Existence of Multiple Positive Solutions for Nonlinear Fractional Boundary Value Problems on the Half-Line

In this paper, we deal with the following nonlinear fractional differential problem in the half-line \({\mathbb{R}^{+}=(0,+ \infty)}\)

$$\left\{\begin{array}{l}D^{\alpha }u(x)+f(x,u(x),D^{p}u(x))=0,\quad x \in \mathbb{R}^{+},\\ u(0)=u^{\prime } \left( 0\right) = \cdots =u^{\left( m-2\right) }(0)=0,\end{array}\right.$$

where \({m\in \mathbb{N}, m \geq 2, m-1 < \alpha \leq m, 0 < p \leq \alpha -1}\), the differential operator is taken in the Riemann–Liouville sense and f is a Borel measurable function in \({\mathbb{R}^{+} \times \mathbb{R}^{+} \times \mathbb{R} ^{+}}\) satisfying certain conditions. More precisely, we show the existence of multiple unbounded positive solutions, by means of Schäuder fixed point theorem. Some examples illustrating our main result are also given.