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

TL;DR

This paper proves the existence of positive solutions for a Hadamard fractional singular boundary value problem of order between 2 and 3, involving a nonlinear function with singularities at boundary points.

Contribution

It establishes the first existence results for positive solutions of a Hadamard fractional boundary value problem with singularities at the boundary.

Findings

Existence of positive solutions proven under certain conditions.

Method applicable to similar fractional boundary value problems.

Addresses singularities at boundary points in fractional differential equations.

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 $\mu$.