The Existence Of Solutions For Conformable Fractional Integro-differential Equation Boundary Value Problems

The fractional derivative is a generalization from the classical one to any order and its model is more widely used than classical integer order model,it is ancient in calculus.In recent decades,fractional integro-differential equations play an important role in the analysis of materials in various fields,including memory and genetic characteristics in various material processes,so fractional integro-differential equations are favored by many scholars and it develops rapidly.Therefore,it is extremely important to study the boundary value problem of fractional integro-differential equations.Among them,the research on the conformable fractional initial value problem has been welcomed by many scholars.This thesis is mainly divided into four chapters:Chapter 1 is the introduction,we give a general overview of the research background and the overall layout of the article.In chapter 2,we investigate the following integral boundary value problem of conformable fractional integro-differential equations with a parameterwhere ??(1,2],?is a positive parameter,T_?is the conformable fractional derivative and I_? is the conformable fractional integral.f: [0,1] × [0,+?)× [0,+?)? [0,+?)is a continuous function.We use a fixed point theorem for monotone operators in ordered Banach spaces,and then establish the existence of positive solutions for the boundary problem.Further,using an iterative sequence to approximate the uniqueness of positive solution and some good properties of positive solution about the parameter ?.In chapter 3,we study the following multiple positive solutions for a system of conformable fractional differential equationswhere ?,??(1,2]?T_?,T_? stand for the comformable fractional derivatives.We intend to give the existence of at least one or two positive solutions for system by fixed point index theory in ordered Banach spaces.In chapter4,we investigate the following integral boundary value problem of conformable fractional differential equationswhere ??(1,2],T_?is the usual conformable fractional derivative,f,g:[0,1]×[0,+?)×[0,+?)?[0,+?) are continuous functions,l(t) is a nonnegative function with l(t)?1.We will investigative the existence and uniqueness of solutions for problem above,and make an iterative to approximate the unique positive solution.The main tool is a general mixed monotone operators method.