Existence and uniqueness of solutions for BVP of nonlinear fractional differential equation (Q1656129)

Summary: In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: \(({}^CD_{0 +}^q u) (t) = f (t, u (t))\), \(t \in (0,1)\), \(u (0) = u'' (0) = 0\), \(({}^CD_{0 +}^{\sigma_1} u) (1) = \lambda (I_{0 +}^{\sigma_2} u) (1)\), where \(2 < q < 3\), \(0 < \sigma_1 \leq 1\), \(\sigma_2 > 0\), and \(\lambda \neq \Gamma (2 + \sigma_2) / \Gamma (2 - \sigma_1)\). The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.