Existence of positive solution for BVP of nonlinear fractional differential equation (Q1723487)

Summary: We consider the following boundary value problem of nonlinear fractional differential equation: \((^C D_{0 +}^\alpha u)(t) = f(t, u(t))\), \(t \in [0,1]\), \(u(0) = 0, u'(0) + u''(0) = 0\), \(u'(1) + u''(1) = 0\), where \(\alpha \in(2,3]\) is a real number, \({}^C D_{0 +}^\alpha\) denotes the standard Caputo fractional derivative, and \(f : [0,1] \times [0, + \infty) \rightarrow [0, + \infty)\) is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.