The Existence Of Solution For The Discrete Fractional Difference Boundary Value Problems With P-Laplacian Operators

In recent decades, due to the fractional order differential equation is widely applied optics and thermal, materials and mechanical systems, signal processing and system identi-fication, control and robotics and other applications, so it gets the wide attention of many scholars. However, follow the emergence of the fractional order differential equation, on the one hand, we are inspired by fractional order differential equations, some scholars begin to study discrete fractional differential equation theory; on the other hand, with the rapid development of computer, discrete data are better able to fit into the computer, but also promote the development of the discrete fractional order difference equations. With the de-velopment of fractional order difference equation, the fractional order difference equation is not limited to pure theoretical research, it can also be applied to biomathematics, statistics, physics and so on. So the fractional order difference equation became one of the hot research topics. At the same time, the problems associated with the fractional difference equation has been studied, which contain discrete fractional order difference problems with p-Laplacian operators. The research of discrete fractional order difference problems with the Laplacian operators is a generalization of the fractional order difference equation, it is studied on the basis of the fractional difference equation theory, so there are many studied for the discrete difference problems with p-Laplacian operators, for example, resonance problems of research, two or more than two points boundary value problems of study, the periodic boundary value problems of study, etc.. This paper mainly studies the existence of solution of multipoint boundary value problem.In the paper, we mainly studies two classes of the fractional multipoint boundary value problems with the p-Laplacian operators. We transformed two classes of equations by the transformation formal. The first class of equation which are transformed are studies by using the fixed point theorem, we prove the existence of positive solutions, thus we get the existence of positive solutions for the original equations; the second class of equation which are transformed are studies by using monotone iterative,we prove the existence of positive solutions and the non-increasing solution. So we get the existence of non-creasing positive solutions for the original equations.This paper consists of five chapters. The first chapter is induction part, it describes the research background and relevance of significance; the second chapter is the preliminary part, it mainly introduces some relevant formulas and lemmas; the third chapter, we mainly the following fractional difference equation multipoint boundary values problem with the p-Laplacian operator Δ(φp,(Δv-1vx(t)))+f(t+v,x(t+v))=0, t ∈[0,T-1]N0 Firstly, the original equation is transformed into the integer order difference equation. Sec-ondly, we establish a Banach space and the cone in this space. We get the expression of the solution by the equation and its boundary conditions. We prove compliance with the conditions on the Banach space and define the corresponding operator. Finally, we give the sufficient condition of the existence of multiple positive solutions for the transformed equa-tion by using the fixed point theorem. It is obtained that the existence of positive solutions of the original equation. The fourth chapter, we mainly the following fractional difference equation multipoint boundary values problem with the p-Laplacian operatorΔφp(t-1Δv-2vx(t)))+f(t+v-1,t+v-εΔv-2ε)= 0, t ∈[0,T-1]N0Firstly, the fractional multipoint boundary value problem is transformed into the integer order difference equation by the transformation formula; secondly, we get the solution ex-pression and some properties by the transformed equation and conditions; finally, we discuss the transformed equation by using the monotone iteration method, so we obtain existence of two non-increasing positive solution for the original equation. The last chapter, it is the conclusion part.