Existence of Periodic Solutions for Shunting Inhibitory Cellular Neural Networks with Neutral Delays

Abstract

This paper considers the existence of periodic solutions for shunting inhibitory cellular neural networks (SICNNs) with neutral delays. By applying the theory of abstract continuation theorem of 𝑘-set contractive operator and some analysis technique, a new result on the existence of periodic solutions is obtained.

1. Introduction

It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. The dynamical behaviors of SICNNs with delays have been widely investigated in recent years. A large number of important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. In particular, there exist many results on the existence and stability of periodic and almost periodic solutions for SICNNs with delays. We refer the reader to [1–4] and references cited therein.

On the other hand, the stability analysis of various neutral delay-differential systems has drawn much research attention [5–7]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar.

Motivated by the above, we consider shunting inhibitory cellular neural networks with neutral delays described by 𝑥̇_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡), (1) where 𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, 𝐶_𝑖𝑗 is the cell at the (𝑖,𝑗) position of the lattice and the 𝑟-neighborhood 𝑁_𝑟(𝑖,𝑗) of 𝐶_𝑖𝑗 is given as 𝑁_𝑟𝑖,𝑗=𝐶_𝑘𝑙||,}.∶max𝑘−𝑖𝑙−𝑗≤𝑟,1≤𝑘≤𝑚,1≤𝑙≤𝑛 (2)𝑥_𝑖𝑗(𝑡) is the activity of the cell, 𝐶_𝑖𝑗, 𝑎_𝑖𝑗(𝑡)≥0 is the passive decay rate of the cell activity, 𝐼_𝑖𝑗(𝑡) is the external input to 𝐶_𝑖𝑗, 𝐶^𝑘𝑙_𝑖𝑗≥0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell 𝐶_𝑖𝑗, and the activity function 𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡)) is a continuous function representing the output or firing rate of cell 𝐶_𝑘𝑙.

By using the Lyapunov functions and linear matrix inequality approach (LMI), most authors studied the asymptotic stability or exponential stability of the equilibrium point (see [7–10]). However, few papers have been published on the existence of periodic solutions or almost periodic solutions for neutral type neural networks ([11–14]). Thus, it is worthwhile to study the existence of periodic solutions for neutral type neural networks with delays. And to the author’s best knowledge, there is no published paper considering the existence of periodic solutions for SICNNs with neutral delays.

The initial conditions of (1) are of the form 𝑥_𝑖𝑗(𝑡)=𝜑_𝑖𝑗(𝑠),𝑠∈(−∞,0],𝑖=1,…,𝑚,𝑗=1,…,𝑛, (3) where 𝜑_𝑖𝑗∈𝐶((−∞,0],𝑅).

For the sake of convenience, we denote Ω₁={1,2,…,𝑚},Ω₂={1,2,…,𝑛},ℎ=max_{[]𝑡∈0,𝜔}|ℎ(𝑡)|,ℎ=min_{[]𝑡∈0,𝜔}|ℎ(𝑡,1)|ℎ=𝜔^𝜔₀ℎ(𝑠)‖ℎ‖𝑑𝑠,₂=^𝜔₀|ℎ(𝑡)|²𝑑𝑡^1/2,ℎ̇₂=^𝜔₀ℎ̇(𝑡)²𝑑𝑡^1/2, (4) where ℎ is a continuous 𝜔-periodic function.

The main purpose of this paper is to obtain some sufficient conditions for the existence of the periodic solutions for system (1). By applying the theory of abstract continuation theorem of 𝑘-set contractive operator and some analysis technique, we derive some new sufficient conditions ensuring the existence of periodic solutions for system (1). Moreover, numerical examples are provided to illustrate the feasibility of the new results.

2. Conditions and Lemma

Throughout this paper, for 𝑖∈Ω₁,𝑗∈Ω₂, we assume that(H₁)𝑎_𝑖𝑗(𝑡),𝐶^𝑘𝑙_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝐼_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡) are continuous 𝜔-periodic functions;(H₂) function 𝜎_𝑖𝑗(𝑡) is nonnegative, bounded, 𝜔-periodic, and continuously differential defined on 𝑅⁺ and inf_𝑡∈𝑅⁺{1−𝜎̇_𝑖𝑗(𝑡)}>0, where 𝜎̇_𝑖𝑗(𝑡) expresses the derivative of 𝜎_𝑖𝑗(𝑡);(H₃) there exists positive constant 𝛼_𝑖𝑗 such that |𝑓_𝑖𝑗(𝑢)|≤𝛼_𝑖𝑗, for all 𝑢,𝑣∈𝑅.

