Asymptotic Dichotomy in a Class of Odd-Order Nonlinear Differential Equations with Impulses

We investigate the oscillatory and asymptotic behavior of a class of odd-order nonlinear differential equations with impulses. We obtain criteria that ensure every solution is either oscillatory or (nonoscillatory and) zero convergent. We provide several examples to show that impulses play an important role in the asymptotic behaviors of these equations. 1. Introduction Impulsive effect, likewise, exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments in time, involving such fields as medicine and biology, economics, mechanics, electronics, telecommunications, and so forth. It has been observed that the solutions of quite a few first-or second-order impulsive differential equations are either oscillatory or (nonoscillatory and) zero convergent (see, e.g., [1–10]). For example, Bainov et al. studied the oscillation properties of first-order impulsive differential equations with deviating arguments [3]. Especially in [4], Chen investigated oscillations of second-order nonlinear differential with impulses, and he promposed that the impulses may change the oscillatory behavior of an equation. Based on [4], the authors were devoted to oscillations of impulsive differential equations (see, e.g., [5–10]). Such a dichotomy may yield useful information in real problems. The implications of this dichotomy are applied to the deflection of an elastic beam [11]. Thus, it is of interest to see whether similar dichotomies occur in different types of impulsive differential equations. One such type consists of impulsive differential equations which are important in the simulation of processes with jump conditions. But papers devoted to the study of asymptotic behaviors of higher order equations with impulses are quite rare. For this reason, Wen et al. studied in [12] the dichotomous properties of the following third-order nonlinear differential equation with impulses: where , such that . On the other hand, in [13], Chen and Wen investigated the oscillatory and asymptotic behaviors for odd-order nonlinear differential equations with impulses of the form where is a positive integer and such that . They obtained some interesting results for assuring that every bounded solution of (2) is either oscillatory or nonoscillatory and zero convergent. In this paper, we will study a class of odd-order nonlinear differential equations with impulses of the form where is a positive integer, and such that , By a solution of (3), we mean a real function defined on such that(I) for ;(II) , and are continuous on ; for and exist, and for