Solutions Of Boundary Value Problems Of Hadamard Fractional-order Differential Equations

The theory of fractional-order differential equations plays an important role in practical applications,such as in biology,physics,medicine and other fields.Using the fixed point index theory of strict set compression operator,Krasnoselskii-Zabreikos fixed point theorem and Cone compression and expansion fixed point theorem,this paper studies the solutions of several kinds of boundary value problems of Hadamard fractional differential equations.In chapter 1,we mainly introduce the research background,research status and some necessary preparatory knowledge of Hadamard fractional differential equations.In chapter 2,we consider the following singular boundary value problem of Hadamard fractional differential equations in abstract space,(?)wherel<? ?2,HD? is the Hadamard fractional derivative,J=[1,e],f:J × E?E is continuous.f(t,x)may be singular at t=1,e and x=0.By constructing a new cone and using the fixed point index theory of strict set compression operator,it is established that there are at least two positive solutions for the corresponding approximation of the boundary value problem.Then,using Ascoli-Arzela theorem and the relative compactness of the sequence of solutions,the sufficient condition is obtained for the existence of multiple positive solutions to the boundary value problem.In chapter 3,we consider the boundary value problem of Hadamard fractional impulse differential equation,(?)where 1<??2,f:J × R?R is continuous.1=t0<t1<…<tk<…<tm<tm+1=e.J=[1,e],J0=[1,t1],J1=(t1,t2],…,Jm=(tm,e],J'=[1,e]\{t1,t2,…,tm}.?(ln tk)2-?x(tk)=(ln tk)2-?x(tk+)-(ln tk)2-?x(tk-),(ln tk)2-?x(tk-)=(ln tk)2-?x(tk),??(ln tk)2-?x(tk)=?(ln tk)2-?x(tk+)-?(lntk)2-?x{tk-),?:=td/dt is the delta derivative.This chapter derives the integral equation which is equivalent to the Hadamard fractional impulse differential equation.Then the sufficient conditions for the existence of solutions are given by using the fixed point theorem of Krasnoselskii-Zabreikos.In chapter 4,we consider the singular multipoint boundary value problems for Hadamard p-Laplace fractional differential equations.(?)where 1<?,??2,?i>0,0<?1<?2<……<?m-2<1,(?),?p(s)=[s|p-2s,p>1,?P-1=?q,(?).f(t,x)may be singular at t=1,e and x=0 or f(t,x)may be singular at t=1,e.By using the properties of Green function and the fixed point theroy in cones,the existence of at least two positive solutions is obtained.