Three point boundary value problems for nonlinear fractional differential equations

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type Dδ0+cu(t) = f(t,u(t), Dσ0+cu(t)), t ∈ [0, T],u(0) = αu(η), u(T) = βu(η),where 1 < δ < 2, 0 < σ < 1, α < 1, α, β ∈ ℝ, η ∈ (0,T), αη(1 − β) + (1 − α) (T − βη) ≠ 01 < δ < 2, 0 < σ < 1, α < 1, α, β ∈ ℝ, η ∈ (0,T), αη(1 − β) + (1 − α) (T − βη) ≠ 0 and Dcδ0+,Dcσ0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.