In order to study (1), we should make some preparations.

Let 𝐸 be a Banach space. For a bounded subset 𝐴⊂𝐸, let 𝛼_𝐸(𝐴)𝐴=inf𝛿>0∶thereisafinitenumberofsubsets_𝑖⊂𝐴suchthat𝐴=_𝑖𝐴_𝑖anddiam𝐴_𝑖≤𝛿 (5) denote the (Kuratowski) measure of noncompactness, where diam (𝐴_𝑖) denotes the diameter of the set 𝐴_𝑖. Let 𝑋, 𝑌 be two Banach spaces and Ω be a bounded open subset of 𝑋. A continuous and bounded map 𝑁∶Ω→𝑌 is called 𝑘-set contractive if for any bounded set 𝑆⊂Ω we have 𝛼_𝑌(𝑁(𝑆))≤𝑘𝛼_𝑋(𝑆), where 𝑘 is a constant. In addition, for a Fredholm operator 𝐿∶𝑋→𝑌 with index zero, according to [15], we define 𝑙(𝐿)=sup𝑎≥0∶𝑎𝛼_𝑋(𝑆)≤𝛼_𝑌(𝐿(𝑆,.))∀boundedsubset𝑆⊂𝑋 (6)

Lemma 1 (see [16]). Let 𝐿∶𝑋→𝑌 be a Fredholm operator with index zero, and let 𝑟∈𝑌 be a fixed point. Suppose that 𝑁∶Ω→𝑌 is a 𝑘-set contractive with 𝑘<𝑙(𝐿), where Ω⊂𝑋 is bounded, open, and symmetric about 0∈Ω. Furthermore, we suppose that(1)𝐿𝑥≠𝜆𝑁𝑥+𝜆𝑟 for 𝑥∈𝜕Ω,𝜆∈(0,1),(2)[𝑄𝑁(𝑥)+𝑄𝑟,𝑥]⋅[𝑄𝑁(−𝑥)+𝑄𝑟,𝑥]<0 for 𝑥∈𝜕Ω∩Ker𝐿, where [⋅,⋅] is some bilinear form on 𝑌×𝑋 and 𝑄 is the project of 𝑌 onto Coker (𝐿).
Then there is a 𝑥∈Ω such that 𝐿𝑥−𝑁𝑥=𝑟.

In order to use Lemma 1 to establish the existence of periodic solutions of (1), we set 𝑌=𝐶_𝜔=()()(𝑡),𝑥𝑥∈𝐶𝑅,𝑅∶𝑥𝑡+𝜔=𝑥𝑥=₁₁,…,𝑥_1𝑛,…,𝑥_𝑖1𝑥,…,_𝑖𝑛,…,𝑥_𝑚1,…,𝑥_𝑚𝑛^𝑇 (7) with the norm defined by ‖𝑥‖=∑^𝑚_𝑖=1∑^𝑛_𝑗=1|𝑥_𝑖𝑗|₀, where |𝑥_𝑖𝑗|₀=max_{𝑡∈[0,𝜔]}|𝑥_𝑖𝑗(𝑡)| and 𝑋=𝐶¹_𝜔=𝑥∈𝐶¹()()(𝑡),𝑥𝑅,𝑅∶𝑥𝑡+𝜔=𝑥𝑥=₁₁,…,𝑥_1𝑛,…,𝑥_𝑖1𝑥,…,_𝑖𝑛,…,𝑥_𝑚1,…,𝑥_𝑚𝑛^𝑇 (8) with the norm defined by ‖𝑥‖₁=max{‖𝑥‖,‖𝑥̇‖}, where ‖𝑥̇‖=∑^𝑚_𝑖=1∑^𝑛_𝑗=1|𝑥̇_𝑖𝑗|₀, |𝑥̇_𝑖𝑗|₀=max_{𝑡∈[0,𝜔]}|𝑥̇_𝑖𝑗(𝑡)|. Then 𝑋, 𝑌 are Banach spaces.

Let 𝐿∶𝑋→𝑌 defined by 𝐿𝑥=𝑑𝑥=𝑥̇𝑑𝑡¹¹,…,𝑥̇_1𝑛,…,𝑥̇_𝑖1𝑥̇,…,_𝑖𝑛,…,𝑥̇_𝑚1,…,𝑥̇_𝑚𝑛^𝑇 (9) and 𝑁∶𝑋→𝑌 defined by the following: 𝑁𝑥_𝑖𝑗=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡), (10) for 𝑖∈Ω₁,𝑗∈Ω₂.

