Positive Solution for a Hadamard Fractional Singular Boundary Value Problem of Order $\mu\in(2,\,3)$

Abstract

In this article, we establish the existence of positive solution for the following Hadamard fractional singular boundary value problem \begin{align*} {}^{H}D_{a^{+}}^{\,\mu}x(t)+f(t,x(t))&=0,\hspace{0.4cm}t\in(a,\,b),\hspace{0.4cm}0<a<b<\infty,\hspace{0.4cm}2<\mu<3,\\ x(a)=a\,x'(a)=x(b)&=0, \end{align*} where $f:(a,\,b)\times(0,\infty)\rightarrow(0,\infty)$ is continuous and singular at $t=a$, $t=b$ and $x=0$. Further, ${}^{H}D_{a^{+}}^{\,\mu}$ is Hadamard fractional derivative of order $\mu$.