Existence Of Solutions For The Boundary Value Problem Of Fractional Differential Equations

In modern mathematics, differential equations is a country with profound theoretical research value, while there are broad value of ap-plied research subjects.As one of its important branch of fractional order differential equations, the boundary value problem is more frontier re-search direction in recent years. Fractional differential equations arise in many engineering and scientific disciplines such as chemistry, economics, physics, control theory, aerodynamics, biophysics,signal and image pro-cessing, and so on. And in this paper,we study the fractional order differential equation,system of equations,and the singular problem by the theory of cones and the fixed point theorem,and applied in some examples.This paper is divided into the following four chapters:Chapter1In this chapter, we study the boundary value problem of fractional differential equations with derivative term. where2<α≤3,is a real number,cD0+αis Caputo’s fractional order derivative and meets: f∈C(I×R+×R+,R+),h∈C+(0,1),whereI=[0,1],the nonlinear term contains derivative term, using two fixed point theorems respec-tively, which get a positive solution of the equations.Chapter2In this chapter, we study the boundary value problem of fractional order differential system of equations of the following: wherel<α,β≤2,0<μ,υ<1,f,9∈C(I×R4,R),,=[0,1],the nonlinear term contains.cD0μ(t),cD0υ(t),and is a problem of equation-s,which get a positive solution.Chapter3In this chapter,we consider the nonlinear singular bound-ary value problem: wherel<α≤2,0≤ξ≤η≤≤1,0≤β,γ≤1,h(t)may be singular at t=0or t=1,which get a posSitive solution.Chapter4In this chapter,we discussed the following boundary val-ue problem of fractionl differential equations:wherel<α≤2,0≤ξ≤η≤1,0≤μ1,μ2≤1,by using the Schauder’s fixed point theorem,we get a positive solution.