Existence of positive solutions to boundary value problem of Caputo fractional differential equation (Q1723470)

Summary: We consider the existence of positive solutions to the nonlinear fractional differential equation boundary value problem \(D_{0 +}^\alpha C u \left(t\right) + f \left(u \left(t\right), u^{\text{'}} \left(t\right)\right) = 0, t \in \left(0,1\right), u \left(0\right) = u \left(1\right) = u'' \left(0\right) = 0\), where \(f : \left.\left[0, + \infty\right.\right) \times \mathbb{R} \rightarrow \left.\left[0, + \infty\right.\right)\) is continuous, \(\alpha \in \left.\left(2,3\right.\right]\), and \(D_{0 +}^\alpha C\) is the standard Caputo differentiation. By using fixed point theorems on cone, we give some existence results concerning positive solutions. Here the solutions especially are the interior points of cone.