Existence Of Solutions For Boundary Value Problems Of Some Dynamic Equations On Time Scales

The theory of time scales was introduced by Stefan Hilger in his Ph.D. Thesis in 1988. This theory not only can unify continuous and discrete analysis, but it also extends these cases to cases "in between". An important branch of dynamic equations on time scales is boundary value problems, due to their striking applications to almost all areas of science, engineering and technology. By researching boundary value problems on time scales, the results not only unify the theory of differential and difference equations and understand deeply the relationship between them, but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. In this paper, the main work is to research the existence of the solutions for some boundary value problems on time scales T, and gives the judge gist of the existence of solutions. The paper contains four chapters.The first chapter is the introduction of this paper. It introduces the background of this paper and the development of time scales. It also introduces some basic theories about time scales. With actual example, we show this research not only has important theory significance, but also has vital practical significance.The second chapter, two fixed point theorems are applied to boundary value problems for some p-Laplacian dynamic equations on time scales. We establish the results of at least two and three positive solutions. An example is introduced to illustrate the main result.In chapter three, we employ the upper and lower solution method to study the existence of an iteration solution for a fourth-order four-point boundary value problem on time scales and provide a theoretical basis for the numerical method. An example is given to prove the main result in this chapter.The last chapter, by use of the fixed point theorem in cone, we study the existence of triple positive solutions for some third-order three-point nonlinear differential equations boundary value problems with p-Laplacian operator on time scales.