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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 &lt; 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 &gt; 0 , where  ω t ≔ t α − 2 1 − t 2 .

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