Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition

This paper deals with the following boundary value problem D α u t = f t , u t , t ∈ 0 , 1 , u 0 = u 1 = D α − 3 u 0 = u ′ 1 = 0 , where 3 < α ≤ 4 , D α is the Riemann-Liouville fractional derivative, and the nonlinearity f , which could be singular at both t = 0 and t = 1 , is required to be continuous on 0 , 1 × ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying u t ≤ c ω t , for   t ∈ 0 , 1   and   c > 0 , where ω t ≔ t α − 2 1 − t 2 .