不具備全局利普希茨條件的時(shí)滯分流抑制型細(xì)胞神經(jīng)網(wǎng)絡(luò)的偽概周期解

張 紅

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不具備全局利普希茨條件的時(shí)滯分流抑制型細(xì)胞神經(jīng)網(wǎng)絡(luò)的偽概周期解

張 紅*

(湖南文理學(xué)院 數(shù)學(xué)與計(jì)算科學(xué)學(xué)院, 湖南 常德, 415000)

研究了不具備全局利普希茨條件的時(shí)滯分流抑制型細(xì)胞神經(jīng)網(wǎng)絡(luò)系統(tǒng), 得到其偽周期解存在性及其唯一性的充分條件, 推廣并改進(jìn)了早期有關(guān)這方面研究的結(jié)果. 通過(guò)構(gòu)造Lyapunov函數(shù)并利用Banach壓縮映像原理, 得到本系統(tǒng)具有指數(shù)型穩(wěn)定性的偽概周期解的充分條件.

指數(shù)型穩(wěn)定; Lyapunov函數(shù); 偽概周期解; 分流抑制系統(tǒng)

自從文獻(xiàn)[1]提出分流抑制系統(tǒng)(SICNNs), 分流型神經(jīng)網(wǎng)絡(luò)已被廣泛應(yīng)用于心理物理、演講、感知、機(jī)器人、自適應(yīng)模式識(shí)別、視覺(jué)圖像處理[2—4]. 研究神經(jīng)動(dòng)力系統(tǒng)不僅涉及到其穩(wěn)定性性能, 而且涉及很多其它動(dòng)態(tài)特性, 如周期振蕩、概周期振蕩特性、混沌現(xiàn)象和分歧問(wèn)題等[5—6]. 最近, 文獻(xiàn)[7—8]研究了如下SICNNs系統(tǒng)的偽概周期解的存在性:

1 偽概周期函數(shù)的基礎(chǔ)知識(shí)和基本結(jié)論

這里是維向量.

2 偽概周期解的存在性和穩(wěn)定性

定理1 若假設(shè)

在R上具有指數(shù)二分性. 結(jié)合引理2可知式(6)存在唯一的偽概周期解:

證

于是有:

定理2證畢.

[1] Bouzerdoum A, Pinter R B. Shunting inhibitory celluar neural networks:derivation and stability analysis[J]. IEEE Trans Circuits Syst, 1993, 40: 215—221.

[2] Liu B, Huang L. Existence and stability of almost periodic solution for shunting inhibitory cellular neural networks with time-varying delays[J]. Chaos Solitons Fract, 2007, 31: 211—219.

[3] Zhou Q, Xiao B, Yu Y, et al. Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays[J]. Chaos Solitons Fract, 2007, 34: 860—866.

[4] Liu B. Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions[J]. Journal of Computational and Applied Mathematics, 2007, 203: 159—168.

[5] 何崇佑. 概周期微分方程[M]. 北京: 高等教育出版社, 1992.

[6] Fink A M. Almost periodic differential equation[M]. Berlin: Spring-Verlag, 1974.

[7] Farouk Ch érif. Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays[J]. Appl Math Comput, 2012, 39: 235—251.

[8] 孫獻(xiàn)德. 具變時(shí)滯分流抑制型細(xì)胞神經(jīng)網(wǎng)絡(luò)的偽概周期解及吸引性[J]. 福州大學(xué)學(xué)報(bào): 自然科學(xué)版, 2011, 39(2): 180—185.

[9] Zhang C Y. Pseudo almost periodic functions and their applications[D]. Ontario: University of Western Ontario, 1992.

[10] Zhang C Y. Pseudo almost periodic solutions of some differential equations[J]. Math AnalAppl, 1994, 181: 62—76.

Pseudo almost periodic solutions of delayed shunting inhibitory cellular neural networks without Global Lipschitz Activaty Functions

ZHANG Hong

(college of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, 415000, China)

In this paper, shunting inhibitory cellular neural networks are studied. Without assuming the global Lipschitz conditions of activaty functions, some new sufficient conditions are obtained for ensuring the existence and uniqueness of pseudo almost periodic solution of this system. Our results improve and generalize those of the previous results. Furthermore,several methods are applied to establish sufficient criteria for the exponential stability of this system. The approaches are based on con-structing suitable Lyapunov functionals and the well-known Banach contraction mapping principle.

exponential stability; Lyapunov functional; pseudo almost periodic solu-tion; SICNNs

10.3969/j.issn.1672-6146.2013.03.001

O 175.14

1672-6146(2013)03-0001-05

email: hongzhang320@aliyun.com.

2013-09-01

湖南省教育廳科研資助項(xiàng)目(11C0915)

(責(zé)任編校:劉曉霞)