Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem

Abstract

In this paper, we consider the following nonlinear fractional three-point boundary value problem

$$D_{0+}^{\alpha }u\left(t\right)+f(t,u\left(t\right))=0\text{,}0<t<1,3<\alpha \le 4\text{,}$$$$u\left(0\right)=u^{\prime }\left(0\right)=u^{″}\left(0\right)=0\text{,}{u}^{″}\left(1\right)=\beta u^{″}\left(\eta \right)\text{,}$$

where $D_{0+}^{\alpha }$ is the standard Riemann–Liouville fractional derivative. By using a fixed point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solution to the above boundary value problem.