Existence and Approximation of Solutions to Three-point Boundary Value Problems for Fractional Differential Equations

In this paper, we study existence and approximation of solutions to some threepoint boundary value problems for fractional differential equations of the type Dq 0+u(t) + f(t, u(t)) = 0, t ∈ (0, 1), 1 < q < 2 u′(0) = 0, u(1) = ξu(η), where 0 < ξ, η ∈ (0, 1) and Dq 0+ is the fractional derivative in the sense of Caputo. For the existence of solution, we develop the method of upper and lower solutions and for the approximation of solutions, we develop the generalized quasilinearization technique (GQT). The GQT generates a monotone sequence of solutions of linear problems that converges monotonically and quadratically to solution of the original nonlinear problem.