Multiple Positive Periodic Solutions for Two Kinds of Higher-Dimension Impulsive Differential Equations with Multiple Delays and Two Parameters

By applying the fixed point theorem, we derive some new criteria for the existence of multiple positive periodic solutions for two kinds of？？ -dimension periodic impulsive functional differential equations with multiple delays and two parameters: ), a.e., , , , , , and ), a.e., , , , , , As an application, we study some special cases of the previous systems, which have been studied extensively in the literature. 1. Introduction Let , , , , , and , respectively. Denote by the set of operators which are continuous for , and have discontinuities of the first kind at the points but are continuous from the left at these points. For each , the norm of is defined as . Let denote the Banach space of bounded continuous functions with the norm , where . The matrix means that each pair of corresponding elements of and satisfies the inequality . In particular, is called a positive matrix if . Impulsive differential equations are suitable for the mathematical simulation of evolutionary process whose states are subject to sudden changes at certain moments. Equations of this kind are found in almost every domain of applied sciences, and numerous examples are given in [1–4]. In recent years, the existence theory of positive periodic solutions of delay differential equations with impulsive effects or without impulsive effects has been an object of active research, and we refer the reader to [5–17]. Recently, in [5], Jiang and Wei studied the following nonimpulsive delay differential equation: They obtained sufficient conditions for the existence of the positive periodic solutions of (1). Motivated by [5], in [6], Zhao et al. investigated the following impulsive delay differential equation: They derived some sufficient conditions for the existence of the positive periodic solutions of (2). In [7], Huo et al. considered the following impulsive delay differential equation: They got sufficient conditions for the existence and attractivity of the positive periodic solutions of (3). Motivated by [5–7], in [8], Zhang et al. studied the following impulsive delay differential equation: They obtained some sufficient conditions for the existence of the positive periodic solutions of (4). However, to this day, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of impulsive functional differential equations with two parameters: with initial conditions: where and , , are -periodic; that is, ,