It is easy to see that 𝐿 is a Fredholm operator with index zero. Clearly, (1) has a 𝜔-periodic solution if and only if 𝐿𝑥=𝑁𝑥+𝑟 for some 𝑥∈𝑋, where 𝑟∶=𝐼(𝑡)=𝐼₁₁(𝑡),…,𝐼_1𝑛(𝑡),…,𝐼_𝑖1(𝑡)𝐼,…,_𝑖𝑛(𝑡),…,𝐼_𝑚1(𝑡),…,𝐼_𝑚𝑛(𝑡)^𝑇. (11)

Lemma 2 (see [17]). The differential operator 𝐿 is a Fredholm operator with index zero and satisfies 𝑙(𝐿)≥1.

Lemma 3. If 𝑘=max_{𝑖∈Ω1,𝑗∈Ω2}{𝑏_𝑖𝑗}<1, then 𝑁∶Ω→𝑌 is a 𝑘-set contractive map, where Ω⊂𝑋.

Proof. The proof of Lemma 3 is similar to Lemma  3 in [14] and will be omitted here.

Lemma 4 (see [13]). Let 𝑡₁,𝑡₂∈[0,𝜔]. If 𝑥∶[0,𝜔]→𝑅 is 𝜔-periodic, then 𝑥(𝑡)≤𝑥𝑡₁+^𝜔₀|𝑥̇(𝑠𝑥(𝑡)𝑡)|𝑑𝑠,≥𝑥₂−^𝜔₀|𝑥̇(𝑠)|𝑑𝑠. (12)

Lemma 5 (see [18]). Suppose that 𝜎∈𝑋 and 𝜎̇(𝑡)<1 for all 𝑡∈[0,𝜔], then the function 𝑡−𝜎(𝑡) has an inverse 𝜈∈𝐶(𝑅,𝑅) with 𝜈(𝑎+𝜔)=𝜈(𝑎)+𝜔, for all 𝑎∈𝑅.

Throughout this paper, we assume that 𝜎_𝑖𝑗∈𝑋 and 𝜎̇_𝑖𝑗(𝑡)<1,𝑖∈Ω₁,𝑗∈Ω₂. So that 𝜎_𝑖𝑗(𝑡) has a unique inverse, and we set 𝜈_𝑖𝑗(𝑡) to represent the inverse of function 𝑡−𝜎_𝑖𝑗(𝑡).

3. Main Results

Set 𝑝²_𝑖𝑗=max_𝑡∈𝑅11−𝜎̇_𝑖𝑗𝜈_𝑖𝑗(𝑡),𝐴=diag𝑎₁₁,…,𝑎_1𝑛,…,𝑎_𝑚1,…,𝑎_𝑚𝑛,𝐵=diag𝑏₁₁𝑝₁₁,…,𝑏_1𝑛𝑝_1𝑛,…,𝑏_𝑚1𝑝_𝑚1,…,𝑏_𝑚𝑛𝑝_𝑚𝑛,𝐷=diag1−𝑏₁₁𝑝₁₁,…,1−𝑏_1𝑛𝑝_1𝑛,…,1−𝑏_𝑚1𝑝_𝑚1,…,1−𝑏_𝑚𝑛𝑝_𝑚𝑛,𝐶=diag_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙₁₁𝛼₁₁,…,_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_1𝑛𝛼_1𝑛,…,_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑚1𝛼_𝑚1,…,_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑚𝑛𝛼_𝑚𝑛,𝐼=𝐼₁₁₂,…,𝐼_1𝑛2,…,𝐼_𝑚12,…,𝐼_𝑚𝑛2^𝑇,𝑍=𝑥₁₁₂,…,𝑥_1𝑛2,…,𝑥_𝑚12,…,𝑥_𝑚𝑛2^𝑇,𝑊=𝑥̇₁₁₂,…,𝑥̇_1𝑛2,…,𝑥̇_𝑚12,…,𝑥̇_𝑚𝑛2^𝑇. (13)

Theorem 6. Suppose that (𝐻₁)--(𝐻₃) hold, furthermore, assume that(H₄)min_{𝑖∈Ω1,𝑗∈Ω2}{1−𝑏_𝑖𝑗𝑝_𝑖𝑗,1−𝑏_𝑖𝑗}>0 and max_{𝑖∈Ω1,𝑗∈Ω2}{𝑎_𝑖𝑗−∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗}>0,(H₅)𝐴−𝐶−𝐵𝐷⁻¹(𝐴+𝐶) is nonsingular 𝑀-matrix, then system (1) has at least one 𝜔-periodic solution.

