Positive Solutions To Some Classes Of Boundary Value Problems Of Fractional Differential Equations

Fractional differential equations arise in many fields, such as physics, chemistry, en-gineering and biological sciences, etc. In recent years, the study of positive solutions for fractional differential equation boundary value problems has attracted considerable atten-tion, and fruits from research into it emerge continuously.This paper is mainly composed of four chapters:Chapter 1 is the introduction. We introduce the development of fractional differential equation boundary value problems and describe the basic concepts and notes.In chapter 2, by using fixed point theorems of general β-concave operators in ordered Banach space, we obtain the existence and uniqueness of positive solutions to the following fractional boundary problem where f(t, x):[0,1] × R+ �' R+is continuous, g(t):[0,1]�' R+is continuous and f(t, x) is increasing in x for each t ∈ [0,1].In chapter 3, by using Krasnoselskii fixed point theorem of sum operators and the properties of the Green function, we obtain the existence of positive solutions to the following fractional boundary problem where 1<α≤ 2,0≤ξ≤η≤1,0≤μ1,μ2≤1.With the condition f(t,x), g(t,x):[0,1] × [0,+∞)�' [0,+∞) is continuous.In chapter 4, by using fixed point theorems of concave operators in partial ordering Banach spaces, we establish the existence and uniqueness of positive solutions to the following fractional differential equations for any given parameter where 1<∝≤2,0≤ξ≤η≤1,0≤μ1,μ≤,λ> 0 is a parameter, f(t,x):[0,1]×R+�' R+ is continuous and increasing in x for each t ∈ [0,1]. And with the condition:Moreover, we present some pleasant properties of positive solutions to the boundary value problem dependent on the parameter.