Existence of Symmetric Positive Solutions for a Caputo Fractional Singular Boundary Value Problem

Abstract

In this article, we establish the symmetric positive existence for the following Caputo fractional boundary value problem \begin{align*} {}^{C}D_{0}^{\,μ}x(t)+f(t,x(t))&=0,\hspace{1cm}t\in(-1,\,1),\hspace{1cm}1<μ\leq2,\\ x(\pm1)=x'(0^{\pm})&=0, \end{align*} where ${}^C{D}_0^{\mu }x(t)={}^C{D}_{0^{+}}^{\mu }x(t)$ for $t\ge 0$, ${}^C{D}_0^{\mu }x(t)={}^C{D}_{0^{-}}^{\mu }x(t)$ for $t\le 0$. Moreover, $f:(-1,1)\times (0,\infty )\to \mathbb{R}$ is continuous and singular at $t=-1$, $t=1$ and $x=0$. Here, ${}^C{D}_{0^{+}}^{\mu }$ and ${}^C{D}_{0^{-}}^{\mu }$, respectively, are Caputo fractional left and right derivatives of order $\mu$.