Proof. We consider the operator equation 𝐿𝑥=𝜆𝑁𝑥+𝜆𝑟, 𝜆∈(0,1), where 𝐿,𝑁 defined by (9) and (10), respectively. Corresponding to system (1), for 𝑖∈Ω₁,𝑗∈Ω₂, we have 𝑥̇_𝑖𝑗(𝑡)=𝜆−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡). (14) Suppose that (𝑥₁₁(𝑡),…,𝑥_1𝑛(𝑡),…,𝑥_𝑚1(𝑡),…,𝑥_𝑚𝑛(𝑡))^𝑇 is a solution of system (14) for a parameter 𝜆∈(0,1). Multiplying both sides of system (14) by 𝑥_𝑖𝑗(𝑡) and integrating over [0,𝜔], from (H₃) and Cauchy-Schwarz inequality, we obtain 𝑎_𝑖𝑗𝑥_𝑖𝑗²₂≤^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥²_𝑖𝑗(𝑡)=𝑑𝑡_𝜔0−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡≤_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗^𝜔₀𝑥²_𝑖𝑗(𝑡)+𝑑𝑡𝑏_𝑖𝑗^𝜔₀𝑥_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝑑𝑡_𝜔0𝑥_𝑖𝑗(𝑡)𝐼_𝑖𝑗(𝑡)𝑑𝑡≤_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗²₂+𝑏_𝑖𝑗𝑥_𝑖𝑗2^𝜔₀𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)²𝑑𝑡^1/2+^𝜔₀𝐼_𝑖𝑗(𝑡)²𝑑𝑡^1/2𝜔₀𝑥_𝑖𝑗(𝑡)²𝑑𝑡^1/2, (15) and according to Lemma 5, we have 𝑥̇^𝜔_0𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)²𝑑𝑡^1/2=^𝜔₀11−𝜎̇_𝑖𝑗𝜈_𝑖𝑗(𝑠)𝑥̇_𝑖𝑗(𝑠)²𝑑𝑠^1/2≤𝑝_𝑖𝑗𝑥̇_𝑖𝑗2,𝑖∈Ω₁,𝑗∈Ω₂. (16) That is, 𝑎_𝑖𝑗𝑥_𝑖𝑗2≤_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2+𝐼_𝑖𝑗2,𝑖∈Ω₁,𝑗∈Ω₂. (17) From the above inequality, we obtain 𝑎_𝑖𝑗−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2≤𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2+𝐼_𝑖𝑗2,𝑖∈Ω₁,𝑗∈Ω₂, (18) which gives (𝐴−𝐶)𝑍≤𝐵𝑊+𝐼. (19) Multiplying both sides of (14) by 𝑥̇_𝑖𝑗(𝑡) and integrating over [0,𝜔], for 𝑖∈Ω₁,𝑗∈Ω₂, we have 𝑥̇_𝑖𝑗²₂=𝜆^𝜔₀𝑥̇_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)×𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗×𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡)≤𝑑𝑡𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗^𝜔₀𝑥̇_𝑖𝑗(𝑡)²𝑑𝑡^1/2×^𝜔₀𝑥_𝑖𝑗(𝑡)²𝑑𝑡^1/2+𝑏_𝑖𝑗^𝜔₀𝑥̇_𝑖𝑗(𝑡)²𝑑𝑡^1/2×^𝜔₀𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)²𝑑𝑡^1/2+^𝜔₀𝐼_𝑖𝑗(𝑡)²𝑑𝑡^1/2𝜔₀𝑥̇_𝑖𝑗(𝑡)²𝑑𝑡^1/2≤𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2𝑥̇_𝑖𝑗2+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗²₂+𝑥̇_𝑖𝑗2𝐼_𝑖𝑗2. (20) Thus 𝑥̇_𝑖𝑗2≤𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2+𝐼_𝑖𝑗2, (21) that is, 1−𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2≤𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2+𝐼_𝑖𝑗2, (22) where 1−𝑏_𝑖𝑗𝑝_𝑖𝑗>0,𝑖∈Ω₁,𝑗∈Ω₂. Then we may rewrite (22) as 