| In recent decades, with the rapid development of natural science such as mathematics, chemistry, physics, economics and control theory, many problems on specific mathemati-cal models have been proposed, which were described by functional differential equations. These problems need us to discuss them. In this paper, we mainly investigate two impor-tant problem further, i.e., zero points distance and positive solution, which are important components of vibration theory. Studying these problems thoroughly and sistematically can reveal the nature of things accurately and enrich the theory of differential equations greatly.These are not only the needs of the development of mathematical theory itself, but also the needs of practice, and are of great value on the practical application.Functional Differential Equations'positive solution problem is an old research direc-tion,which many scholars are doing research in this area and achieved rich results at home and abroad. On the existence of positive solutions for boundary value problems is a very hot research area.In this paper the main work is study several different equations, and obtain a number of different results. Some of the results are optimization of the original, and others the result are promotion of the original.This article is divided into four parts, Its structure is as follows.In part 1, the necessary background knowledge and prior preliminary was presented.The part 2, the zero points distance of solution of the first order differential equations for some time are discussed and adjacent the upper bound are obtained. Some examples are given to illustrate out results.In part 3, by using the conclusions of the existed theorems, we obtain several several decision theorems on the second order impulsive delay equations.The part 4, by using the conclusions of the existed theorems from literature [27]-[28], we obtain several decision theorems on multiple positive solutions for second order differential equations.Some examples are given. |
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