In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.

In this article, a generalized model of neural networks involving time-varying delays and impulses is considered. By establishing the delay differential inequality with impulsive initial conditions and using the properties of M-cone and eigenspace of the spectral radius of nonnegative matrices, some new sufficient conditions for global exponential stability of impulsive delay model are obtained. The results extend and improve the earlier publications. An example is given to illustrate the theory.

Using the relationship of a polynomial and its associated polynomial, we derived a necessary and sufficient condition for determining all roots of a given polynomial on the circumference of a circle defined by its associated polynomial. By employing the technology of analytic inequality and the theory of distribution of zeros of meromorphic function, we refine two classical results of Cauchy and Pellet about bounds of modules of polynomial zeros. Sufficient conditions are obtained for the polynomial whose Cauchy's bound and Pellet's bounds are strict bounds. The characteristics is given for the polynomial whose Cauchy's bound or Pellet's bounds can be achieved by the modules of zeros of the polynomial.

The authors analyze asymptotic behavior of the partial functional differential equations, and the sufficient conditions on existence of global attractor of a class of reaction-diffusion equations with delays are given.

We will be concerned with the invariant and asymptotic properties of Volterra difference equations with delays. Sufficient conditions for determining the invariant and attracting sets of the equations are obtained. Examples are given to illustrate the obtained results.