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

Since monotone methods have been initiated by Kamke 1 and Müler 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. There are many interesting results, for example, 6, 8–12, 14–17 for competitive cooperative or type-K competitive cooperative systems and 6, 7, 13 for delayed systems. In particular, for the scalar delay differential equations of the form