Existence Of Positive Solutions Of Boundary Value Problem For The Fractional Order Coupled System With P-Laplacian Operator

In recent years,the theory of fractional differential equations boundary value has been widely used in different fields of science,it's theoretical significance and application value are obvious.Therefore,the fractional differential equation has become an important topic for many scholars,so do the research is more thorough,the results are more comprehensive.The coupled system as an important research of boundary value problems for fractional differential equations,much attention has been focused on the study of the studying the existence and uniqueness of solutions.At the same time,the coupled system has become a hot topic in many research mathematicians.In this paper,we will study the existence of positive solutions for four classes of fractional coupled systems with p-Laplacian operators.According to their respective boundary conditions,and combined with Banach compression mapping principle,Guo-Krasnosel'skii fixed point theorem on cones,Schauder fixed point theorem and Leray-Schauder fixed point theorem.Finally,we obtain the existence,uniqueness and multiplicity of positive solutions.The contents of article are as follows:In the first chapter,we give the definition and research background of the fractional differential equation.What's more,we give some fixed point theorems in the third section.In the second chapter,establishing a Caputo type fractional coupled system with p-Laplacian operators,with the properties of Green functions.Then we study the uniqueness of positive solutions of coupled systems by applying Banach compression mapping principle.In the third chapter,we structure the Green functions and get it's properties,then by using Schauder fixed point theorem,we study the sufficient condition for the existence of positive solutions for the coupled system of fractional differential equation.In the fourth chapter,we consider the Riemann-Liouville type fractional differential equation coupled system with p-Laplacian operators with integral boundary value condition.Obtained the properties of the Green function,then turn differential equation into integral equation.By using Guo-Krasnosel'skii fixed point theorem on cones,we prove the existence,multiplicity and nonexistence of positive solutions,finally an example is given to verify it.In the fifth chapter,we discuss the existence and uniqueness of positive solutions for coupled system of Riemann-Liouville type fractional differential equation with p-Laplacian.Firstly,we obtained the corresponding Green function,and turn differential equation into integral equation.Secondly,we prove the existence of positive solutions by using Banach contraction mapping principle,Leray-Schauder fixed point theorem and Guo-Krasnosel'skii fixed point theorem on cones.Finally,two examples are provided to illustrate our main results.