Positive Solution For Boundary Value Problems Of Fractional Differential Equation

There are a lot of methods to study the fractional order boundary value problems.Using nonlinear analysis technology is a common method to study the existence of thesolutions and the right solutions of the problem of the nonlinear fractional order diferentialequations.For example, Krasnosel'skii fixed point theorem, Leggett-Williams fixed point theo-rem and other fixed point theorem, Leray-Schauder theory, and the solution method suchas Lower and upper solutions.This paper is organized as follows:The first part of this paper, we deal with the following nonlinear fractional boundaryvalue problemD₀^α+u(t)+f (t, u(t))=0, t∈[0, T], n <α≤n+1,Ï…(0)=Ï…'(0)=Ï…''(0)==Ï…⁽ⁿ¹⁾(0)=Ï…⁽ⁿ¹⁾(T)=0,where D₀^α+is the standard Riemann-Liouville fractional derivative, n is a natural number,T is a integer.In previous literature, the variable t of the above boundary value problem only requestin the interval [0,1]. And α requirement in the interval [1,2] or (2,3] or (3,4) and so on, itis not enough for actual need. Here using lower and upper solution method and Schauderfixed-point theorem to extend the time interval to [0,T] and the order number expand toa (n, n+1]. And thus gain the existence of the solution and some properties of the abovefractional boundary value problem.The second part of this paper, we first introduce of the basic knowledge of the qcalculus, then we study the existence of positive solutions of the nonlinear q-fractionalboundary value problemD_q^αυ(t)+f (t, Ï…(t))=0, t∈[0,1],2<α≤3,Ï…(0)=(D_q^Ï…)(0)=Ï…(1)=0, where Dqu is the q-derivative of a function u.We use the inside of the cone fixed point theorem and considering Dirichlet typeboundary conditions to the boundary value problem of the existence of the positive solu-tions and some properties.