Monotone and concave positive solutions to a boundary value problem for higher-order fractional differential equation (Q642708)

Summary: We consider the boundary value problem for the nonlinear fractional differential equation \[ D^\alpha_{0+}u(t) + f(t, u(t)) = 0,\;0 < t < 1,\;n - 1 < \alpha \leq n,\;n > 3, \] \[ u(0) = u'(1) = u''(0) = \cdots = n^{(n-1)}(0) = 0, \] where \(D^\alpha_{0+}\) denotes the Caputo fractional derivative. By using a fixed point theorem, we obtain some new results for the existence and multiplicity of solutions to a higher-order fractional boundary value problem. The interesting point lies in the fact that the solutions here are positive, monotone and concave.