Existence and uniqueness of positive solutions to higher-order nonlinear fractional differential equation with integral boundary conditions

In this article, we consider the nonlinear fractional order three-point boundary-value problem $$displaylines{ D_{0+}^{alpha} u(t) + f(t,u(t))=0, quad 0 < t < 1,cr u(0) = u'(0) = dots = u^{(n-2)}(0)=0, quad u^{(n-2)}(1) = int_0^eta u(s)ds, }$$ where $D_{0+}^{alpha}$ is the standard Riemann-Liouville fractional derivative, $n-1 < alpha leq n$, $n geq 3$. By using a fixed-point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.