𝐷𝑊≤(𝐴+𝐶)𝑍+𝐼⟹𝑊≤𝐷⁻¹(𝐴+𝐶)𝑍+𝐷⁻¹𝐼. (23) Substituting (23) into (19), we get (𝐴−𝐶)𝑍≤𝐵𝐷⁻¹(𝐴+𝐶)𝑍+𝐷⁻¹𝐼⟹+𝐼𝐴−𝐶−𝐵𝐷⁻¹(𝐴+𝐶)𝑍≤𝐵𝐷⁻¹𝐼+𝐼. (24) By (H₅), we have 𝑍≤𝐻⁺≜(ℎ⁺₁₁,…,ℎ⁺_1𝑛,…,ℎ⁺_𝑚1,…,ℎ⁺_𝑚𝑛)^𝑇. That is, 𝑥_𝑖𝑗2≤ℎ⁺_𝑖𝑗,𝑖∈Ω₁,𝑗∈Ω₂, (25) where 𝐻⁺=(𝐴−𝐶−𝐵𝐷⁻¹(𝐴+𝐶))⁻¹(𝐵𝐷⁻¹𝐼+𝐼). Substituting (25) into (23), we have 𝑊≤𝐺⁺≜(𝑔⁺₁₁,…,𝑔⁺_1𝑛,…,𝑔⁺_𝑚1,…,𝑔⁺_𝑚𝑛)^𝑇. That is, 𝑥̇_𝑖𝑗2≤𝑔⁺_𝑖𝑗,𝑖∈Ω₁,𝑗∈Ω₂, (26) where 𝐺⁺=𝐷⁻¹(𝐴+𝐶)𝐻⁺+𝐷⁻¹𝐼.
Integrating both sides of (14), from 0 to 𝜔, we obtain 0=−^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)+𝑑𝑡_𝜔0−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡)𝑑𝑡. (27) Using Cauchy-Schwarz inequality, we have ^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡≤_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}^𝜔₀𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗(𝑡+)𝑑𝑡_𝜔0𝑏_𝑖𝑗𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)𝑑𝑡+^𝜔₀𝐼_𝑖𝑗(𝑡≤)𝑑𝑡_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2√𝜔+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2√𝜔+𝐼_𝑖𝑗2√𝜔. (28) In view of (14), we have 𝑥̇^𝜔_0𝑖𝑗(𝑡)𝑑𝑡≤^𝜔₀𝑎_𝑖𝑗𝑥_𝑖𝑗(𝑡)𝑑𝑡+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}^𝜔₀𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗(𝑡+)𝑑𝑡_𝜔0𝑏_𝑖𝑗𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)𝑑𝑡+^𝜔₀𝐼_𝑖𝑗(𝑡≤)𝑑𝑡𝑎_𝑖𝑗𝑥_𝑖𝑗2√𝜔+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2√𝜔+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2√𝜔+𝐼_𝑖𝑗2√𝜔. (29) From Lemma 4, for any 𝑡₁^𝑖𝑗,𝑡₂^𝑖𝑗∈[0,𝜔], 𝑖∈Ω₁,𝑗∈Ω₂, we have ^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡≤^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗𝑡₁^𝑖𝑗+𝑑𝑡_𝜔0𝑎_𝑖𝑗(𝑡)^𝜔₀𝑥̇_𝑖𝑗(𝑡)𝑑𝑡𝑑𝑡,_𝜔0𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡≥^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗𝑡₂^𝑖𝑗−𝑑𝑡_𝜔0𝑎_𝑖𝑗(𝑡)^𝜔₀𝑥̇_𝑖𝑗(𝑡)𝑑𝑡𝑑𝑡. (30) Dividing by ∫^𝜔₀𝑎_𝑖𝑗(𝑡)𝑑𝑡 on the two sides of the above inequalities, we obtain that 𝑥_𝑖𝑗𝑡₁^𝑖𝑗≥1∫^𝜔₀𝑎_𝑖𝑗(𝑡)𝑑𝑡^𝜔0𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡−^𝜔₀𝑥̇_𝑖𝑗(𝑡𝑥)𝑑𝑡,_𝑖𝑗𝑡₂^𝑖𝑗≤1∫^𝜔₀𝑎_𝑖𝑗(𝑡)𝑑𝑡^𝜔0𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡+^𝜔₀𝑥̇_𝑖𝑗(𝑡)𝑑𝑡. (31) Let 𝑡_𝑖𝑗,𝑡_𝑖𝑗∈[0,𝜔],𝑖∈Ω₁,𝑗∈Ω₂, such that 𝑥_𝑖𝑗(𝑡_𝑖𝑗)=max_{𝑡∈[0,𝜔]}𝑥_𝑖𝑗(𝑡),𝑥_𝑖𝑗(𝑡_𝑖𝑗)=min_{𝑡∈[0,𝜔]}𝑥_𝑖𝑗(𝑡), then by (25)–(31), we have 