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

The boundary value problem of fractional integro-differential equations comes from some practical problems in applied mathematics and physics.In recent decades,the topic of fractional integro-differential equations has become an important and popular research field.Fractional integro-differential equations play an important role in describing real-world materials.Therefore,it is of great value and practical significance to study the boundary value problems of fractional integro-differential equations.This thesis is mainly divided into four chapters:Chapter 1 is the introduction.We give a general overview of the relevant research background and the overall layout of the article.In chapter 2,we investigate the following fractional integro-differential equation with boundary conditions where n-1<a ?n,n?N,n? 2,t ? J:=[0,+?),a ? C(J,J),f? C[J x J x JJ],?>0,Da is the Riemann-Liouville fractional derivatives,(Tu)(t)=?0t k(t,s)u(s)ds with k(t,s)E C[E,J],E={(t,s)E R2|0<s<t}.We will give the existence and uniqueness of positive solutions and make an iterative to approximate the unique positive solution.The methods used here are some properties of normal cones and a recent fixed point theorem for monotone operators.Also,we can get the existence and uniqueness of positive solutions for the following problem where n-1<a?n,n?N,n?2,t?J,a? C(J,J),f?C[J×J×J×J,J],?>0,T is same problem above,(Su)(t)=?0?h(t,s)u(s)ds with h(t,s)?C[J×J,J].In chapter 3,we study the following fractional integro-differential equation with multi-point boundary conditions where a?(n-1,n],n?N,n?3,?>0 is a parameter,f is a continuous function,ai,?i? R,i=1,...,m(m ? N),0<?1<…<?m?1,p?[1,n-2],q?[0,p],D0?+is the Riemann-Liouville derivative of order a,Tu(t)=?0t K(t,s)u(s)ds,Su(t)=?01H(t,s)u(s)ds,t?[0,1].For any fixed parameter ?>0,we will investigate the existence and uniqueness of positive solutions for problem above and establish some clear properties of positive solutions with respect to the parameter ?.Our methods used here are a fixed point theorem and some properties of eigenvalue for general mixed monotone operators.In chapter 4,we investigate the following multi-point fractional integro-differential e-quation where ??(n-1,n],n ? N,n? 3,ai? 0,0<?1<…<?m?1,p?[1,n-2],q?[0,p],b>0,(Tu)(t)=?0t K(t,s)u(s)ds,(Su)(t)=?01L(t,s)u(s)ds for t?[0,1],with K(t,s),L(t,s)? C([0,1]×[0,1,[0,+)).We will investigative the existence and uniqueness of solutions for problem above,and make an iterative to approximate the unique positive solution.The method used here is a recent fixed point theorem of increasing ?-(h,r)-concave operators defined on special sets in ordered spaces.