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],β>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],β>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,λ>0 is a parameter,f is a continuous function,ai,ξi∈ R,i=1,...,m(m ∈ N),0<ξ1<…<ξm≤1,p∈[1,n-2],q∈[0,p],D0α+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 λ>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 λ.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 α∈(n-1,n],n ∈ N,n≥ 3,ai≥ 0,0<ξ1<…<ξm≤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 Ψ-(h,r)-concave operators defined on special sets in ordered spaces.