Existence Of Positive Solutions To Boundary Value Problems Of P-Laplacian Nonlinear Fractional Differential Equations

In recent years,the boundary value problems(BVPs)of fractional differential equations have been paid more and more attention by researchers at home and abroad.Besides its application in mathematics,it is widely used in the aspects of fluid mechan-ics,the conductance of the biological system,the neural fraction model,the fractional regression model and so on.At present,there are many achievements in the study of the boundary value problem for fractional differential equations.In this paper,we mainly study the existence of positive solutions to different classes of nonlinear fractional differential equations with p-Laplacian,which is divided into four chapters.In the first chapter,we introduce the history of fractional calculus and its extensive application,and the status of the study on BVPs of nonlinear fractional differential equations with p-Laplacian.In chapter 2,we study the existence of positive solutions to the following boundary value problem using the properties of Green's function and Guo-Krasnosel'skii fixed-point theorem,Leggett-Williams fixed-point theorem.where2<a<3,1<??2,cD0?+,cD0?+ are Caputo fractional derivatives.?p(s)=|s|p-2s,p>1,?p-1=?q,1/p+1/q=1.f:[0,1]×[0,?)?[0,?)is continuous.In chapter 3,we discuss the existence of positive solutions to a class of integral boundary value problems of p-Laplacian fractional differential equations with singular terms.where 1<?,??2,0<??1,?-?-1>0.D0+?,D0+?,D0?+ are the standard Riemann-Liouville derivatives.?01 x(t)dA(t)is denoted by a Riemann-Stieltjes integral,A(t)is a function of bounded variation.The ?p is defined as:?p(s)=|s|p-2,p>1,?p-1=?q,1/p+1/q=1.Here f:[0,1]x(0,+?)?[0,+?)is continuous,f(t,x)is sin-gular at x=0,i.e.lim x?0+ f(·,x)=+?.By the fixed point theorem of mixed monotone operator,we prove the existence of positive solutions for this boundary value problem.