𝑥_𝑖𝑗𝑡_𝑖𝑗≤1𝑎_𝑖𝑗𝜔^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡+^𝜔₀𝑥̇_𝑖𝑗(𝑡≤1)𝑑𝑡𝑎_𝑖𝑗√𝜔_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2+𝐼_𝑖𝑗2+√𝜔𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗2+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑥̇_𝑖𝑗2+𝐼_𝑖𝑗2≤1𝑎_𝑖𝑗√𝜔_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗ℎ⁺_𝑖𝑗+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑔⁺_𝑖𝑗+𝐼_𝑖𝑗2+√𝜔𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗ℎ⁺_𝑖𝑗+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑔⁺_𝑖𝑗+𝐼_𝑖𝑗2≜𝑀⁽¹⁾_𝑖𝑗. (32)
Similarly, 𝑥_𝑖𝑗𝑡_𝑖𝑗1≥−𝑎_𝑖𝑗𝜔^𝜔₀𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)𝑑𝑡+^𝜔₀𝑥̇_𝑖𝑗(𝑡1)𝑑𝑡≥−𝑎_𝑖𝑗√𝜔_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗ℎ⁺_𝑖𝑗+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑔⁺_𝑖𝑗+𝐼_𝑖𝑗2+√𝜔𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗ℎ⁺_𝑖𝑗+𝑏_𝑖𝑗𝑝_𝑖𝑗𝑔⁺_𝑖𝑗+𝐼_𝑖𝑗2=−𝑀⁽¹⁾_𝑖𝑗. (33)
Thus, for all 𝑖∈Ω₁,𝑗∈Ω₂, |𝑥_𝑖𝑗(𝑡)|≤𝑀⁽¹⁾_𝑖𝑗, which yields that |𝑥_𝑖𝑗|₀≤𝑀⁽¹⁾_𝑖𝑗. Again from (14), we get 𝑥̇_𝑖𝑗0≤𝑎_𝑖𝑗𝑥_𝑖𝑗0+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑥_𝑖𝑗0+𝑏_𝑖𝑗𝑥̇_𝑖𝑗0+𝐼_𝑖𝑗≤𝑎_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑀⁽¹⁾_𝑖𝑗+𝑏_𝑖𝑗𝑥̇_𝑖𝑗0+𝐼_𝑖𝑗, (34) which gives |𝑥̇_𝑖𝑗|₀≤((𝑎_𝑖𝑗+∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗)𝑀⁽¹⁾_𝑖𝑗+𝐼_𝑖𝑗)/(1−𝑏_𝑖𝑗)≜𝑀⁽²⁾_𝑖𝑗.
Denote 𝑀⁽¹⁾=∑^𝑚_𝑖=1∑^𝑛_𝑗=1𝑀⁽¹⁾_𝑖𝑗, 𝑀⁽²⁾=∑^𝑚_𝑖=1∑^𝑛_𝑗=1𝑀⁽²⁾_𝑖𝑗, 𝑀⁽³⁾=max_{𝑖∈Ω1,𝑗∈Ω2}{|𝐼̃_𝑖𝑗|/(𝑎_𝑖𝑗−∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗)}. Let 𝑀>max{𝑀⁽¹⁾,𝑀⁽²⁾,𝑀⁽³⁾}, Ω={𝑥∈𝐶¹_𝜔∶‖𝑥‖₁<𝑀}, and define a bounded bilinear form [⋅,⋅] on 𝐶_𝜔×𝐶¹_𝜔 by [𝑦,𝑥]=∫^𝜔₀𝑦(𝑡)𝑥(𝑡)𝑑𝑡. Also we define 𝑄∶𝑦→Coker(𝐿) by 𝑦→∫^𝜔₀𝑦(𝑡)𝑑𝑡. Obviously, {𝑥∣𝑥∈Ker𝐿∩𝜕Ω}={𝑥∣𝑥≡𝑀or𝑥≡−𝑀}. Without loss of generality, we may assume that 𝑥≡𝑀. Let 𝑃_𝑖𝑗=−𝑎_𝑖𝑗𝑀−∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑓_𝑖𝑗(𝑀)𝑀+𝐼̃_𝑖𝑗,𝑄_𝑖𝑗=𝑎_𝑖𝑗𝑀+∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑓_𝑖𝑗(−𝑀)𝑀+𝐼̃_𝑖𝑗, 𝑖∈Ω₁,𝑗∈Ω₂. Thus [𝑄𝑁(𝑥)+𝑄(𝑟),𝑥][𝑄𝑁(−𝑥)+𝑄(𝑟),𝑥]=𝑀²𝜔²⋅𝑃₁₁𝑄₁₁⋮𝑃_𝑚𝑛𝑄_𝑚𝑛. (35)
Considering assumption (H₄), we know 𝑀>max_{𝑖∈Ω1,𝑗∈Ω2}{|𝐼̃_𝑖𝑗|/(𝑎_𝑖𝑗−∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗)}>0, and it follows that 𝑎_𝑖𝑗𝑀>∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝛼_𝑖𝑗𝑀+𝐼̃_𝑖𝑗>∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗|𝑓_𝑖𝑗(𝑀)|𝑀+𝐼̃_𝑖𝑗. So we have [𝑄𝑁(𝑥_𝑖𝑗)+𝑄(𝑟_𝑖𝑗),𝑥_𝑖𝑗][𝑄𝑁(−𝑥_𝑖𝑗)+𝑄(𝑟_𝑖𝑗),𝑥_𝑖𝑗]<0, for all 𝑖∈Ω₁,𝑗∈Ω₂.
Therefore, by using Lemma 1, we obtain that (1) has at least one 𝜔-periodic solution. The proof is complete.

