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Abstract
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In this paper, by using fixed-point methods, we study
the existence and uniqueness of a solution for the
nonlinear fractional differential equation boundaryvalue
problem Dαu(t)=f (t, u(t)) with a Riemann–
Liouville fractional derivative via the different
boundary-value problems u(0)=u(T), and the threepoint
boundary condition u(0)=β1u(η) and u(T)=
β2u(η), where T >0, t ∈ I =[0, T], 0<α<1, 0<η
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