A Survey On Existence Results For Boundary Value Problems Of Singular Nonlinear Differential Equations

Fractional calculus has a long history. As early as in1695, the concept offractional differential was already mentioned in the correspondence of Leibniz andL’Hospital. During the past three centuries, the research of fractional calculus theorywas mainly concentrated in the pure theoretical field of mathematics. However, in therecent several decades many scholars in succession pointed out that fractionalcalculus is very suitable to characterize materials and processes with memory andhereditary properties, which were often neglected in the classical models．Nowadays,fractional differential equation models are increasingly used to describe the problemsin acoustics, thermal systems, material mechanics， signal processing, systemidentification, control theory, robotics and other applied fields．This thesis is divided as follows:The first part is an introduction, briefly presents the research history anddevelopment status of the fractional calculus and fractional differential equations, andsome past research works about the existence of positive solutions of the fractionaldifferential equations.The second part studies a singular nonlinear semipositone Sturm-Liouvilleboundary value problem. We redefine the nonlinear part f （y）, and make the singularboundary value problem transform into a nonsingular positone boundary valueproblem, and then prove the existence of a positive solution for the original singularnonlinear boundary value problem by using the cone fixed point theorem as well asknowledge of functional analysis.The third part discusses the positive solution existence for Dirichlet boundaryvalue problem of a singular nonlinear fractional differential equation. We study itsnonlinear partf f(t,x（t）,D₀₊^βx（t）) and have it transform into a nonsingular boundaryvalue problem, and then prove the existence of a positive solution x_n for eachboundary value problem with redefined nonlinear part f_n(t,x（t）,D₀₊^βx（t）)（n∈N） andfinally we give the existence of a positive solution for the original Dirichlet boundaryvalue problem via the limit properties of a sequence of functions on compact sets.