The Existence And Uniqueness Of Positive Solutions To Several Classes Of Impulsive Differential Equations

In this paper, we discussed the initial value problem and the boundary value problems of several classes impulsive differential equations. The proof of our main results is based on the cone and partial ordering theory. By using fixed point theorems with concavity、convexity and mixed monotonicity, the existence and uniqueness of positive solutions for the relevant equations are obtained. The new results extend and improve some conclusions in the relevant literatures, and provide some methods to solve the practical problems. The overall structure of this paper is as follows:In the Chapter 1, we introduce the background and status of the research problems, and the necessity and practical significance of this paper. And we state the main conclusions of this paper.In the Chapter 2, we consider the two-point boundary value problems of a second order impulsive differential equations with dependence on first order derivative: By taking advantage of fixed point theorem of τ-φ concave operator, the sufficient conditions for existence and uniqueness of positive solutions are established. To illustrate the correct of the method, moreover, related an examples is given for illustrations.In the Chapter 3, we consider a class of fourth-order impulsive differential equation with nonlinear boundary conditions:By using fixed point theorem for mixed monotone operators with sum form, the sufficient conditions which guarantee the existence and uniqueness of monotone positive solutions to the impulsive differential equation are established. And an example is given to illustrate our main result.In the Chapter 4, we study the initial value problem of a fourth-order impulsive integro-differential equation:We show the existence and uniqueness of positive solutions to our problem for each given control functions. Also, we consider the control problem of positive solutions to our equations. The proof of our main results is based on a fixed point theorem of generalized α-oncave operators. Then we give an example to illustrate the main result.