The Research Of Existence Of Positive Solutions For Fractional Differential Equations Boundary Value Problems

The boundary value problems of fractional differential equations arise in a variety ofdifferent areas of applied mathematics and physics. Recently, many authors have studiedthe boundary value problems of fractional differential equations and obtained manyachievements. On the basis of the existing literatures, this dissertation discusses theexistence of positive solutions for specific fractional differential equations boundary valueproblems from various angles by different fixed point theorems. This paper is divided intofour chapters, and the main contents are as follows:The introduction part introduces the research background, research significance andthe current research status of the overseas and domestic on the boundary value problemsof fractional differential equations, and at last the main contents in this paper.By using the Avery-Peterson fixed point theorem on a cone, discuss the existence ofpositive solutions for singular fractional differential equation boundary value problemswith first-order derivative, where the differential refers to the Caputo fractional differential.The existence of at least three positive solutions for this boundary value problems isobtained by defining appropriate cone and operator, also the specific properties of Green’sfunction.For the existence of positive solutions of fractional differential equation boundaryvalue problems with sign changing nonlinearities, by using the Avery-Peterson fixed pointtheorem, the existence of at least three positive solutions is obtained for this boundaryvalue problems by calculating the Green’s function, discussing it’s properties, andconstructing appropriate cone and operator.By using the cone expansion and cone contraction fixed point theorem, discuss theexistence of positive solutions for systems of fractional differential equations boundaryvalue problems.