4. Examples

In this section, we give two examples to demonstrate the results obtained in previous sections.

Consider the following shunting inhibitory cellular neural networks with neutral delays: 𝑥̇_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗𝑥_𝑘𝑙𝑡−𝜏_𝑘𝑙(𝑡)𝑥_𝑖𝑗(𝑡)+𝑏_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗𝑡−𝜎_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡), (36) where 𝑖=1,…,𝑚,𝑗=1,…,𝑛.

Example 1. Let 𝑚=𝑛=2, 𝑓_𝑖𝑗(𝑢)=tanh𝑢=(𝑒^𝑢−𝑒^−𝑢)/(𝑒^𝑢+𝑒^−𝑢). We take 𝑎_𝑖𝑗3×3=√4+0.2sin3𝑡5+0.1cos𝑡6+sin,𝐶2𝑡5−cos𝑡_𝑖𝑗3×3=||,𝑏0.3−0.1sin𝑡1+0.2cos𝑡0.6sin𝑡0.3−0.2sin𝑡_𝑖𝑗3×3=,𝐼0.03sin𝑡0.10.150.1sin2𝑡_𝑖𝑗3×3=1.6cos𝑡+sin𝑡3𝑡tan2,𝜏0.5_𝑖𝑗3×3=,𝜎sin𝑡cos𝑡0.25cos𝑡7sin𝑡sin0.1𝑡_𝑖𝑗3×3=.1+sin0.5𝑡0.8cos𝑡0.3sin𝑡sin0.2𝑡 (37) Obviously, (H₁)–(H₃) hold and _{𝐶𝑘𝑙∈𝑁1()1,1}𝐶^𝑘𝑙₁₁=_{𝐶𝑘𝑙∈𝑁1()1,2}𝐶^𝑘𝑙₁₂=_{𝐶𝑘𝑙∈𝑁1()2,1}𝐶^𝑘𝑙₂₁=_{𝐶𝑘𝑙∈𝑁1()2,2}𝐶^𝑘𝑙₂₂=2.7, (38)𝑝₁₁=1.414, 𝑝₁₂=2.236, 𝑝₂₁=1.195, and 𝑝₂₂=1.118.
Let 𝛼_𝑖𝑗=1, we have 𝐴=diag(3.8,4.9,5,4), (),(),(),𝐵=diag0.042,0.224,0.179,0.112𝐶=diag2.7,2.7,2.7,2.7𝐷=diag0.958,0.776,0.821,0.888𝐴−𝐶−𝐵𝐷⁻¹(𝐴+𝐶)=diag(0.815,0.006,0.621,0.455). (39)
It is easy to see that (H₄) and (H₅) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one 2𝜋-periodic solution.

