Invariant and attracting sets of non-autonomous impulsive neutral integro-differential equations

Due to the plentiful dynamical behaviors, integro-differential equations with delays have many applications in a variety of fields such as control theory, biology, ecology, medicine, etc [1, 2]. Especially, the effects of delays on the stability of integro-differential equations have been extensively studied in the previous literature (see [3]-[9] and references cited therein). Besides delays, impulsive effect usually exist in many evolution processes in which the states exhibit abrupt changes at certain moments, such as threshold phenomena in biology, bursting rhythm models in medicine and frequency modulated systems, etc. In recent years, the theory of impulsive integro-differential equations with delays has attracted wide attention and lots of significant results on existence, initial (boundary) value problems and stability have been reported [10]-[20]. Some results for impulsive neutral differential equations with delays have been published. For instance, in [21], the exponential stability for impulsive neutral differential equations with finite delays has been studied by using differential inequality technique. In [22, 23], some stability conditions based on Lyapunov-Krasovkii functional method have been established for impulsive neutral differential equations with finite delays. In [24], authors studied the exponential stability for impulsive neutral integro-differential equations with delays by developing a singular integro-differential inequality. However, in general, the results about impulsive neutral differential equations with delays are still scarce due to some theoretical and technical difficulties. Additionally, it worth noting that those results in previous literature [21]-[24] have only focused on the stability of the equilibrium point for autonomous impulsive neutral differential equations with delays. However, under impulsive perturbation, the equilibrium point sometimes does not exist in many real physical systems, especially in nonlinear and non-autonomous dynamical systems. Therefore, an interesting and more general issue is to discuss the invariant set and attracting set of non-autonomous impulsive systems. Some important progress has been made in the techniques and methods for determining the invariant and attracting sets of delay differential equations [25, 26], impulsive differential equations with delays [27] and neutral differential equations [28]. Until now the corresponding problems for impulsive neutral differential (or integro-differential) equations with delays have not been considered. Motivated by the above discussion, we will investigate the asymptotic behaviors of solutions for a non-autonomous impulsive neutral integro-differential equation with time-varying delays in this