Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations

In this paper, we study the existence of solutions for the boundary value problem of the following nonlinear fractional differential equation: D0+α[x(t)f(t,x(t))]+g(t,x(t))=0, 0<t<1, x(0)=x(1)=x′(0)=0, where 2<α≤3 is a real number and D0+α is the Riemann-Liouville fractional derivative. By a fixed point theorem in Banach algebra, an existence theorem for the boundary value problem of the above fractional differential equation is proved under both Lipschitz and Carathéodory conditions. Two examples are presented to illustrate the main results.