The Solutions Of Fractional Differential Equations On Infinite Intervals And Hadamard Type Fractional Boundary Value Problems

Fractional differential equations are mathematical extensions of integer differential e-quations.Fractional differential equations with boundary value problems are extremely important applications in many fields such as physics,chemistry,engineering,and biologi-cal sciences.In recent decades,with the development of research,the mathematical model,described by boundary value problems of fractional differential equations has been proposed many times in many practical cases.Therefore,the study of such problems has great value in promoting the resolution of practical problems.This thesis mainly studies the existence and uniqueness of solutions for the boundary value problems of fractional differential equations on infinite interval and system of Hadamard type fractional differential equations with boundary value problems.The thesis has four chapters.The main contents are as follows:Chapter 1 is the introduction.We introduce the research background and current situation of the problems as well as the overall layout of the thesis.In chapter 2,under some new conditions we use different methods,and investigate the m-point boundary value problem for fractional differential equations on an infinite interval(?)where 2<?<3,?>0 is a parameter,a:[0,+?)?[0,+?)and f:[0,1]x[0,+?)?[0,+?)are continuous,0<?1<?2<…<?m-2<+?,?i?0,i=1,2,...,m-2,and m-20<(?)?i?i?-1<?(?),D0+?-1u(+?):=limt?+? D0+?-1u(t)exists.We establish the existence and uniqueness of positive solutions for any fixed parameter ?>0,and we get some good properties of positive solutions which depend on the parameter.In chapter 3,we study a class of nonlinear fractional differential equations m-point boundary value problems on an infinite interval where 2<?<3,a,b:[0,+?)?[0,+?),f,g:[0,+?)×[0,+?)?[0,+?)are continuous,0<?1<?2<…<?m-2<+?,?i>0,i=1,2,...,m-2,satisfie 0<(?)?i?i?-1<?(?),D0+?-1u(+?):=limtt?+?D0+?-1u(t)Using some fixed point theorems of sum of two operators in semi-ordered Banach spaces,we prove the existence and uniqueness of positive solutions to the boundary value problem for a class of fractional differential equationsIn chapter 4,we use a recent fixed point theorem for?—(h,r)-concave operators to study new Hadamard fractional differential system with four-point boundary conditions where a,b are two parameters with 0<ab(log?)?-1(log?)?-1<1,?,??(n-1,n]are two real numbers and n?3,f,g?C([1,e]×(-?,+?),(-?,+?)),lf,lg are constants,we establish the existence and uniqueness of solutions for systemIn chapter 5,we discuss a Hadamard differential systems with fractional integral con-ditions where f,g?C([1,e)×(-?,+?),(-?,+?)),a,b are constants,?i,?j>0,i=1,...,m,j=1,...,n,?,?>0.By the use of increasing ?-(h,r)-concave operators,we not only obtain the existence and uniqueness of solutions,but also we construct convergent sequences to approximate the unique solution.