Existence Of Positive Solutions For Boundary Value Problems Of Nonlinear Fractional Differential Equations

Fractional calculus has been a familiar concept for people in the math-ematicians and engineers nowadays.It is about integral and differential theory of an arbitrary order.Fractional calculus refers to the generaliza-tion of integer-order integrals and derivatives.It is one of most active domains of nonlinear functional analysis at present. The integral and multi-point boundary value problems are the hot spot which have been discussed in recent years,and become two very important domalins of differential equation research at present.In this paper,we discuss two kinds of integral and a class three-point boundary value problems and give some conditions of the existence of positive solutions by using the cone theory,the fixed point theory as well lower and upper solutions,the Avery-Peterson fixed point theorems,superlinear condition and the fixed point theory.The thesis is divided into four sections according to contents.Chapter 1 Preference,we introduce the main contents of this paper.Chapter 2 We use the fixed point theory as well upper and lower solutions to investigate the following nonlinear fractional integral bound-ary value problem where α>2,n-1<α≤n,η∈(0,1],i∈N,0≤i≤n-2,△-λ/αηα>0,i is a fixed constant, f:[0,1]×[0,+∞)j[0,+∞)is continuous.D0+∝ is the standard Riemann-Liouville derivative.By using the fixed point theory as well low- er and upper solutions,We obtain the existence of positive solution for nonlinear fractional integral boundary value problem(2.1.1).Chapter 3 We will study the following three-point boundary value problems of higher-order fractional differential equations. where cD0+∝ is the Caputo’s fractional derivatives of order α,and the function f:[0,1]×R+×R�'R+ is continuously differentiable.Here, 0≤q≤p,0<ξ<1,2≤n-1<α<n.By virtue of Auery-Peterson fixed pointed theorems,the existence of positive solution for the following boundary value problem is guaranteed.Chapter 4 By employing superlinear condition and the fixed point theory,we study the existence of positive solutions for the nonlinear fractional integral boundary value problems where 2<α≤3,0<η≤1,0<λ,λ≠a,0≤ληα/α<1. h(t)is allowed to be singular at t=0 and t=1. We make an exhaustive study of the sign of the related Green’s function and obtain the exact values for which it is positive on the whole square of definition. The existence of positive solutions follows from the definition of suitable cones on Banach spaces.