Type-K Exponential Ordering with Application to Delayed Hopfield-Type Neural Networks

Bin-Guo Wang

doi:10.1155/2012/580482

Abstract

Order-preserving and convergent results of delay functional differential equations without quasimonotone condition are established under type-K exponential ordering. As an application, the model of delayed Hopfield-type neural networks with a type-K monotone interconnection matrix is considered, and the attractor result is obtained.

Highlights

Since monotone methods have been initiated by Kamke 1 and Muler 2, and developed further by Krasnoselskii 3, 4, Matano 5, and Smith 6, the theory and application of monotone dynamics have become increasingly important see 7–18
It is well known that the quasimonotone condition is very important in studying the asymptotic behaviors of dynamical systems
We introduce a type-K exponential ordering and establish order-preserving and convergent results under the weak quasimonotone condition WQM see Section 2 and apply the result to a network model with a type-K monotone interconnection matrix

Summary

Introduction

Since monotone methods have been initiated by Kamke 1 and Muler 2 , and developed further by Krasnoselskii 3, 4 , Matano 5 , and Smith 6 , the theory and application of monotone dynamics have become increasingly important see 7–18. It is well known that the quasimonotone condition is very important in studying the asymptotic behaviors of dynamical systems If this condition is satisfied, the solution semiflows will admit order-preserving property. The right hand side of 1.1 must be strictly increasing in the delayed argument This is a severe restriction, and so the quasimonotone conditions are not always satisfied in applications. A typical example is a Hopfield-type neural network model with a type-K monotone interconnection matrix, which implies that the interaction among neurons is excitatory and inhibitory For this purpose, we introduce a type-K exponential ordering and establish order-preserving and convergent results under the weak quasimonotone condition WQM see Section 2 and apply the result to a network model with a type-K monotone interconnection matrix.

Type-K Exponential Ordering

Delayed Hopfield-Type Neural Networks