Example 2. Let 𝑚=𝑛=3, 𝑓_𝑖𝑗(𝑢)=(1/2)sin𝑢. We take 𝑎_𝑖𝑗3×3=,𝐶4+sin3𝑡5+cos𝑡6−sin𝑡5+sin3𝑡6+sin2𝑡4+sin𝑡6+cos5𝑡4+sin𝑡5+cos𝑡_𝑖𝑗3×3=||||||||||||||||||,𝑏0.1cos3𝑡0.2cos𝑡0.1sin𝑡0.2sin2𝑡0.3sin𝑡0.4cos𝑡0.1cos5𝑡0.1sin𝑡0.2sin𝑡_𝑖𝑗3×3=,𝐼0.01sin𝑡0.02cos5𝑡0.01sin2𝑡0.02sin3𝑡0.03sin2𝑡0.01sin5𝑡0.03cos3𝑡0.01sin𝑡0.02cos3𝑡_𝑖𝑗3×3=,𝜏2+sin𝑡cos𝑡sin𝑡sin𝑡cos𝑡1+cos𝑡cos𝑡sin2𝑡sin𝑡+cos𝑡_𝑖𝑗3×3=,𝜎0.1sin𝑡0.5cos𝑡0.6sin𝑡0.8sin𝑡0.1sin𝑡0.4sin𝑡0.5cos𝑡0.15sin𝑡0.55cos𝑡_𝑖𝑗3×3=.0.6cos𝑡0.2cos𝑡0.15sin𝑡0.2sin𝑡0.75sin𝑡0.4cos2𝑡0.8cos𝑡0.9sin𝑡0.75sin𝑡 (40) Obviously, (H₁)–(H₃) hold and _{𝐶𝑘𝑙∈𝑁1()1,1}𝐶^𝑘𝑙₁₁=0.8,_{𝐶𝑘𝑙∈𝑁1()1,2}𝐶^𝑘𝑙₁₂=1.3,_{𝐶𝑘𝑙∈𝑁1()1,3}𝐶^𝑘𝑙₁₃=1,_{𝐶𝑘𝑙∈𝑁1()2,1}𝐶^𝑘𝑙₂₁=1,_{𝐶𝑘𝑙∈𝑁1()2,2}𝐶^𝑘𝑙₂₂=1.7,_{𝐶𝑘𝑙∈𝑁1()2,3}𝐶^𝑘𝑙₂₃=1.3,_{𝐶𝑘𝑙∈𝑁1()3,1}𝐶^𝑘𝑙₃₁=0.7,_{𝐶𝑘𝑙∈𝑁1()3,2}𝐶^𝑘𝑙₃₂=1.3,_{𝐶𝑘𝑙∈𝑁1()3,3}𝐶^𝑘𝑙₃₃=1, (41)𝑝₁₁=1.581, 𝑝₁₂=1.118, 𝑝₁₃=1.085, 𝑝₂₁=1.118, 𝑝₂₂=2, 𝑝₂₃=2.236, 𝑝₃₁=2.236, 𝑝₃₂=3.162, and 𝑝₃₃=2.
Let 𝛼_𝑖𝑗=1/2, we have 𝐴=diag(3,4,5,4,5,3,5,3,4), (),(),(),𝐵=diag0.016,0.022,0.011,0.022,0.06,0.022,0.067,0.032,0.04𝐶=diag0.4,0.65,0.5,0.5,0.85,0.65,0.35,0.65,0.5𝐷=diag0.984,0.978,0.989,0.978,0.94,0.978,0.933,0.968,0.96𝐴−𝐶−𝐵𝐷⁻¹()().𝐴+𝐶=diag2.545,3.245,4.439,3.399,3.777,2.268,4.266,2.229,3.313 (42)
It is easy to see that (H₄) and (H₅) hold. Therefore, all conditions of Theorem 6 are satisfied, then system (36) has at least one 2𝜋-periodic solution.

5. Conclusion

In this paper, a class of neutral type shunting inhibitory cellular neural networks is investigated by means of an abstract continuous theorem of 𝑘-set contractive operator and some analysis technique. Sufficient conditions are derived to ensure the existence of periodic solutions. To the best of our knowledge, the results presented here have not appeared in the related literature. Also, illustrative examples are given to demonstrate the effectiveness of the results obtained.

Acknowledgments

The author is grateful to the referees for their suggestions on the first draft of the paper. This work was supported by the scientific research fund of Fujian Provincial Education Department (JA11309).