The Existence Of Positive Solutions For Several Classes Of Nonlinear Fractional Differential Equations

Fractional calculus as a branch of calculus, Generally speaking fractional calculus is theexpansion of the integer-order calculus theory. From1695, the study of fractional calculus hasexperienced three hundred years, but the early fractional calculus is found only in the field oftheoretical mathematics, so in a long period of time, the study of the fractional calculus did notget the attention of the researchers. Until the mid-seventies of the last century, the fractionalcalculus and fractional differential equations both in the theory and in the application have arapid development, the application also will be more and more widely. Beginning to show thegeneral promotion of the fractional theory of ordinary differential equations and functionaldifferential equations. And the fractional calculus has a widely range of applications inViscoelastic Mechanics, Biology, Engineering, Mechanics, Chemistry, Economics, Statistics,Stochastic Processes, and many other fields. At the same time, the fractional calculus hasreceived the attention of many scientists, therefore it has become a hot issue in recent years.Due to the applications of fractional calculus in Applied Mathematics, Biophysics, MaterialMechanics, gradually people paid more attention to fractional calculus. Especially in the1982American-French mathematician B.B. Mandelbrot pointed out the fact that a large number offractal dimension in the nature and many of the technical sciences firstly, and the phenomenon ofself-similarity existed between the whole and the parts, fractional calculus as a basic andpowerful tool in Fractal geometry and Fractal dimension dynamics have a rapid development.In this paper, we study the existence of positive solutions for several classes of nonlinearfractional differential equations. Including the existence, nonexistence and multiplicity ofpositive solutions for a class of fractional differential equations of boundary value problem、theexistence of positive solutions for two classes of mixed fractional differential equations ofboundary value problem and the existence of positive solutions for a coupled system offractional differential equations. Some new existence theorems are obtained, and some examplesare presented to verify the main results of the paper.