The Solution Of Fractional Differential Equation And The Study Of A Fixed Point Theorem

Fractional differential equation is an important part of math field and the problem of its positive solutions has been developed for many years. This paper mainly studys the existence of positive solutions for singular multi-point boundary value problems with dependence on the parameter and also discuss the existence of positive solutions for singular fractional differential equation with a sign mea-sure. At last, we study generalized n-dimensional T-coincidence point theorems under (φ,φ)-contractive condition in complete partially ordered metric spaces.This paper is divided into three chapters according to contents.In chapter 1, we deal with the existence of positive solutions for the following singular multi-point boundary value problems:The differences between [24] and this chapter are that q(t) may be singular at t= 0,1 and the parameter b is drawn to the boundary value. Then we study the existence of positive solutions under the parameter b by using the Guo-Krasnosel’skii fixed point theorem.In chapter 2, the existence of positive solutions for the following singular differential equation with a sign measure is studied:The boundary value in this chapter is wider than the multiple points boundary value and the integral boundary value in paper [36] and [31]. And the nonlinear term f(t, u) may be singular at t= 0 and/or t= 1. We find the properties of the first eigenvalue of a singular fractional differential equation by using the spectral analysis of the relevant linear operator. Then we study the existence of positive solutions of a singular fractional differential equation with help of the fixed point index theorem.In chapter 3, we generalize the contract conditions of [10]and [16] as follow: the number of the variables of F is promoted to n, and F and g are commute. We present the existence and uniqueness of the common (?)-coincidence point under (φ,φ) contract condition in complete partially ordered metric spaces.