A New Existence Theory for Positive Periodic Solutions to a Class of Neutral Delay Model with Feedback Control and Impulse

We acquire some sufficient and realistic conditions for the existence of positive periodic solution of a general neutral impulsive -species competitive model with feedback control by applying some analysis techniques and a new existence theorem, which is different from Gaines and Mawhin's continuation theorem and abstract continuation theory for -set contraction. As applications, we also examine some special cases, which have been studied extensively in the literature, some known results are improved and generalized. 1. Introduction In this paper, we consider the existence of the positive periodic solution of the following impulsive -species competition system with multiple delays and feedback control: with the following initial conditions: where , , , , , , , , , , , and are continuous -periodic functions; are continuous？？ -periodic functions with . The growth functions are not necessarily positive; since the environment fluctuates randomly, in some conditions, may be negative. Consider the following: ; and , , and . And and represent the birth rate and the harvesting (or stocking) rate of at time , respectively. When , it stands for harvesting, while means stocking. For the ecological justification of (1) and the similar types, refer to [1–14]. In 1991, in [1], Gopalsamy et al. have established the existence of a positive periodic solution for a periodic neutral delay logistic equation where , and are positive continuous -periodic functions with and is a positive integer. In 1993, in [2], Kuang proposed an open problem (Open problem 9.2) to obtain sufficient conditions for the existence of a positive periodic solution of the following equation: In [3], Li tried to give an affirmative answer to the previous open problem; however, there is a mistake in the proof of Theorem 2 in？？[3]. With the aim of giving a right answer to the previous open problem,？？[4–6] also have investigated the previous question. However, it is more complex to check the sufficient conditions of the system？？[5, 6]. ？Moreover, in？？[7], Li studied the existence of positive periodic solution of the neutral Lotka-Volterra equation with several delays where , and are positive continuous -periodic functions and are nonnegative constants. Recently, in [8], Lu and Ge investigated a neutral delay population model with multiple delays: They applied the theory of abstract continuous theorem of -set contractive operator and some analysis techniques to obtain some sufficient conditions for the existence of positive periodic solutions of the model (6). It is of course very interesting to study