The Existence Of Solutions For Fractional Boundary Value Problems And Second Order Functional Differential Equations

Many problems in the real world can be solved by models of differential equations,however many equations can't find exact solutions.Therefore,the existence of solutions of differential equations is an important part of the qualitative theory of differential equations.Based on this,the existence of solutions of four kinds of equations is studied by using the fixed point theory.In Chapter 1,describe the research background,main work and preparatory knowledge.In Chapter 2,the existence of solutions of the fractional differential equation is studied(?)where ?>0 is a parameter,3<??4 is a real number,D0+? is the standard Riemann-Liouville differentiation.The existence of at least one positive solution of the equation is studied by using the fixed point theorem on the ordered interval.Schauder's fixed point theorem is used to study the existence of at least one nontrivial solution of the equation,the suficient conditions for the existence of solutions are given.In Chapter 3,the existence of solutions of the Caputo fractional difference equation is studied(?)where ?>0 is a parameter,?Cxv(t)is the standard Caputo difference.In the equation,we raise the range of real numbers v to 3<v?4.Firstly,the Green's function of the equation is calculated and its properties axe given.Secondly,the existence of at least one positive solution is studied by using the fixed point theorem on the ordered interval,the existence of at least one positive solution and two positive solutions is studied by using the Guo-Krasnosel'skii fixed point theorem,the existence of at least one nontrivial solution is studied by using Schauder's fixed point theorem,the sufficient conditions for the existence of solutions are given.In Chapter 4,the existence of solutions of the fractional q-difference equation is studied(?)where ?>0 is a parameter,3<??4 is a real number,Dq? is the standard Riemann-Liouville q-differentiation.The existence and nonexistence of positive solutions depending on different value range of parameters are studied by using the Guo-Krasnosel'skii fixed point theorem,the sufficient conditions for the existence and nonexistence of positive solutions are given.In Chapter 5,the existence of solutions of a class of second order functional differential equation is studied(?)where ?>0 is a parameter,b'>0 is a real number.The existence and uniqueness of nontrivial solutions of the equation are obtained by using the fixed point theorem of ?-(h,e)concave operator,the sufficient conditions for the existence and uniqueness of nontrivial solutions are given.