The Positive Solutions For A Boundary Value Problem Of Nonlinear Fractional Differential Equations

In this paper,we mainly investigate the existence of positive solutions for boundary value problem of fractional differential equations involving Riemann-Liouville derivative.Firstly,we study the boundary value problem of the following fractional differential equation: where4<α≤5is a real number,Dα is Riemann-Liouville fractional derivative,and f∈C[0,1].By means of lower and upper solution method,the sufficient conditions are obtained for the above fractional boundary value problem.As an application,an example is presented to illustrate the result.Then using five fixed theorem,we get the sufficient conditions when this problem has at least three positive solutions.Furthermore,we discuss the existence of positive solutions to the following (n,p) boundary value problem where f∈C[0,1], n-1<α≤n,0≤p≤n-1.By means of the fixed-point theorem on cone expansion and compression,the conditions is obtained when this problem has at least one positive solution.Then we discuss the existence of at least three positive solutions by Legget-William’s fixed-point theorem.