Positive Solutions For Two Types Of Boundary Value Problems Of Nonlinear Fractional Differential Equations

Fractional differential equation is the generalization of integer order differential equation,which is one of the most active fields in application of nonlinear analysis.Fractional differential equation theory has been applied in many fields such as math-ematics,physics,engineering and so on.The existence and uniqueness of solutions for fractional differential equation has attracted the great attention of more and more researchers.In this paper,we use the fixed point theory,the method of upper and low-er solutions,u0-positive operator,monotone iterative technique and so on,to study the existence of solutions for nonlinear fractional differential equations with integral boundary conditions.This paper is divided into four chapters:The first chapter introduces the present situation of fractional differential equa-tions,and gives the relative definitions and lemmas.The second chapter researches the coupled system of fractional differential equa-tions with integral boundary conditions where 1<?,? ? 2,?,??L[0,1],are nonnegative and f,g ?C[0,1]x[0,+?),[0,+?),a,b ? C((0,1),[0,+?)),?,? are the parameters,and 0<?,?? 1,D0+? is a orders the standard Riemann-Liouville fractional derivative.By building product space and using the fixed point theory,we obtain the uniqueness of positive solution for integral boundary value problem.The third chapter researches the nonlinear higher-order fractional differential e-quations integral boundary value problem where n-1<?<n,n? 3,? is the parameter,0<?<?,CD0+?i is the standard Caputo fractional derivative,i=1,2,· · ·,n-1,i-1<?i<i.The nonlinear term f:[0,1]x Rn ?[0,+?)is continuous.By using the monotone iterative technique based on the method of upper and lower solutions,we obtain maximum and minimum solutions for integral boundary value problem.The fourth chapter researches the fractional differential equation with integral boundary conditions where 2<?<3 is a real number,p ? C((0,1),[0,+?)),and ?01p(t)dt<+?.f?C([,1]× R?[0,+?)).??C((0,1),[0,+?)),and 0??01?(t)t?-1dt<1.By using the u0-positive operator and combining with the properties of the Green function,we obtain the existence and uniqueness of positive solutions for integral boundary value problem.