The Properties Of Solutions To Several Types Of Differential Equations With Non-local Boundary Conditions

In this paper,we investigate the properties of the positive solution for the two classes of fractional differential equations with nonlocal boundary conditions by using the fixed point theory and mixed monotone method,the thinking method of functional analysis,we obtain the existence,uniqueness and multiplicity results of the problem.Some examples are given to demonstrate the validity of our main results.It has certain application value and theoretical significance.The thesis is divided into two chapters:The chapter 1,we investigate the uniqueness、existence and multiplicity of posi-tive solutions for the following class of nonlinear singular fractional differential equations involving integral boundary value conditions:where are the standard Riemann-Liouville derivatives and,is continuous and a,h∈C（0,1）,A is a function of bounded variation,₀^∫ηh（s）D₀^β2₊u（s）dA（s）,₀^∫1a（s）D₀^β3₊u（s）dA（s）denote the Riemann-Stieltjes integral with respect to A.The uniqueness of positive solution is derived by the fixed point theorem of mixed monotone operator.The Guo-Krasnosel’skill fixed point theorem is utilized to derive the existence and multiplicity of positive solutions for the fractional differential equation.The main features of the present paper are as follow.Firstly,the nonlinearity f is allowed to depend on higher derivatives of unknown functions and we allow f to be singular at t=0,1 and x_i=0（i=0,1,...,n-2）.Secondly,the fractional derivatives in the boundary conditions can be different,at the same time,the higher derivatives in the the nonlinearity f are also different from the fractional derivatives in the integral boundary conditions.The boundary conditions involving fractional derivatives of unknown function are more general cases,which covers the multi-point boundary conditions and integral boundary conditions as special cases.Thirdly,the given conditions f₀,f_∞and（H₃）,（H₅）are quite different from other papers and are more weaker and wider.The chapter 2,we investigate the existence result as well as multiplicity result of positive solutions for the following a class of nonlinear singular semipositone fractional differential equations with multi-point boundary conditions:where are the standard Riemann-Liouville derivatives and is a continuous function and f（t,u）may be singular at t=0,1and u=0.By employing Krasnoselskii’s fixed point theorem,we obtain the existence of positive solutions for a class of singular fractional differential equations with a sign-changing nonlinearitiy,subject to multi-point fractional boundary conditions.This article admit the some new features.Firstly,compared with[19],the fractional derivatives in the boundary conditions can be different,the fractional derivatives of order q_iare related to coefficients a_i.That is to say that BVP（2.1.1）has the more general form.secondly,the nonlinear term f permit singularities on the time and the space variable at the same time and change its sign.Thirdly,the method exploited in this paper is different from that in[19,34,42]in essence,the techniques used in this paper are the approximation method to overcome the singularity and obtain multiple positive solutions.To the best of our knowledge,no contribution exists considering the multiple positive solutions for the BVP（2.1.1）.