Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters

We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature. 1. Introduction First, we give the following definitions. Let 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 . 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; numerous examples are given in [1–3]. In recent years, in [4–11], many researchers have obtained some properties of impulsive differential equations, such as oscillation, asymptotic behavior, stability, and existence of solutions. However, to this day, still no scholars investigate the existence of multiple positive periodic solutions for impulsive functional differential equations with infinite delay and two parameters. Motivated by this, in this paper, we mainly consider the following impulsive functional differential equations with two parameters: where and , are two parameters, ,？？ , ？？ are -periodic, that is, , , , is an operator on (here denotes the Banach space of bounded continuous operator with the norm , where , ？ , (here represents the right limit of at the point ), , that is, changes decreasingly suddenly at times . is a constant, , , , and . We assume that there exists an integer such that , , , where . Models of forms (1) and (2) have been proposed for population dynamics (single species growth models), physiological processes (such as production of blood cells, respiration, and cardiac arrhythmias), and other practical problems. Equations (1) and (2) are very general and incorporate many famous mathematical models extensively studied in the literature [12–21]. In this paper, we will study the existence of positive periodic solutions in more cases than the previously mentioned papers and obtain some easily verifiable sufficient criteria. Throughout the paper, we make the following