Order Impulsive Boundary Value Problems

As an important aspect of di?erential equations, impulsive di?erential equations bound-ary value problems have gained more attention because it has wide practical background.There are many methods in solving these kinds of problems, however, functional analysis isthe most wide-ranging approach and the upper and lower solutions coupled with monotonetechnique are more convenient and e?ective.In this paper, we investigate several classes of boundary value problems for second di?er-ential equations: di?erential systems, impulsive di?erential systems and impulsive functionaldi?erential equations. And for impulsive functional di?erential equations we consider di?er-ence boundary conditions and di?erence definition of the upper and lower solutions.The thesis encompassing four chapters is divided into three parts. The part I is the firstchapter. Which mainly considers historical background, research objectives and theoreticalsignificance.The second chapter is the part II. Here we discuss the existence of solutions for boundaryvalue problems of second order (impulsive) di?erential systems. By generalizing the definitionof lower and upper solutions, using Brouwer and Schauder's fixed point theorems, we getthe existence of at least one solution.The third and fourth chapters constitute the last part. We consider second impul-sive functional di?erential boundary value problems with the nonlinear boundary conditionsx(0) = x(T) + k1, g(x (0),x (T)) = 0 and g(x(0),x(T)) = 0, h(x (0),x (T)) = 0.In the third chapter, we discuss the boundary value problems with the first boundaryconditions. Firstly, we establish impulsive di?erential inequalities as comparison principles.And we prove the existenceness and uniqueness of the solutions for the linear impulsivefunctional equations. Next, by using the comparison principles, the monotone iterativetechnique and the method of upper and lower solutions we obtain the existence for theboundary value problems of second impulsive functional di?erential equations with the firstboundary conditions. Finally, we give an example to illustrate. In the fourth chapter, we consider the boundary value problems for second impulsivefunctional di?erential equations with the second boundary conditions. Introduce two defi-nitions of upper and lower solutions, that is , the classical ones and the extended ones. Weobtain the extremal solutions of the boundary value problems by using comparison princi-ples and monotone iterative technique in the case of the classical upper and lower solutionsand the extended ones, respectively. Finally, we give two examples to explain the existenceresults for the extremal solutions in the case of the classical upper and lower solutions.