Existence results for boundary value problems of nonlinear fractional differential equations

In this paper, we consider the existence of solutions for the nonlinear fractional differential equation CD0+αu(t)+rCD0+α−1u(t)=f(t,u(t)),t∈(0,1) with the boundary value conditions u(0)=u(1),u(ξ)=η,ξ∈(0,1), where CD0+α and CD0+α−1 are the standard Caputo derivative with 1<α≤2,r≠01<α≤2,r≠0. By using the contraction mapping principle and the Schauder fixed point theorem, some existence results are obtained. In addition, Lemma 2.6 in this paper is a valuable tool in seeking solvability of the fractional differential equations.