 3 . 2 9 ) By the definition of 𝐿 − 1 , 𝑥 ∗ ∈ 𝑊 𝑝 1 , 1 ( [ 0 , 𝜔 ] , 𝑅 𝑛 ) . Moreover, by Definition 2.1 and (3.29), 𝑥 ∗ ( ⋅ ) is a solution of the boundary-value problem (2.5), that is, the neural network (2.1) has an 𝜔 -periodic solution. The proof is completed.

4. Global Asymptotical Stability of Periodic Solution

Theorem 4.1. Suppose that 𝐻 1 and the following assumptions are satisfied.
𝐻 3 : for each 𝑖 ∈ 1 , … , 𝑛 , there exists a constant 𝑙 𝑖 > 0 , such that for all two different numbers 𝑢 , 𝑣 ∈ 𝑅 , for all 𝛾 𝑖 ∈ 𝐾 [ 𝑔 𝑖 ( 𝑢 ) ] and for all 𝜁 𝑖 ∈ 𝐾 [ 𝑔 𝑖 ( 𝑢 ) ] 𝛾 𝑖 − 𝜁 𝑖 𝑢 − 𝑣 ≥ − 𝑙 𝑖 . ( 4 . 1 )
𝐻 4 : 𝐴 = d i a g ( 𝑎 1 , … ,