Existence Of Solutions For Two Classes Multi-Point Boundary Value Problems With P-Laplacian On Time Scales

The theory of dynamic equations on time scales was put forward by Stefan-Hilger in 1988. When the time scale theory did not appear, we can use differential equation to describe some continuous phenomenon and some continuous processes and we use difference equation to learn some discrete phenomenon and some discrete change processes. But for some mixed state mathematical model we can’t give answer. The theory provides a new method to study some phenomenon that not regular. Since the time scale theory, the theory has caused the attention of many scholars.Due to the development of the time scale theory is very fast, so the theories on the time scale are becoming more and more mature. The existence of solutions for the above problems is one of the problems that many scholars pay attention to. The proof method also varied, such as: fixed point theorem on cone, monotone iterative method, the upper and lower solution method, the Leggett-Williams fixed point theorem, Leray—Schauder nonlinear selection theorem, etc. and it has a lot of valuable conclusions. By reading a lot of references, we can find that most of the authors are studied the existence of the solutions of the first order or second order on time scales. For the third order dynamic equations on time scales, Few research results on the existence of solutions of the multi-point boundary value problems. Certainly, the research results on With p—Laplacian operator of the existence of third order multipoint boundary value problem solution are even fewer.In the paper, we mainly study the existence of positive solutions of two classes boundary value problems with the p—Laplacian operators. For the first boundary value problem, I used two ways to prove that the sufficient conditions for the existence of multiple positive solutions. The first method is the Leggett—Williams fixed point theorem, the second method is an fixed point theorem on cone. For the second class of equation which is proved by the Leray—Schauder nonlinear alternative, we prove the sufficient conditions for the existence of at least one positive solution.There are five chapters in this paper. The first chapter is introduction, simply describes the historical background and the research value of the time scale theory; The next chapter is the preliminary part, it mainly introduces some relevant definitions and theorems; The third chapter, we mainly studies the following third order boundary value problems with the p—Laplacian operators At first, According to the relevant theory and properties on time scales, we get the solution expression. Secondly, we establish a proper Banach space and the cone for this equation and define the corresponding operator Q. Then I used two ways to prove that the sufficient conditions of the existence of multiple positive solutions. The first method is the Leggett-Williams fixed point theorem, the second method is another fixed point theorem. In the end, I gave two practical examples to prove the main results.The forth chapter, we mainly studies the Second class boundary value problem with the p—Laplacian operators At first, Depending on the relevant theory and properties on time scales, we get the solution expression. Secondly, we establish a proper Banach space and the cone for this equation and define the corresponding operator R. Then, we prove the sufficient conditions for the existence of at least one solution by Leray—Schauder nonlinear alternative theorem. The last chapter, it is the conclusion and references.